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Numerical Methods for NonlinearElliptic Differential Equations

A Synopsis

Klaus BohmerUniversity of Marburg

1

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With love to my late wife Inge,my daughters Annette and Christine, and

my grandchildren Joachim, Dorothea, Leon, Luca, and Rebekka

∗ To God Alone Be The Glory, Organ St. Michaelis, Luneburg

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Contents

Preface xiv

PART I ANALYTICAL RESULTS

1 From linear to nonlinear equations, fundamental results 31.1 Introduction 31.2 Linear versus nonlinear models 31.3 Examples for nonlinear partial differential equations 101.4 Fundamental results 13

1.4.1 Linear operators and functionals in Banach spaces 131.4.2 Inequalities and Lp(Ω) spaces 181.4.3 Holder and Sobolev spaces and more 201.4.4 Derivatives in Banach spaces 27

2 Elements of analysis for linear and nonlinear partial ellipticdifferential equations and systems 322.1 Introduction 322.2 Linear elliptic differential operators of second order, bilinear forms

and solution concepts 362.3 Bilinear forms and induced linear operators 452.4 Linear elliptic differential operators, Fredholm alternative and regular

solutions 542.4.1 Introduction 542.4.2 Linear operators of order 2m with C∞ coefficients 582.4.3 Linear operators of order 2 under Ck conditions 642.4.4 Weak elliptic equation of order 2m in Hilbert spaces 69

2.5 Nonlinear elliptic equations 772.5.1 Introduction 772.5.2 Definitions for nonlinear elliptic operators 792.5.3 Special semilinear and quasilinear operators 812.5.4 Quasilinear elliptic equations of order 2 882.5.5 General nonlinear and Nemyckii operators 962.5.6 Divergent quasilinear elliptic equations of order 2m 1002.5.7 Fully nonlinear elliptic equations of orders 2, m and 2m 108

2.6 Linear and nonlinear elliptic systems 1132.6.1 Introduction 1132.6.2 General systems of elliptic differential equations 1142.6.3 Linear elliptic systems of order 2 118

viii Contents

2.6.4 Quasilinear elliptic systems of order 2 andvariational methods 125

2.6.5 Linear elliptic systems of order 2m,m ≥ 1 1322.6.6 Divergent quasilinear elliptic systems of order 2m 1372.6.7 Nemyckii operators and quasilinear divergent systems

of order 2m 1402.6.8 Fully nonlinear elliptic systems of orders 2 and 2m 146

2.7 Linearization of nonlinear operators 1472.7.1 Introduction 1472.7.2 Special semilinear and quasilinear equations 1492.7.3 Divergent quasilinear and fully nonlinear equations 1512.7.4 Quasilinear elliptic systems of orders 2 and 2m 1572.7.5 Linearizing general divergent quasilinear and fully

nonlinear systems 1582.8 The Navier–Stokes equation 163

2.8.1 Introduction 1632.8.2 The Stokes operator and saddle point problems 1632.8.3 The Navier–Stokes operator and its linearization 167

PART II NUMERICAL METHODS

3 A general discretization theory 1733.1 Introduction 1733.2 Petrov–Galerkin and general discretization methods 1753.3 Variational and classical consistency 1853.4 Stability and consistency yield convergence 1893.5 Techniques for proving stability 1943.6 Stability implies invertibility 2033.7 Solving nonlinear systems: Continuation and Newton’s method

based upon the mesh independence principle (MIP) 2053.7.1 Continuation methods 2053.7.2 MIP for nonlinear systems 206

4 Conforming finite element methods (FEMs) 2094.1 Introduction 2094.2 Approximation theory for finite elements 212

4.2.1 Subdivisions and finite elements 2124.2.2 Polynomial finite elements, triangular and

rectangular K 2144.2.3 Interpolation in finite element spaces, an example 2214.2.4 Interpolation errors and inverse estimates 2294.2.5 Inverse estimates on nonquasiuniform triangulations 2334.2.6 Smooth FEs on polyhedral domains, with O. Davydov 2384.2.7 Curved boundaries 250

4.3 FEMs for linear problems 257

Contents ix

4.3.1 Finite element methods: a simple example, essential tools 2584.3.2 Finite element methods for general linear equations and

systems of orders 2 and 2m 2644.3.3 General convergence theory for conforming FEMs 266

4.4 Finite element methods for divergent quasilinear ellipticequations and systems 273

4.5 General convergence theory for monotone and quasilinearoperators 277

4.6 Mixed FEMs for Navier–Stokes and saddle point equations 2814.6.1 Navier–Stokes and saddle point equations 2814.6.2 Mixed FEMs for Stokes and saddle point equations 2824.6.3 Mixed FEMs for the Navier–Stokes operator 286

4.7 Variational methods for eigenvalue problems 2884.7.1 Introduction 2884.7.2 Theory for eigenvalue problems 2894.7.3 Different variational methods for eigenvalue problems 292

5 Nonconforming finite element methods 2965.1 Introduction 2965.2 Finite element methods for fully nonlinear elliptic problems 298

5.2.1 Introduction 2985.2.2 Main ideas and results for the new FEM:

An extended summary 2995.2.3 Fully nonlinear and general quasilinear elliptic equations 3055.2.4 Existence and convergence for semiconforming FEMs 3085.2.5 Definition of nonconforming FEMs 3115.2.6 Consistency for nonconforming FEMs 3175.2.7 Stability for the linearized operator and convergence 3195.2.8 Discretization of equations and systems of order 2m 3325.2.9 Consistency, stability and convergence for m, q ≥ 1 3365.2.10 Numerical solution of the FE equations with

Newton’s method 3415.3 FE and other methods for nonlinear boundary conditions 3455.4 Quadrature approximate FEMs 346

5.4.1 Introduction 3465.4.2 Quadrature and cubature formulas 3485.4.3 Quadrature for second order linear problems 3505.4.4 Quadrature for second order fully nonlinear equations 3575.4.5 Quadrature FEMs for equations and systems

of order 2m 3615.4.6 Two useful propositions 367

5.5 Consistency, stability and convergence for FEMs withvariational crimes 3685.5.1 Introduction 3685.5.2 Variational crimes for our standard example 370

x Contents

5.5.3 FEMs with crimes for linear and quasilinearproblems 380

5.5.4 Discrete coercivity and consistency 3875.5.5 High order quadrature on edges 3905.5.6 Violated boundary conditions 3925.5.7 Violated continuity 3995.5.8 Stability for nonconforming FEMs 4065.5.9 Convergence, quadrature and solution of FEMs

with crimes 4115.5.10 Isoparametric FEMs 414

6 Adaptive finite element methods, by W. Dorfler 4206.1 Introduction 420

6.1.1 The model problem 4216.1.2 Singular solutions 4216.1.3 A priori error bounds 4236.1.4 Necessity of nonuniform mesh refinement 4256.1.5 Optimal meshes – A heuristic argument 4256.1.6 Optimal meshes for 2D corner singularities 4276.1.7 The finite element method–Notation and

requirements 4286.2 The residual error estimator for the Poisson problem 430

6.2.1 Upper a posteriori bound 4306.2.2 Lower a posteriori bound 4326.2.3 The a posteriori error estimate 4336.2.4 The adaptive finite element method 4346.2.5 Stable refinement methods for triangulations in R2 4366.2.6 Convergence of the adaptive finite element method 4386.2.7 Optimality 4426.2.8 Other types of estimators 4476.2.9 hp finite element method 448

6.3 Estimation of quantities of interest 4496.3.1 Quantities of interest 4496.3.2 Error estimates for point errors 4496.3.3 Optimal meshes–A heuristic argument 4516.3.4 The general approach 451

7 Discontinuous Galerkin methods (DCGMs), with V. Dolejsı 4557.1 Introduction 4557.2 The model problem 4597.3 Discretization of the problem 461

7.3.1 Triangulations 4617.3.2 Broken Sobolev spaces 4627.3.3 Extended variational formulation of the problem 4637.3.4 Discretization 469

7.4 General linear elliptic problems 472

Contents xi

7.5 Semilinear and quasilinear elliptic problems 4747.5.1 Semilinear elliptic problems 4747.5.2 Variational formulation and discretization of

the problem 4757.5.3 Quasilinear elliptic systems 4777.5.4 Discretization of the quasilinear systems 478

7.6 DCGMs are general discretization methods 4827.7 Geometry of the mesh, error and inverse estimates 486

7.7.1 Geometry of the mesh 4877.7.2 Inverse and interpolation error estimates 487

7.8 Penalty norms and consistency of the Jσh 491

7.9 Coercive linearized principal parts 4947.9.1 Coercivity of the original linearized principal parts 4947.9.2 Coercivity and boundedness in Vh for the Laplacian 4957.9.3 Coercivity and boundedness in Vh for the general linear

and the semilinear case 4997.9.4 Vh-coercivity and boundedness for quasilinear problems 502

7.10 Consistency results for the ch, bh, �h 5037.10.1 Consistency of the ch and bh 5037.10.2 Consistency of the �h 505

7.11 Consistency properties of the ah 5077.11.1 Consistency of the ah for the Laplacian 5077.11.2 Consistency of the ah for general linear problems 5117.11.3 Consistency of the semilinear ah 5147.11.4 Consistency of the quasilinear ah for systems 5187.11.5 Consistency of the quasilinear ah for the equations of

Houston, Robson, Suli, and for systems 5237.12 Convergence for DCGMs 5277.13 Solving nonlinear equations in DCGMs 532

7.13.1 Introduction 5327.13.2 Discretized linearized quasilinear system and

differentiable consistency 5327.14 hp-variants of DCGM 538

7.14.1 hp-finite element spaces 5397.14.2 hp-DCGMs 5407.14.3 hp-inverse and approximation error estimates 5407.14.4 Consistency and convergence of hp-DCGMs 542

7.15 Numerical experiences 5467.15.1 Scalar quasilinear equation 5467.15.2 System of the steady compressible Navier–Stokes

equations 554

8 Finite difference methods 5608.1 Introduction 5608.2 Difference methods for simple examples, notation 562

xii Contents

8.3 Discrete Sobolev spaces 5668.3.1 Notation and definitions 5668.3.2 Discrete Sobolev spaces 569

8.4 General elliptic problems with Dirichlet conditions, and theirdifference methods 5728.4.1 General elliptic problems 5728.4.2 Second order linear elliptic difference equations 5748.4.3 Symmetric difference methods 5818.4.4 Linear equations of order 2m 5838.4.5 Quasilinear elliptic equations of orders 2, and 2m 5848.4.6 Systems of linear and quasilinear elliptic equations 5868.4.7 Fully nonlinear elliptic equations and systems 587

8.5 Convergence for difference methods 5888.5.1 Discretization concepts in discrete Sobolev spaces 5898.5.2 The operators Ph, Q

′h 5918.5.3 Consistency for difference equations 5948.5.4 Vh

b -coercivity for linear(ized) elliptic difference equations 6008.5.5 Stability and convergence for general elliptic difference

equations 6048.6 Natural boundary value problems of order 2 610

8.6.1 Analysis for natural boundary value problems 6118.6.2 Difference methods for natural boundary value

problems 6138.7 Other difference methods on curved boundaries 622

8.7.1 The Shortley–Weller–Collatz method for linear equations 6238.8 Asymptotic expansions, extrapolation, and defect corrections 626

8.8.1 A difference method based on polynomial interpolation forlinear, and semilinear equations 627

8.8.2 Asymptotic expansions for other methods 6308.9 Numerical experiments for the von Karman equations,

with C.S. Chien 633

9 Variational methods for wavelets, with S. Dahlke 6359.1 Introduction 6359.2 The scope of problems 6379.3 Wavelet analysis 639

9.3.1 The discrete wavelet transform 6409.3.2 Biorthogonal bases 6449.3.3 Wavelets and function spaces 6469.3.4 Wavelets on domains 6479.3.5 Evaluation of nonlinear functionals 652

9.4 Stable discretizations and preconditioning 6539.5 Applications to elliptic equations 6599.6 Saddle point and (Navier–)Stokes equations 664

9.6.1 Saddle point equations 6649.6.2 Navier–Stokes equations 666

Contents xiii

9.7 Adaptive wavelet methods, by T. Raasch 6699.7.1 Nonlinear approximation with wavelet systems 6729.7.2 Wavelet matrix compression 6759.7.3 Adaptive wavelet–Galerkin methods 6789.7.4 Adaptive descent iterations 6809.7.5 Nonlinear stationary problems 683

Bibliography 686

Index 733

Preface

Nonlinear problems play an increasingly important role in mathematics, science andengineering. Thus, an exciting interplay originates between the different disciplines,stimulated by the insight that linear problems often provide only poor approximationsfor the often nonlinear original problems. This applies particularly for all the bifurca-tion and induced dynamical regimes in nonlinear differential equations. In fact, thisbook, A ‘Synopsis’, and the following Numerical Methods for Bifurcation and CenterManifolds in Nonlinear Elliptic and Parabolic Equations [120], are motivated by theseinterrelations and their numerical realization.

A major part of these phenomena is governed by elliptic and the corresponding par-abolic partial differential equations and systems of orders 2 and 2m. For these nonlinearequations, bifurcation and the related, hence local, dynamics are mainly governedby the underlying elliptic partial differential equations and a small dimensional timedependent model, the center manifold. So we can avoid most of the problems of timediscretization of parabolic partial differential equations and concentrate on the spacediscretization of elliptic partial differential equations. The problem of convergence ofthe solutions of space discretized elliptic problems to those of the original nonlinearproblems is the core of this book.

Astonishing is the fact that in the many excellent books on numerical methods forelliptic problems, see, e.g. below, nonlinear phenomena are either totally omitted oronly treated by one or two important examples. Most books concentrate on just oneof the many available discretization methods. Certainly, one of the reasons is the factthat by linearization many of the nonlinear equations and their numerical methods canbe reduced to the linear case. However, for many problems, e.g. all types of bifurcationscenarios, a closer relation between the original nonlinear problem and its nonlineardiscretization is mandatory.

For linear problems the existence, uniqueness and regularity of solutions are wellknown. For nonlinear problems this is no longer correct and the results are widelyspread in a huge number of textbooks, monographs and much more original papers.They are obtained by very different methods. So we have to summarize and formulatethose analytical results, necessary for the following proofs for and the convergencerates of the numerical methods. In particular, we need a systematic listing of condi-tions guaranteeing the existence, uniqueness and regularity for the solutions of thesenonlinear elliptic problems, and exact conditions for the linearization. We includehints to available textbooks and monographs for this area and to special problems,not covered here. With a few exceptions, we do not refer to nor list the results oforiginal papers.

In contrast to other books, the goal here is a systematic study, in particular of thejoint structure, of the discretization methods for linear and nonlinear elliptic problems.

Preface xv

Necessarily this has to start with the linear foundations but has to proceed to fullynonlinear problems, where the highest derivatives occur nonlinearly. In particular,this last highlight is possible, since recently the author, probably for the first time,has published a finite element method and proved its convergence for this wideclass of general nonlinear problems, cf. Bohmer [118, 119]. So, we prove stability andconvergence for the numerical methods below and for all the linear and nonlinearproblems studied in Chapter 2.

It cannot be the goal of this book to provide an encyclopedia of all available spacediscretization methods for elliptic problems, since new methods are emerging all thetime. Rather we have chosen the most important examples of different types. Newmethods will probably fit into our framework, certainly with appropriate generaliza-tions and modifications.

It would be pretty unrealistic to assume that a standard reader would study thewhole book. Rather, a scientist would look for the class of elliptic problems and thecorresponding numerical method most interesting for him. Therefore this preface givesa much extended survey of the content of each chapter and thus allows an appropriatechoice. We formulate two examples at the end of this preface for a goal-orientedselection of material from the book. The book presents the result that these typesof short studies are possible.

To give a feeling for the type of our elliptic problems, we formulate ahierarchy of increasingly nonlinear model problems, generalizing the LaplacianΔu =

∑ni=1 ∂

i∂i u = g, with g independent of the solutions u. Its obvious lin-ear generalization in Rn is

∑ni,j=0(−1)j>0∂

j(aij∂i u) = g. We use the notation

∂j u := ∂u/∂xj , ∂0 u := u, ∇u := (∂ju)nj=1 ∇2u := (∂i∂ju)n

i,j=1, and (−1)j>0 =−1 for j > 0 and else = 1. Special semilinear forms are Δu + f(u) = 0 or,the more general, Δu + f(x, u,∇u) = 0. Quasilinear equations in divergence formare −∑n

i=0 ∂i(ai(x, u,∇u)) = 0. Fully nonlinear problems are the most difficult:

a(x, u,∇u,∇2u) = 0. These second order equations are generalized in Chapter 2to equations of order 2m and to systems of order 2m,m ≥ 1, and the generalizedLaplacian. We study them on polygonal or curved domains mainly with generalizedDirichlet boundary conditions. For m = 1 we admit (nonlinear) boundary conditions,induced by the (nonlinear) differential operators.

The goal for our discretization approach is the proof of the unique existence ofdiscrete solutions for all these previous problems, their computability and convergenceto the exact solutions. We reach this goal for finite element methods, FEMs, andadaptive forms, difference, mesh-free and spectral methods. For discontinuous Galerkinmethods it allows linear to quasilinear equations and systems of order 2. Due to theopen problem of calculating complicated nonlinear operators, wavelet methods arerestricted to semilinear equations and systems of order 2m,m ≥ 1.

So this monograph is essentially oriented towards theoretical numerical mathemat-ics. The only problems computed here are quasilinear and steady compressible Navier–Stokes equations in Section 7.15 and the van Karman equations in Section 8.9. Allthe other case studies, e.g. the dew drops on a spider’s web, the local dynamics in theearth mantel, and weather models, and the beginning chaos in Navier–Stokes equationsrequire the tools of numerical symmetry breaking. They are delayed to [120].

xvi Preface

Linearization techniques are the main tools for both books. The trick which makesit work relies upon the bounded invertibility of the derivative of the operator ina neighborhood of a solution. If this is assumed, it may be shown that even anequibounded invertibility, hence stability, holds for the chosen discretizations. Mostprevious approaches for quasilinear problems rely upon monotonicity results for theequation at hand. This limits the nonlinearities and also the discretizations which maybe used. The new approach permits a unified convergence theory for the previouslymentioned problems and discretizations. Nevertheless, each of these combinationsrequires a specific theoretical “lifting” to establish convergence and stability. Theauthor is pretty sure that by modifying this lifting the techniques in this book can beapplied to the other space discretization methods as well. Indeed, it has worked verywell starting with FEMs and proceeding to all our other methods.

The linearization technique for the previous FEMs and all the following methodsdoes not allow convergence results for monotone and quasilinear operators in Wm,p(Ω)for 1 < p < 2, cf. Subsection 4.3.3 and Lemma 2.77, Section 2.7, and Chapters 4 and5. However it yields, for 2 ≤ p <∞, convergence of the expected order with respectto the discrete Hm(Ω) norms for all types of elliptic problems.

On the other hand, the monotone operator techniques yield convergence results withdifferent orders for 1 < p <∞ with respect to discrete Wm,p(Ω) norms, cf. Theorem4.67. Strongly monotone operators or quasilinear operators in H2(Ω) allow specificresults. In fact, many important applications are modeled by monotone operators.So we embed the monotone operator approach, independent of linearization, into ourgeneral discretization theory. We do not aim for a full proof of these results. Ratherwe combine the excellent presentation in Zeidler [678], with Chapter 3. We formulatethe results in Section 4.5 such that the other discretization methods, in particularthe nonconforming methods in Chapters 5 and 7, are included. These results seem toextend the state of the art in several directions.

These two independent approaches, here linearization and monotone operator tech-niques, complement each other very appropriately. Both yield complementing resultsfor the existence, uniqueness and convergence of the discrete solutions for all previousdiscretizations and for quasilinear to fully nonlinear problems.

Back to the linearized problems. These are split into a coercive part and a com-pact perturbation, thus allowing the highly successful perturbation techniques. Manymethods are nonconforming, so the approximating functions violate the boundaryand/or smoothness properties of the exact solutions. Then the relations betweenthe strong and the weak form of the (linearized) problem are essential. A so-calledanticrime transformation masters these difficulties for the convergence theory. For theadaptive FEMs in Chapter 6, Dorfler uses related techniques.

These techniques allow the above-mentioned formulation of a new class ofFEMs, difference, mesh-free and spectral methods for fully nonlinear problems,cf. [118–120] This is different from the earlier handling of quasilinear problemsvia monotonicity arguments. They do not seem to be applicable to fully nonlinearproblems. The next book [120] is the motivation for this book. There all the bifurcationand local dynamical scenarios are again governed by linearization of the correspondingproblems. So the possibility of linearization has to be carefully studied. The class of

Preface xvii

nonlinear problems amenable to these techniques is characterized in Section 2.7. Itessentially requires the above-mentioned boundedly invertible linearized operator.

Finally, many methods for solving nonlinear discrete problems are based upon thecorresponding linear problems, e.g. all types of Newton and continuation techniques.Furthermore, we obtain the Fredholm alternative and can prove stability and con-vergence in a unified way for the different types of space discretization methods andthe large class of linear and nonlinear problems characterized before. Despite thisunified convergence theory, the different methods presented here, cf. the followingparagraphs, and in [120], require carefully distinguishing their different approximationcharacteristics.

So we have many good reasons for linearization as the main tool.It is a fascinating phenomenon that an appropriate structure allows very similar

arguments for a wide range of problems.The most frequently used method are FEMs, including nonconforming FEMs.

Some of the techniques in FEMs can be modified for the other methods as well.Adaptive FEMs employ a posteriori estimates for determining problem-adapted FEMs.Discontinuous Galerkin methods, DCGMs, are closely related to and modify FEMs bypenalty terms. The oldest, the finite difference method, is still a favorite in bifurcationproblems. Wavelets are one of the quite recent methods with still many open technicalproblems. Only spectral methods allow the numerical realization of infinite symmetrygroups. The mesh-free or radial basis methods allow a very flexible positioning of theircenters, again very advantageous for maintaining the symmetry of problems in theirdiscretization. Therefore both methods are omitted here and used in Bohmer [120] asdemonstration examples for the general discretization theory employed in both books,cf. Chapter 3 and the new approach in [120].

We omit several interesting methods, e.g. some generalizations of finite elements, thefinite volume, cf. Bey [90] and Knabner and Angermann [445]. For boundary integralmethods, several interesting books are available by Steinbach and Rjasanov [556,595]and Hsiao and Wendland [413]. They have been applied to nonlinear problems bylinearizing the original problem and combining Newton–Kantorowich methods andcontinuation. The corresponding linear equations can then be solved with thesemethods. The same techniques have recently been used for wavelet and mesh-freemethods, now included in our convergence theory. For boundary integral methods wedid not see this possibility.

Originally this book was planned as part of [120]. As a basis for numerical methodsin bifurcation of nonlinear differential equations, we had to summarize the stateof the art of numerical methods here for elliptic problems. But, for most of themethods studied here, these results turned out to be rather incomplete towardsnonlinearity, in particular full nonlinearity. So we had to extend many known resultsfor nonlinear problems into different and missing directions. This was possible bysystematically using the above general discretization theory. It even allowed the proofof the convergence results for nonlinear boundary operators. These are problems whichseem to be not much discussed in the literature. Forced by this wide range of problems,we split the presentation into two essentially independent books.

xviii Preface

As mentioned, most of the many excellent books on numerical methods for PDEsare concentrated on linear problems. In particular, they intensively discuss the cor-responding linear solvers. Many standard solvers for large nonlinear problems areessentially based upon linear solvers. The discretization methods presented here areso-called “linear” methods. This is the essential condition for numerical methodsadapted to bifurcation and for the mesh independence principle (MIP). This statesthat the Newton method applied to the discrete problem converges, for a smallenough “step size”, h, essentially as fast and independent of h, as that applied tothe analytic problem. This implies quadratic and, e.g. linear convergence for theoriginal and modified Newton’s method. So the convergence of these iteration methodsto discrete solutions is guaranteed for our wide range of discretization methods andnonlinear problems. The other tools are continuation methods. Both methods solvesequences of linear problems. The nonlinearity only enters via the computation of thedefects or residuals of the approximate discrete solution in the nonlinear problem.A combination of Newton with continuation techniques provides efficient solversfor nonlinear discrete systems. Usually this is combined with available multilevelmethods.

Since these linear solvers are pretty different for the different types of the abovemethods and problems, a synopsis of the linear solvers does not make too muchsense. Hence, we do not discuss linear solvers. Modifications for bifurcation andapproximations for the eigenvalues and invariant subspaces for the linearized operatorsare delayed to [120]. So this synopsis on numerical methods for nonlinear ellipticproblems essentially studies stability, consistency, convergence, and computability.

The following references, here and throughout the book, are chosen to supportspecific points in the text. It would be totally impossible to list all relevant books andpapers. The different discretization methods are studied in hundreds or even thousandsof papers. So we usually only cite books, unless specific points have to be made.This implies that the contributions of many scientists represent extremely importantcontributions to the area in this book without being mentioned here. One exampleare the numerical methods for the Navier–Stokes equations, with 3,731 citations inOctober 2009 in the Zentralblatt.

According to the previous discussion, this book is organized as follows.Part I is devoted to analytical results. Chapter 1 demonstrates, for the simple

mechanical example of a bent rod, the change in the character from linear to nonlinearregimes. This is followed by several examples for different types of nonlinear ellipticdifferential equations, e.g. in mathematics – the Monge–Ampere equation – in science –the reaction–diffusion systems – and in engineering – the von Karman and the Navier–Stokes equations. These problems and methods require the analytical results necessaryfor our (space-) discretization methods. The necessary tools from functional analysisand calculus in Banach spaces are listed in Section 1.4.

In Chapter 2, we summarize the indicated analytical results. We include generallinear, special semilinear, semilinear, quasilinear and fully nonlinear elliptic differentialequations and systems of orders 2 and 2m, e.g. the Navier–Stokes equations. An earlyform of such a collection of results for engineering problems are the two volumes ofAmes [26,27]. Numerical methods are studied there as well. Extensions to science and

Preface xix

the necessary analytic and group theoretic tools are collected in Rogers and Ames [557].The essentials are existence, uniqueness and regularity of their solutions and propertiesof the linearizations. This linearization is applicable to nearly all the nonlinear ellipticproblems discussed before (see Section 2.7). Excluded are nonlinear problems that arenot elliptic in the standard sense, but monotone. Their FEMs are discussed in Section4.5. The linearization and its bounded invertibility for the included problems yield theFredholm alternative and the stability of space discretization methods, respectively.

Linearization is not always appropriate for the analysis of nonlinear problems.Different techniques have to be used for other aspects of nonlinear problems: exis-tence, uniqueness and other properties of solutions are obtained by methods, usu-ally independent of linearization. They are appropriately studied via monotone andNemyckii operators, systematically described, e.g. in Zeidler’s impressive “multigraph”[675]–[678]. Other examples are discussed, e.g. by Haslinger et al. [393], forunilateral boundary value problems, by Feistauer and Zenısek [316, 317], andby Zenısek [681], with monotone and quasimonotone methods and variationalcrimes. Haslinger et al. [393] consider mainly mechanical contact problems viainequalities.

Often maximum principles, and different inequalities are employed, cf. Gilbargand Trudinger, [346]. In a fascinating series of papers, [215–228] Crandall, Lionsand colleagues study viscosity solutions, which we do not consider here. In Caf-farelli and Glowinski, [154], they are applied to numerical methods for the Pucciequation.

Another example is the huge area of practically relevant problems, e.g. in fluidand solid mechanics and in modeling. In solid mechanics the nonlinearity is com-plicated by additional corner and edge singularities and different scales for theinteresting scenarios. Here and in related problems, adaptive mixed FEMs are manda-tory, cf. e.g., Braess, [135–137] and with Carstensen, and Reddy, [138], Johannsenet al. and Kawohl et al. [421, 438] Stein/Sagar, [594]. Essential phenomena, basedupon a posteriori error estimates, employ the nonlinearity directly. Some of theseproblems are indicated in Chapter 6, but we do not study them explicitly. Thequestion how well these problems can be solved with lineraization techniques isstill open.

We turn to Part II numerical methods. Most standard monographs or textbooks,present the convergence theory of space discretizations only for one specific discretiza-tion method and only linear problems. In other books, e.g., Brenner and Scott, [142],FEMs for nonlinear problems are essentially described by a few examples. The booksof Glowinski, partially with Lions and Tremolieres, and surveys [350, 351, 353, 354],systematically study numerical methods for variational inequalities and nonlinearvariational problems. Again nonlinear problems play an important role in Glowinski’smany proceedings edited, e.g. with Absi, Angrand, Bertin, Bristeau, Chen, Dervieux,Desideri, Fortin, Larrouturou, Lascaux, Le Tallec, Lions, Liou, Mantel, Neittanmaki,Pekka, Periaux, Pouletty, Tezduyar, Tong, Tremolieres, Veysseyre, Zhang. We only listthe first and last one [355, 356]. Glowinski discusses special fully nonlinear problemswith Dean and recently with Caffarelli and Guidoboni, Juarez, Pan [154,272–276,352].

xx Preface

The last two papers use viscosity solutions. These allow other problems than ourlinearization, e.g. the Pucci equation.

Numerical methods for nonlinear problems are intensively studied in China as well,cf. e.g. the proceedings edited by Chen, Chow, Glowinski, Kako, Li, Periaux, Shi,Tong, Ushijima, Widlund, Zhang. e.g. [586,642]. Different methods are discussed, e.g.in Temam, [621–624], Feistauer, Felcman and Straskraba, [315]. [621–624] considers forthe Navier–Stokes problem, finite difference and finite element methods. [315] discuss,for compressible flow problems, in addition finite volume and DCGMs, with proofs inNovotny, Straskraba, [515].

Finite difference and finite element methods for linear problems are discussed byHackbusch [387]. Different strategies are presented for proving convergence, e.g. M-matrix techniques for finite differences versus the standard bilinear forms for FEMs.Grossmann et al. [374–376] mainly study linear problems, approximated by difference,FE and boundary integral methods, cf. Sloan, Suli and Vandewalle [592]. Convergenceresults for nonlinear boundary operators seem to be nearly totally missing.

In the earlier approaches, studying general convergence theories, e.g. as in [28–30, 32, 371, 441, 528–532, 547, 596, 675–678], spectral or FEMs with variational crimes,DCGMs, wavelet and mesh-free methods are necessarily missing. Sometimes finitedifference and FEMs are studied via outer and inner approximation methods, e.g.in [528–532, 678]. Conforming FEMs are considered as inner approximation schemes,and classical difference methods as external approximation schemes, cf. Temam,Vainikko, Schumann and Zeidler [574, 622, 645, 647, 678]. More complicated meth-ods, e.g. nonconforming FEMs, and systems and fully nonlinear equations are notincluded.

Here we employ a new theory, combining generalized Petrov–Galerkin methods andone of the classical theories, here that by Stetter [596]. This yields, in Chapter 3, thenecessary existence, stability, convergence and computing results for all these problemsand discretization methods in a unified way. In particular, it avoids distinguishing innerand external approximation schemes. The standard “consistency and stability implyconvergence” result is proved.

Chapter 3, with its general discretization theory, strongly depends upon linearizationand requires discussing its limitations and advantages. As previously mentioned,linearization for quasilinear problems, defined on Sobolev spaces Wm,p(Ω), is limitedto 1 ≤ p <∞. We obtain stability and convergence results for 1 ≤ p ≤ 2 discreteWm,p(Ω) norms, but for 2 ≤ p <∞ only with respect to discrete Hm(Ω) norms. Thisis complemented by the monotone convergence approach mentioned above.

Based upon the previous analytical existence, uniqueness, and regularity results,obtained by any other method, linearization allows proving, in a unified way and for amajor part of these problems, the desired stability, convergence, and Fredholm results.In fact, the stability of the nonlinear system is a consequence of the stability of thelinearized discrete equation. For general fully nonlinear problems this seems, untilnow, the only possible way at all. Finally, we have already mentioned the appropriatecombination of Newton and continuation techniques in the MIP.

Most problems considered in this book can be studied in their weak form. Thenthe spaces of ansatz functions, including the exact solution, and of test functionsare identical. For the exceptional nondivergent quasilinear and the fully nonlinear

Preface xxi

problems this is no longer correct, cf. Subsection 2.5.7 and Sections 5.2 and 5.3. SoChapter 3 is formulated to include these weak cases.

Additionally, we assume that the derivative of the nonlinear operator, evaluatedin the exact (isolated) solution, is boundedly invertible. We do have to defend thiscondition: The arguments are all closely related to the numerically necessary conditionof a (locally) well conditioned problem. This is often guaranteed by monotonicityarguments, cf. Subsections 2.5.5 and 2.5.6. For an existing locally unique solution,Frechet differentiability is satisfied for many practically relevant problems. Exceptionsmight be unilateral problems, which seem to have been studied until now mainly forlinear problems, see e.g. Haslinger, Hlavacek and Necas [393].

As a consequence of the Taylor formula and the Fredholm alternative for theseoperators, the bounded invertibility of the derivative is usually, but not necessarily,satisfied as well. Otherwise a nontrivial kernel indicates bifurcation, see below, orthe second derivative has to be positive or negative definite, or other combinationswith higher derivatives have to be satisfied to fit the above monotonicity arguments.Thus there will be (exceptional) operators with locally unique solutions violating thecondition of bounded invertibility. For our numerical approaches we are luckily able tomonitor this danger. If this exceptional case is detected, the convergence for the fol-lowing numerical methods simply is not proved. Furthermore, except for monotonicityand iteration techniques, cf. Zeidler [678], Section 26.4, or small dimensional systems,there are not too many numerical methods for solving the nonlinear discrete equationwithout requiring a boundedly invertible derivative in one form or the other. Thelatter, in particular, has to be correct for linear solvers and the Newton method for thediscrete problem. Unless the derivative is nearly singular, the corresponding discretelinear equation is stable, hence boundedly invertible. This fact is often monitoredautomatically.

There is another important reason for assuming the bounded invertibility of thederivative, strongly related to the Fredholm alternative for the linearized elliptic oper-ator. If it were violated, small perturbations of the original problem, e.g. correspondingto round-off errors, could be embedded into a parameter-dependent problem. Thiswould usually have a bifurcation point or another singularity. The standard numericalmethods would have to be replaced by numerical methods for bifurcation, see e.g.Bohmer [113–115,120] Caloz and Rappaz [156], and Cliffe, Spence and Tavener [180].

Conversely, the bounded invertibility of the linearized original operator as a con-sequence of the stability of the linearized discrete operator is proved in Section3.6. This last result is particularly interesting: For the more complicated ellipticproblems the available analytical results become more and more lucanary. Thencontinuation techniques can be combined with the results in Section 3.6, relatingthe stability of the linearized Ah with the invertibility of A, and with methodssimilar to the classical existence results, obtained e.g. by Courant, Friedrichs andLewy [213]. This allows research trips into analytically unknown areas, e.g. systemsof the fully nonlinear Monge–Ampere equations. All these results apply to the abovewide class of nonlinear problems and the chosen important types of space discretizationmethods.

So there are enough good and numerically necessary reasons for assuming a bound-edly invertible derivative.

xxii Preface

There is a totally different other reason for this assumption in several discretizationmethods, see above. For example, for boundary integral methods a boundary integralformulation of nonlinear problems seems to be difficult. For wavelet and mesh-freemethods stability and convergence results for nonlinear problems have not been avail-able. So the different Newton–Kantorowich and continuation techniques are appliedto the analytical problem. For these originating linear problems the above-mentioneddiscretization methods yield convergence. However, most scientists in the numericalcommunity seem to prefer a discretization of the nonlinear problem compared to thislinear detour. For wavelet and mesh-free methods we succeeded in proving convergencefor a wide range of nonlinear problems in Chapter 9 and in [120].

We want to return to stability and convergence of these discretizations. Stabilityis reduced in Chapter 3 to a few well-known basic concepts from functional analysisand approximation theory. We combine coercive bilinear forms or monotone operators,their compact perturbations, bordered systems (for bifurcation and center manifolds),exact and approximate projectors with interpolation, best approximation and inverseestimates for approximating spaces. This allows a unified proof for stability for theabove discretization methods and operator equations. Stability for the linear operatorsuffices. Its principal part is essentially coercive, hence its discretization is stable. Thestability of the linear operator Ah itself, with A, a compact perturbation of its principalpart, turns out to be a consequence of the bounded invertibility of A. Examples are thevon Karman or the Navier–Stokes equations. Sections 2.7 and 2.8 show that our ellipticlinear and nonlinear equations and systems do satisfy this condition. Consistencyis simple for conforming FE, wavelet and spectral methods. For the nonconformingcases we have to employ appropriate tricks. Often the relation between the weak andstrong form is essential. The necessary conditions and results, including the meshindependence principle, are formulated.

Following this rather extended discussion of stability and convergence for discretiza-tion methods and their differences compared to other approaches, we present shortsummaries of the results for our different methods in the following chapters. Thecorresponding introductions formulate extended summaries.

It is certainly justified considering the nowadays most important FEMs moreextensively than the others. In Chapter 4 we start the discussion with conformingFEMs. Thus the ansatz and test functions satisfy the appropriate regularity andboundary conditions. We start by summarizing the well-known approximation theoryof FEs. Linear to quasilinear problems allow strong and weak formulations. Theoriginal and discrete forms have to be compared. In fact, the weak, simultaneouslyvariational forms are directly used for these problems to yield the conforming FEMs.Based upon Chapter 3 the proof of consistency, stability and convergence is prettysimple. Consistency is nearly obvious. Stability is proved by compactly perturbing thecoercive principal parts, or for the Navier–Stokes problem, the Stokes operator. We donot study super convergence results, cf. Bank, Xu and Griewank, Reddien and Jiang,Shi, Xue, [66,368,417].

However, the strong requirements for conforming FEMs with respect to ansatz andtest functions, exactly evaluated test integrals, and restrictions to weak forms is asevere drawback. This is solved by considering nonconforming FEMs in Chapter 5.

Preface xxiii

Only to these nonconforming FEMs do we apply and prove convergence for the fullspectrum of linear to fully nonlinear equations and systems of orders 2 and 2m.Weak forms are no longer possible for fully nonlinear problems. So specific techniquesare applied. They relate the strong form of the fully nonlinear system to the weakform of the linearized equations with stability properties. We present, with O. Davy-dov, special differentiable FEs, needed for these problems, cf. Subsection 4.2.6. Sonew FEMs are formulated for general, nondivergent quasilinear and fully nonlinearequations and systems, see Bohmer [118, 119]. In particular for nonlinear problemsand nonconforming FEMs, the relation between classical and variational consistencyhas to be discussed. This technique can even be used for proving convergence fornonlinear boundary operators. The necessary quadrature approximations for all FEMsare described. Variational crimes for FEs violating regularity and boundary con-ditions are studied in Section 5.5 in R2 for linear and quasilinear problems. Thisallows high order conforming and nonconforming versions, particularly importantfor bifurcation, and yields several new results. An essential tool is the anticrimetransformation.

As a consequence of the dominant role of FEMs, the presentation of numericalsolutions of five classes of problems is restricted to FEMs: the variational methods foreigenvalue problems are presented in Section 4.7. The convergence theory for monotoneoperators as applied to quasilinear problems is described in Section 4.5. Section 5.2 isdevoted to FEMs for fully nonlinear elliptic problems. In Section 5.3 we study FEMsfor nonlinear boundary conditions. Convergence results for these problems are hardly,if at all, discussed in the literature. Quadrature approximate FEMs are the topic inSection 5.4. We thus close several gaps in the literature. The reason for this restrictionto FEMs is very simple. These stability, convergence and computing results are validessentially for all the other methods as well. In fact, the proofs for all these resultsremain correct for all the other methods. This could be shown by only slightly andobviously modifying our unified approach on the basis of the specific consistency andstability results available for each method. This claim has to be slightly restricted forDCGMs and wavelet methods. Our DCGM approach is available only for problems oforder 2, and is not possible for fully nonlinear elliptic problems. Wavelet methods arelimited to semilinear problems.

In the meantime, for many of the difficult problems in applications, e.g. the Navier–Stokes problem in fluid dynamics, or in elastomechanics and electromagnetism, mixedFEMs are combined with nonconforming FEs, cf. Braess [135, 136] and Brezzi andFortin [144]. Totally new potentials seem to develop within the mesh-free methods,cf. [120].

In Chapter 6, W. Dorfler describes the idea of a posteriori error estimation andadaptive mesh refinement. The main points are the description of the adaptive finiteelement algorithm and its convergence and the sketch of the optimal complexity of thisalgorithm with respect to an approximation class of solutions. The techniques in thischapter, presented essentially for linear elliptic problems, can be combined with thosefor nonlinear problems in the previous chapters. As indicated in the last subsection,this might allow generalizing many results to nonlinear problems. Interesting resultsfor nonlinear problems are due to Verfuhrt [657].

xxiv Preface

Discontinuous Galerkin methods (DCGMs) with V. Dolejsı in Chapter 7 are anothertype of nonconforming FEMs. For avoiding rather highly technical conditions andproofs we restricted DCGMs to quasilinear equations and systems of order 2. DCGMsviolate boundary conditions and continuity of the piecewise polynomials. These viola-tions are compensated by additional penalty functions in the discrete weak form. Theabove technique in FEMs applied to fully nonlinear problems seems to be impossiblefor DCGMs.

In Chapter 8, difference methods are treated, as the other methods, by applyingour new approach to the difference equations in weak form. This is different frommost earlier approaches, either based upon the strong forms or studied as externalapproximation schemes. There weak forms have been considered in many papers andbooks, cf. [574, 622, 647, 678], and in lectures, cf. Bank [62]. Our results again areformulated for linear to quasilinear equations and systems of orders 2m,m ≥ 1. Incontrast to the earlier M -matrix approach, the discretization errors are here estimatedwith respect to a discrete Sobolev norm of order m for a problem of order 2m,similar to our FEMs results. This allows efficient defect correction methods as anappropriate strategy for formulating high order methods as in Bohmer [105–109] andHackbusch [384]. Particularly interesting are symmetric forms, again for equations andsystems of order 2m,m ≥ 1. For cuboidal domains with faces parallel to the axes theboundary condition are exactly reproduced. This implies convergence of order 2 and2k for k defect corrections, compared to order 1 and k for the unsymmetric forms.Only partially do we prove the results for the difference methods for two reasons.The techniques are very similar to FEMs, including the case of violated boundaryconditions in Sections 8.2–8.6. The special methods for curved boundaries in Section8.7 are known only for special cases and require highly technical proofs based upontotally different techniques. They do not seem to be applied too much these days. Wefinally apply, with C.S. Chien, difference, combined with extrapolation methods, tothe von Karman equation.

Wavelet methods are applied in Chapter 9 with S. Dahlke for the first time inthis generality and appropriate for nonlinear problems and bifurcation. This presentsan obvious extension of the first papers on wavelets for nonlinear elliptic problems,Bohmer and Dahlke [121, 122]. This contrasts with the recent books on wavelets,combined with elliptic PDEs, by Cohen [195] and Urban [640, 641]. The most far-reaching results for linear problems until now were obtained for self-adjoint andsaddle point problems requiring a nonsingular linear operator A. For these casesproofs for convergence are available, directly employing the special properties ofwavelets. As a consequence of the difficulties with evaluating nonlinear functions andoperators with wavelet arguments, cf. Subsection 9.3.5, general quasilinear and fullynonlinear problems have to be excluded. With this exception, the whole spectrumof corresponding wavelet methods is shown to be stable and convergent. Againthe corresponding linearized operator has to be boundedly invertible. T. Raaschfinishes this chapter with adaptive wavelet methods and presents some numericalresults.

Spectral methods in Bohmer [120] inherit, for correct choices of approximatingspaces, even Γ-equivariance for infinite symmetry groups, e.g. O(3)×D4, from the

Preface xxv

original to its discrete bifurcation problem. This is the key condition for reproducingbifurcation scenarios by discretization. Mesh-free methods have been successfullyapplied to nonlinear problems. Convergence proofs have been totally missing for morethan 20 years. So our linearization approach again shows its power by proving, onlyvery recently, the missing convergence for these problems and the standard mesh-freemethods in [120]. Due to the rather free placement of their “centers”, they are veryappropriate for problems with symmetries as well. So we delay the presentation ofspectral and mesh-free methods to [120].

We discuss dicretization methods for problems in Rn, n ≥ 2. The analysis for FEMsworks well in Rn. The methods in Section 5.5 seem to be the only type of FEMs, ana-lytically restricted to R2, losing accuracy in possible generalizations to R3. However,there is a severe restriction of FEMs on triangulations essentially to Rn, n ≤ 3, causedby the tremendously increasing complexity of the software, in particular for adaptiveFEMs. This situations is maintained for DCGMs. FEMs and difference methods oncuboidal subdivisions allow general Rn, but do not admit adaptivity. The waveletmethods in Chapter 9 have not yet reached the level of FEMs and DCGMs. For thespectral methods in [120] we have excluded domain decomposition techniques, usuallynot necessary for equivariant problems. So all these methods in our presentation allowconvergence in Rn, but the problem of highly technical software is not yet solved.

The situation is totally different for the mesh-free methods in [120]. The analysis ispossible for Rn, n ≥ 1, but the approximation quality strongly increases with increasingn. The software is nearly independent of n. So for many cases mesh-free methods mightbe the future methods of choice for n ≥ 3. In [120] we give a proof for convergenceof these methods. This is an approach totally different from Chapter 3 in this book.For the first time it shows convergence of meshfree methods applied to nonlinearproblems. This can be applied to the other methods as well and allows modificationsand adaptivity for some of them.

Summarizing, we obtain stability and convergence for the different types of spacediscretization methods and all the elliptic linear and nonlinear equations and systemsof orders 2 and 2m. This synopsis allows the extensions and generalizations describedin the previous paragraphs, partially much beyond the present state of the art. Thusit fills a gap in the available literature.

This book is aimed at three groups of people. Firstly, graduate students andscientists who want to study and to numerically analyze nonlinear elliptic problemsin mathematics, science and engineering. They will find the necessary tools andconvergence results. This simultaneously serves as a solid foundation for many inthe huge number of papers in scientific computing in this area. Finally, the boundedinvertibility of the linearized original operator as a consequence of the stability ofdiscrete operator in Section 3.6 can be combined with methods similar to the classicalexistence results as in Courant, Friedrichs and Lewy [213], for indicating new existenceresults. All these methods apply to the above wide class of nonlinear and evenfully nonlinear problems and the important types of space discretization methods. Inparticular, many of the existence, uniqueness and regularity results are only known forthe case Hm

0 (Ω). Generalizations to the Wm,p0 (Ω,Rq) situation would be worthwhile.

In fact, results as in Theorem 2.122, form the basis for numerical methods for the

xxvi Preface

Wm,p0 (Ω,Rq) setting, opening the possibility for research into new problems outside

the previous scope.Secondly, we have previously indicated, and update that later on, several gaps in

our numerical results. These might stimulate future research. For example, FEMsviolating continuity and boundary conditions and DCGMs are restricted here to order2, these FEMs even to R2. Possible tools might be complementary boundary conditionsand higher order cubature formulas. Asymptotic expansions for the discretizationerror and defect correction methods on this basis can probably be generalized withour techniques to FE and the other methods as well. Extensions of our resultsto other methods, e.g. the excluded finite volume methods, might be worthwhile.The complicated problems in elastomechanics and electromagnetism, approximatedby mixed FEMs combined with nonconforming FEs, might find some new tools incombination with adaptive FEMs. These adaptive FEMs are presented here essentiallyfor the Laplacian with some hints for generalizations. So there are still interesting areasfor further research.

Thirdly, material for many different graduate courses advanced seminars or researchprojects can be chosen. The starting point is the motivation for nonlinear problemsin Chapter 1. The next step would lead to the existence, uniqueness and regularityresults for the desired class of elliptic problems in Chapter 2. The next Chapter 3would be mandatory for all elliptic problems and their numerical methods. Then e.g.a DCGM for a quasilinear operator, F , is chosen. If the Frechet derivative, F ′, ofthis operator is coercive, then stability and convergence of the DCGM is completelyproved in Chapter 7. For a more general, only boundedly invertible F ′, the proof forstability for the nonconforming FEM has to be updated to the DCGM. This straightforward job is left as a (not quite trivial) task to the interested reader. Or the newnumerical methods for fully nonlinear problems allow different new research projects.In geometry a surface with a rather general prescribed Gauss curvature, closely relatedto the Monge–Ampere equation, could be numerically solved for the first time even inhigher dimensions. It would be desirable to extend the deep results for Monge–Ampereequations to systems. Again our numerical methods for fully nonlinear problems canbe combined with continuation methods indicating new possibilities. These examplesshow the wide range of applications from pure mathematics to science and engineering.

So the author hopes that this book will stimulate further research, e.g., ellipticproblems on manifolds. Concentrating my work for decades on numerical methodsfor general defect corrections and bifurcation of PDEs, and only recently on FEMs,certainly some important results for the different discretization methods will haveescaped me. In a huge project of ca. 1500 pages in both books, I will have missed a lotof misprints and hopefully only a few minor errors. So I will gratefully accept critics,suggestions, or remarks.

Acknowledgements: I am very grateful to many people who have strongly sup-ported and influenced my books or encouraged me in their genesis. First of allmy co-authors of this volume, C.S. Chien, S. Dahlke, O, Davydov, W. Dorfler,V. Dolejsı, and T. Raasch have done a great job. It was always fascinating howinitial ideas developed to the final forms. Then my good friends and cooperators

Preface xxvii

E. Allgower and Z. Mei, cf. the Bibliogrphy, were particularly influential for manyresults in this and the next book. The co-authors of [120], E. Allgower, P. Bel-tram, P. Chossat, A. Cliffe, G. Dangelmayr, C. Geiger, K. Georg(†), N. Jangle,D. Janovska, V. Janovski, Z. Mei, R. Schaback, K.-H. Schild, B. Schmitt, J. Tauschand my other friends and colleagues P. Antonietti, M. Buhmann, E. Doedel,P. DuChateau, B. Fiedler, J. Gwinner, S. Feistauer, W. Hackbusch, R. OMalley,K. Nickel(†), L. Schumaker, Z.-C. Shi, A. Spence, W. Wendland, G. Wittum, E.Zeidler, stimulated the progress with many discussions. Last but not least most of myPh.D. students, J. Bosek, U. Garbotz, N. Sassmannshausen, A. Schwarzer,R. Sebastian, and additional guests in Marburg, P. Ashwin, W. Govaerts, A. Hoy,and S. F. McCormick provided essential results.

The Deutsche Forsungsgemeinschaft (DFG), and the Deutscher Akademischer Aus-tauschdienst (DAAD), provided the necessary additional financial support.

The Philipps University Marburg offered a stimulating atmosphere for teachingand research. Essential parts of this book had been used in year long lectures.Without the help of my many masters and Ph.D. students in particular [120] wouldhave been impossible. N. Sassmannshausen contributed good ideas to Chapter 3.Several colleagues in our Department, W. Gromes, V. Mammitzsch, C. Portenier,G. Schumacher, H. Upmeier, were always ready to answer my many questions. Mostimportant was the support by our group numerical mathematics, mainly by S. Dahlke,J. Kappei, T. Raasch and B. Schmitt. F. Muth has done a great job. She has typedapproximately half of the early forms of Chapters 1, 2, 4, 5.

Very supporting was Oxford University Press’ highly professional, very constructive,kind and personal advice by S. Adlung, E. Suli, A. Warman, M. Johnstone, P. Hendryand many others.

Finally, the love, caring, and encouragement of my late wife through all the manyyears with successes and intrigues is the basis for this book.MarburgMay 2010

Klaus Bohmer


Part I

Analytical Results


1

From linear to nonlinear equations,fundamental results

1.1 Introduction

Why do linear models describe nonlinear reality sometimes correctly and sometimestotally insufficiently? To get a feeling and foster the insight for this difference the firsttwo examples are chosen as very simple mechanical problems, deduced by physicalreasoning. In this chapter we do not intend to give a rigorous analysis of the prob-lems, but rather motivate, by the following examples, the rigorous analysis in laterchapters. So we do not care at the moment to present the mathematical equationsin the appropriate function spaces, nor do we refer much to physics, chemistry orbiology. Usually varying parameters play a dominant role. The first examples evenindicate bifurcation effects in differential equations. These phenomena are the goal ofBohmer [120]. We end this chapter by summarizing the necessary basic results for thisbook.

1.2 Linear versus nonlinear models

To answer the introductory question we show why the following linear model cannotdescribe the situation correctly. For more explicit mechanical arguments, see, e.g.Troger and Steindl [637].

Example 1.1. Bending rod with perpendicular loadLet a vertically positioned rod be clamped at the lower end and be free at the upperend, see Figure 1.1. A load P at the end of the rod, originally perpendicular to therod, forces it to bend sideways. Its displacement x is proportional to P , for small P ,i.e. the following linear relationship holds:

x = cP for an appropriate c. (1.1)

However, in a simple experiment we observe that the rod will deform nonlinearly oreven break when the load P passes a certain critical value. In other words, for largeP , the linear law (1.1) does not represent the “reality” appropriately.

So we have to incorporate the “nonlinear” effects of deformation. We want to derivea nonlinear mathematical/physical model for the stationary position of the bent rod.There is a strong analogy between the relatively unknown strain energy of a rod and

4 1. From linear to nonlinear equations, fundamental results

j (s)

P > 0P = 0

Figure 1.1 Rod with load perpendicular to the rod.

the well-known kinetic energy E of a moved body. This is described by the formula E =m/2 (du(t)/dt)2 with the mass m, the velocity, v = du(t)/dt, the distance traversed,u(t), and the time, t. Analogously, we obtain for the (local) strain energy, U, at a pointwith arc length s on the deformed rod the rule

U =α

2

(dϕ

ds(s))2

.

Here α is the bending stiffness and ϕ(s) represents the angle between the vertical axisand the tangent in the point with arc length s (see Figure 1.1). The total bendingenergy, Ubend, of the deformed rod is given by

Ubend =α

2

∫ L

0

(dϕ

ds

)2

ds for ϕ ∈ C1[0, L],

with the total length of the rod, L. The potential energy, due to the moving of the tip ofthe rod in the direction P , is given by −Pd(L), where d(L) indicates the displacementat the end of the rod. Figure 1.1 shows that

d(L) =∫ L

0

sinϕ(s)ds.

Therefore, the total energy, i.e. the so-called potential, is the sum of these two terms,see Troger and Steindl [637]:

V (ϕ) :=α

2

∫ L

0

(dϕ

ds

)2

ds− P

∫ L

0

sinϕ ds for ϕ ∈ C1[0, L]. (1.2)

One of the fundamental principles of mechanics states that a position of equilibrium ofa mechanical system, here the equilibrium of the rod bent by ϕ, is characterized by aminimum of its potential, here V , i.e. for every small function ψ we have necessarily

V (ϕ + ψ) ≥ V (ϕ). (1.3)

1.2. Linear versus nonlinear models 5

We claim that this minimizing function ϕ is characterized by a nonlinear boundaryvalue problem with the differential equation

G(ϕ, λ) :=d2ϕ

ds2+ λ cosϕ = 0, for λ := P/α, (1.4)

and the boundary conditions, indicated by b in the corresponding subspace,

C2b [0, L] :=

{ϕ ∈ C2[0, L]; ϕ(0) =

dϕ

ds(L) = 0

}. (1.5)

This represents a nonlinear model for the equilibrium or stationary position of thebent rod. Now we find, by the standard linearization of (1.2), for small ψ and sinψ =ψ +O(‖ψ‖2), compare Definition 1.41 and Theorem 1.42,

0 ≤ V (ϕ + ψ)− V (ϕ) = α

∫ L

0

(dϕ

ds

dψ

ds

)ds− P

∫ L

0

ψ cosϕds +O(‖ψ‖2).

Here we use the standard notation

g(ψ) = O(‖ψ‖α) is equivalent to (1.6)

‖g(ψ)‖/‖ψ‖α remains bounded for ‖ψ‖ → 0, and

g(ψ) = o(ψ) ⇐⇒ lim‖ψ‖→0

= 0,

with ‖ψ‖ indicating any norm of ψ. By partial integration we obtain

0 ≤ V (ϕ + ψ)− V (ϕ) = α

(ψdϕ

ds

) ∣∣∣L0−∫ L

0

ψ

(αd2ϕ

ds2+ P cosϕ

)ds +O(‖ψ‖2).

(1.7)For the situation in Figure 1.1, ϕ has to satisfy the boundary conditions, see (1.5),

ϕ(0) =dϕ

ds(L) = 0, (1.8)

since the rod at the endpoint is not bent. The first term in (1.7) vanishes if we require,according to (1.8), ψ(0) = (dϕ/ds)(L) = 0. This has to be satisfied for every smallperturbation, ψ, for a rod, ϕ + ψ, still clamped at s = 0. Thus

0 ≤ V (ϕ + ψ)− V (ϕ) = −∫ L

0

ψ

(αd2ϕ

ds2+ P cosϕ

)ds +O(‖ψ‖2). (1.9)

We started with ϕ,ψ ∈ C2[0, L], but find ψ ∈ C[0, L] with ψ(0) = 0 sufficient for (1.9).These changes in differentiability will play an increasingly important role for morecomplicated existence results, see e.g. Example 1.7 below. We have split the incrementV (ϕ + ψ)− V (ϕ) into a linear operator (here a functional), acting upon ψ, and ahigher order remainder term O(‖ψ‖2), sometimes denoted as h.o.t.

We have to show that the minimizing solution ϕ of (1.3) indeed solves the nonlineardifferential equation (1.4). Using the standard argument of the fundamental lemma of


the calculus of variations, we assume, contrary to (1.4), that

d2ϕ

ds2+ λ cosϕ �≡ 0, for ϕ ∈ C2[0, L] and λ := P/α.

Choosing dψ0 := −(d2ϕ/ds2 + λ cosϕ) ∈ C0[0, L] leads to

c0 := −∫ L

0

ψ0

(αd2ϕ

ds2+ P cosϕ

)ds = α

∫ L

0

(ψ0)2ds > 0.

The remainder term in (1.9) satisfies

O(‖μψ0‖2) = μ2c1, with c1 := O(‖ψ0‖2).

Hence, for small enough μ we have |μc1| < c0/2, hence c0 + μc1 > c0/2, and

V (ϕ + μψ0)− V (ϕ) > μc0/2 for μ > 0,

V (ϕ + μψ0)− V (ϕ) < μc0/2 for μ < 0.

This contradicts the fact that ϕ is a minimum for V , thus showing (1.4), indeed anonlinear model for the stationary position of the bent rod.

To study the relationship of (1.4) and (1.5) to (1.1), we start the discussion withthe case of small angles ϕ. From cosϕ ≈ 1 and (1.4) we obtain

d2ϕ

ds2+ λ cosϕ ≈ d2ϕ

ds2+ λ = 0.

Hence ϕ ≈ −λs2/2 + μs + ν and, by (1.5), we find ν = 0, μ = Lλ = LP/α, and

0 ≤ ϕ(s) = λ(−s2/2 + Ls) ≤ ϕ(L) = PL2

2α� 1 for small P.

This implies in particular, that a nontrivial solution exists for every λ �= 0.Similarly we may use sinϕ ≈ ϕ and find

d(L) =∫ L

0

sinϕ(s)ds ≈∫ L

0

ϕ(s)ds = λ

(−s3

6+ L

s2

2

)∣∣∣∣L0

= PL3

3α; (1.10)

thus we have reproduced the form (1.1).When we proceed to larger values of P , and hence ϕ, the solution of the boundary

value problem (1.4), (1.5) requires the use of numerical methods or elliptic functions.Physical arguments indicate that the rod will break at a distance s, where, for

increasing P , for the first time, the tension σ = σ(s) on the surface of the rod becomeslarger than the critical tension σcrit of the material used. Furthermore, it is well knownthat the curvature κ = dϕ/ds is proportional to σ = cdϕ/ds. If we assume |ϕ(s)| <π/2, we derive from (1.4) and (1.5) that

d2ϕ

ds2(s) = −λ cosϕ < 0.


Hencedϕ

ds(0) >

dϕ

ds(s) >

dϕ

ds(L) = 0 for 0 < s < L.

So the rod will break for s = 0 if P is so large that cdϕ/ds(0) > σcrit. Certainly thelinear relation (1.1) can no longer describe this situation correctly.

In particular, the linear model (1.1) is not appropriate, since we have assumedat different stages that either ϕ or ψ is small. Only in this case does neglecting theh.o.t.s makes sense, otherwise they will strongly influence the solutions for a nonlinearmodel. �

In Example 1.1, showing the need for nonlinear models, the solutions for the linearand nonlinear problem were pretty similar for small values of P . The next Example 1.2deals with a qualitative difference in the behavior of solutions of linear and nonlinearproblems.

Example 1.2. Rod with axial loadAgain, we consider deformations of a thin inextensible rod of length L. This time, westudy an axial load P . The rod is pinned at one end (for a horizontally positioned rodthis is denoted as a simply supported end); the other end moves freely in the verticaldirection. So, the rod tends to deform in a plane. We perform the experiment indicatedin Figure 1.2. If we slowly increase P , starting with P = 0, the rod stays unchanged.For a further increase of P beyond a critical P0, deviations of the rod from the verticalposition are observable. These deviations increase in size with P.

For modeling the new situation we can use the same bending energy as in Exam-ple 1.1. The total length equals the total arc length, s = L , of the rod. The potentialenergy is again given as −Pd(L) with the displacement at the upper end of the rod,d(L), here in the vertical direction. Figure 1.2 shows that, instead of the original L,

P P

(s)j

Figure 1.2 Rod with axial load.


contrasting to (1.10), the upper end has come down by

d(L) =∫ L

0

(1− cosϕ)ds.

With this d(L) we obtain the total energy V (ϕ) and the difference V (ϕ + ψ)− V (ϕ),see also (1.2), as

V (ϕ) := α

∫ L

0

(dϕ

ds(s))2

ds/2− P

∫ L

0

(1− cosϕ)ds for ϕ ∈ C2[0, L], (1.11)

V (ϕ + ψ)− V (ϕ) = α

(ψdϕ

ds

) ∣∣∣L0−∫ L

0

ψ

(αd2ϕ

ds2+ P sinϕ

)ds +O(‖ψ‖2).

An analogous argument as in Example 1.1 shows that the bent rod in the stationaryposition necessarily minimizes V (ϕ) in (1.11). The first term in V (ϕ + ψ)− V (ϕ) ≥ 0vanishes, typically, for the following boundary conditions.

dϕ

ds(0) =

dϕ

ds(L) = 0. (1.12)

Equation (1.12) represent the so-called natural or Neumann boundary conditions asin Figure 1.2: the rod is not bent at the endpoints, but the lower endpoint is fixed,and the upper moves along the vertical axis. Imposing (1.12), the minimized V (ϕ) in(1.11) yields

0 ≤ V (ϕ + ψ)− V (ϕ) = −∫ L

0

ψ

(αd2ϕ

ds2+ P sinϕ

)ds +O(‖ψ‖2).

The minimizing function is characterized by a now different boundary value problem

G(ϕ, λ) :=d2ϕ

ds2+ λ sinϕ = 0 with λ := P/α, and

dϕ

ds(0) =

dϕ

ds(L) = 0. (1.13)

Obviously, ϕ ≡ 0 represents a trivial solution for (1.13), independent of λ,L. ThisG(ϕ, λ) = 0 in (1.13) is a nonlinear equation for ϕ. The question of whether (1.13) hasa (unique) solution or not has to be answered by mathematical results. For the caseof elliptic partial differential equations we will present those results in Chapter 2. If a(unique) solution ϕ of (1.13) exists, similarly for (1.4), (1.5) in Example 1.1, then ananalogous argument shows that ϕ minimizes V (ϕ) and thus solves our problem. �

These two examples show totally different behavior, as a consequence of the differentnonlinearities in the differential equations, different boundary conditions, and loads.In Example 1.1 a solution ϕ ≡ 0 is possible only for λ = 0 = P . Already, a very smallP causes displacements d(L) > 0. The experiments in Example 1.2 show that up toa certain P = P0 we only have the trivial solution ϕ(s) ≡ 0. If P increases beyondP0, the rod starts to buckle in the form of a, at the beginning, sinusoidal function.If we were to fix the rod at its middle point, a much higher P would be necessaryto deform the rod. So why and when are nontrivial solutions generated? To explainthese observations, we again study the nontrivial solutions for (1.13), originating from


the trivial state ϕ(s) ≡ 0. We start with a very small function ϕ and consider theapproximation

sinϕ = ϕ− ϕ3/3! + ϕ5/5! + · · · = ϕ + h.o.t.

The small nontrivial solutions ϕ of (1.13) have to satisfy the linearized problem

d2ϕ

ds2+ λϕ = 0,

dϕ

ds(0) =

dϕ

ds(L) = 0, λ = P/α, (1.14)

see (1.4). This linear boundary eigenvalue problem has nontrivial solutions of the formϕ = c cos

√λ s if and only if

dϕ

ds(L) = −c

√λ sin

√λL = 0 =

dϕ

ds(0), (1.15)

hence if and only if√λL = nπ, λ = (nπ/L)2 and ϕ(s) = c cosnπs/L. In this case

the linearized operator has a nontrivial kernel or null space. This observation showsthat only for certain discrete values of P = α(πn/L)2 may the linearized problem(1.14) have a nontrivial solution, originating in the trivial solution. This seems to beand in fact is a necessary condition for a nontrivial solution “bifurcating from thetrivial solution” of the original problem, see Figure 1.3. We will study “bifurcationproblems”, their local dynamical behavior, and their numerical approximation forPDEs in Bohmer [120].

These very simple examples indicate two features, essential for further studies. Fora nonlinear operator its linearized form provides essential information. If “bifurcationphenomena” of the original nonlinear problem are interesting the null space of itslinearized operator plays a major role. This is one of the reasons why in the followingchapters the problem of linearization and its corresponding Fredholm alternative,hence results about the null space of its linearized operator, are so important.

As we indicated already in the Preface, linearization is an essential feature of ourlater convergence proofs for all the numerical methods studied in the two books.Nevertheless, there are different opinions about combining linearization and numerical

λ

L /2

±⎟⎢j⎟ ⎢

−L /2

Figure 1.3 Global structure of bifurcating solution branches.


methods. Some use linearizations for the original analytical problems. The corre-sponding numerical methods are only applied to linear problems. Hence a nonlinearconvergence theory is unnecessary. The many papers on nonlinear convergence theoryshow that most mathematicians disagree with this idea. On the other hand, adaptivemixed FEMs, e.g. for mechanical problems, still seem to require the full nonlineartechniques.

1.3 Examples for nonlinear partial differential equations

We consider mathematical models for important problems in different sciences. We finda spectrum of parameter-dependent semi- to fully nonlinear partial elliptic differentialequations. As usual in mathematics and physics, they are appropriately simplifiedmodels. We do not go into details, but only give the corresponding equations and themeaning of the occurring functions and usually omit the exact function spaces in thisintroduction. Most of these examples will be more intensively discussed in the latercase studies in this book and in Bohmer [120]. For many other interseting, howevermostly linear applications, cf. Dautray and Lions [256,257].

Example 1.3. Chladny sound figuresAssume that we have a thin flexible square plate, horizontally fixed at its center[13–16]. On the surface of the plate we uniformly distribute tiny particles, for example,sand. Above the plate we fix a source of sound waves with variable frequency relatedto the λ in (1.16). We want to examine the reaction of the thin plate for varying λ.The pressure of the sound waves acts as a load on the surface of the plate. In the caseof resonance, a vibration with (unmoved) nodal lines is induced on the plate. Therethe tiny particles are collected. Thus, the distribution of these particles with respectto varying λ illustrates the vibration of the plate. Let Ω := [0, L]× [0, L] representthe plate and u = u(x, y) denote the maximal deviation of the vibration from thetrivial flat position of the plate at the point (x, y). Then u(x, y) satisfies the stronglysimplified mathematical model equation, compare (1.13) and Theorem 2.55,

G(u, λ) := Δu + λ sinu = 0 in Ω = [0, L]× [0, L] defined on (1.16)

C2b (Ω) :=

{u ∈ C2(Ω);

∂u

∂n= 0 on ∂Ω

}. (1.17)

Here Δ := ∂2 /∂x2 + ∂2 /∂y2 is the Laplacian operator and ∂u/∂n the normalderivative in the outer direction of ∂Ω. For arbitrary λ the trivial flat state u(x, y) ≡ 0represents a solution of (1.16), (1.17). As in Example 1.2 we linearize Equation (1.16)with respect to u at (u, λ) = (0, λ), cf. Definition 1.41 and Theorem 1.42, and obtain,similarly to (1.14), (1.15), with the trivial solution v = 0 for arbitrary λ,

∂uG(0, λ)v = Δv + λv = 0 ⇐⇒ Δv = −λv in Ω for v ∈ C2b (Ω). (1.18)

1.3. Examples for nonlinear partial differential equations 11

The −λ, allowing nontrivial solutions in the kernel of ∂uG(0, λ), are the

eigenvalues λ0 = −λ = −(m2 + n2)π2/L2 of Δ for all m,n ∈ N0 s.t. (1.19)

v(x, y) = vm,n(x, y) := cosm

Lπ x cos

n

Lπy, and v(x, y) = vn,m(x, y) are (1.20)

the eigenfunctions of Δ, hence, Δv = Δ(cos

m

Lπ x cos

n

Lπ y)

= −m2 + n2

L2π2 v.

Therefore, we obtain, e.g. for (m,n) = (1, 1), (m,n) = (2, 3) and (m,n) = (5, 5),(m,n) = (1, 7) the eigenvalues λ0 = −2π2/L2,−13π2/L2, and −50π2/L2 with one,two and three linearly independent solutions vm,n, vn,m. All their linear combinationssolve (1.18) as well, for fixed λ0. Hence, these λ0 for the sound well are candi-dates for bifurcations to nontrivial u, and for the generation of various Chladnyfigures.

Figure 1.4 shows v1,3 − v3,1, one of the many possible linear combinations withspecific symmetry properties. This Chladny problem is a typical example of multiplebifurcations in nonlinear problems with symmetries. �

Example 1.4. Buckling of a plateConsider a plate P under compression of the form Ω ∈ R2. It is defined by the Airy

stress function w(x, y) of the plate P at the point (x, y) and the deviation or deflectionu(x, y) for P from its flat trivial state. They are modeled by the von Karman equations.

0.50 1.51 2 2.5 3x

0

0.5

1

1.5y

2

2.5

3

cos (x)*cos (3*y)-cos (3*x)*cos (y)

Figure 1.4 Chladny sound figures on [0, π] × [0, π]. The straight (!) lines through the origin

are nodal lines (fixed, elevation 0); the curves show elevation lines. Notice the high symmetry,

and neglect the wriggles near the origin.


A mathematical justification for this model is given by Berger and Five [82–84], and,as the first terms in an asymptotic expansion, by Ciarlet [175]:

Gs(u,w) :=

⎛⎝Δ2u −[u,w]

Δ2w + 12 [u, u]

⎞⎠ =

⎛⎝ f

0

⎞⎠ in Ω ⊂ R2. (1.21)

Here λ is the external load, Δ2 = ΔΔ is the two-dimensional biharmonic operator, Ωthe shape of the plate, and the bracket operator [·, ·] is defined by

[u, v] = uxxvyy − 2uxyvxy + uyyvxx.

This semilinear system will be discussed in Subsection 2.6.6, cf. (2.403) ff. Bifur-cation regimes should be avoided in realistic constructions. But the mode jumpingis perhaps one of the most noteworthy features of experimental and mathematicalstudies about the buckling of plates. Equation (1.21) is taken up as a case study inChapter 8. �

Example 1.5. Reaction–diffusion equationsSemilinear systems of reaction–diffusion equations are typical models in nuclearreactor physics, and in chemical and biological reactions. In particular, many chemicalreactions, ecological systems and mechanisms of pattern formation are described math-ematically as systems of semilinear differential equations. Here we study stationaryequations with ∂�u/∂t = 0 of the form

∂�u

∂t= G(�u, α) = DΔ�u + �f(�u, α) = 0 in Ω× (0, T ). (1.22)

In this process the domain Ω ⊂ Rn represents the area where the chemical reactionoccurs or a biological/ecological system lives. �u : Rn → Rq represents various (inter-mediate) components in a chemical reaction or species of a biological system. It isa smooth vector function representing the reaction effect of the system. The vectorα ∈ Rp defines the control parameters in the system. The components of D ∈ Rq×q,a diagonal matrix, describe the diffusion rate of each species and �f : Rn × Rp → Rq

their reactions. Appropriate boundary conditions are combined with (1.22) for eachgiven situation. They can be any of Dirichlet, Neumann, or Robin type, or evenin some dynamical forms. The interaction of diffusion and reaction, measured byD and �f(�u, α), can yield rich and interesting phenomena. This is demonstratedin case studies in Bohmer [120]. An example used in many numerical studies isD = 1 = q, �u = u, f(u, α) = αf(u), see Theorem 2.58. �

Example 1.6. Three quasilinear and fully nonlinear geometric problemsThe solutions of these problems are functions u : Ω → R with prescribed boundaryconditions u = φ on ∂Ω.

The minimal surface equation, see Example 2.72 and Gilbarg and Trudinger [346],p. 339, has the quasilinear form

Gsu = Gu = (1 + |Du|2)Δu +n∑

i,j=1

∂iu∂ju∂i∂ju = 0 on Ω. (1.23)

1.4. Fundamental results 13

The Monge–Ampere equation is a fully nonlinear problem. For general n and forn = 2, see Example 2.78 and [346], pp. 441, 471, it has the form

0 = G(u) := detD2u− f(x, u,Du) = uxxuyy − u2xy − f(x, u,Du) for n = 2. (1.24)

These G in (1.24) and in (1.25) are uniformly elliptic in an u1 ∈ C2(Ω) if D2u1 isstrictly positive definite and symmetric in Ω.

Equation for a surface with prescribed Gauss curvature: now let K = K(x) at x ∈ Ωbe the prescribed Gauss curvature. Then (1.24) has the special form, see Example 2.78and [346], p. 442,

G(u) := detD2u−K(1 + |Du|2)(n+2)/2 = 0, K(x) > 0. (1.25)

�Example 1.7. Navier–Stokes equationsOne of the most stimulating equations for research in mathematics and physics isthe Navier–Stokes equation. It models, e.g. the flow of water around a moving vessel,and the air around the wing of an airplane. In this context (1.26) with small valuesof 0 < ν are interesting. Then boundary layers and turbulence develop, modeling theup-wind properties. Mathematically, it was the first extremely important system ofnonlinear differential equations. It required an update of the definition of ellipticor parabolic equations, and for small 0 < ν the boundaries between parabolic andhyperbolic problems are no longer well defined. Furthermore, by varying 0 < ν andother parameters, extremely interesting regimes of bifurcation and the correspondingdynamics and chaotic phenomena originate, cf. [120]. Still, for a lot of these problemssatisfactory answers are missing.

∂�u

∂t= G(�u, p) : =

⎛⎝−νΔ�u +n∑

i=1

ui∂i�u +∇ p

div �u

⎞⎠ =(�f0

)in Ω

∫Ω

pdx = 0, �u = 0 on Γ = ∂Ω. (1.26)

Here �u, p and �f denote velocity, pressure and forcing term of an incompressiblemedium, respectively,

�u = (u1, . . . , un)T, �f = (f1, . . . , fn)T : Ω ⊂ Rn → Rn, p : Ω → R, n ≤ 3,

and the divergence and gradient are div �u := ∂u1/∂x1 + · · ·+ ∂un/∂xn and grad p =∇ p = (∂p/∂x1, . . . , ∂p/∂xn)

T

. The equation∫Ωpdx = 0 guarantees a unique p. Here

we only study the stationary form with ∂�u/∂t ≡ 0. We will discuss its so-called centermanifold approximations as a case study in Bohmer [120]. �

1.4 Fundamental results

1.4.1 Linear operators and functionals in Banach spaces

Continuous or bounded and compact linear operators are among the most importantconcepts in functional analysis. We use them in both books very intensively.


Definition 1.8. Dense, compact sets, linear bounded, compact operators:

1. A subset U of a Banach space X (a linear space, closed with respect to its norm)is called dense, if every x ∈ X is the limit of a sequence {xn}n∈N ⊂ U . A Uis called precompact (or relatively compact) and compact, if every sequence{xn}n∈N ⊂ U contains a convergent subsequence {xni

}i,ni∈N and additionallysatisfies limi→∞ xni

∈ U , respectively.2. The linear operator L : X → Y is called bounded (or equivalently continuous)

and compact if the image of the unit sphere {x ∈ X : ‖x‖ ≤ 1} is bounded andprecompact, respectively. Hence, for a bounded L there exists C ∈ R+ such that

‖Lx‖Y ≤ C‖x‖X for all x ∈ X and ‖L‖ := ‖L‖Y←↩X := inf{C} (1.27)

is called (and is) a norm for L. The set of all these continuous linear operatorsis denoted as L(X ,Y). Furthermore, L = 0 ⇔ ∀x ∈ X : Lx = 0, and

for L ∈ L(X ,Y) we call N (L) := {x ∈ X : Lx = 0} and (1.28)

R(L) := {y ∈ Y : ∃ x ∈ X : y = Lx}

the kernel or null space and range of L, respectively.3. Analogously we define bounded linear and bilinear forms, l : X → R, x→ 〈l, x〉 :=

l(x) and b : X × Y → R, l = 0, b = 0, and their norms. We define the dual spaceas

X ′ := {l : X → R : l is a bounded linear form}. (1.29)

A subset U ⊂ X is precompact, if its closure, U , is compact. The compactness of asubset U is equivalent to the following property: Any open covering, U ⊂ ∪i∈IUi, Ui

open in X , I an index set, contains a finite subcovering. For finite-dimensional Banachspaces, X , compactness is equivalent to U being a closed, bounded set. For infinite-dimensional Banach spaces compactness only implies, but is not equivalent to thisproperty. A change from ‖ · ‖X , ‖ · ‖Y to equivalent norms does not change L(X ,Y),but ‖L‖. Now we have the following

Proposition 1.9. Simultaneously with Y, L(X ,Y) is a Banach space. The functionalin (1.27) is indeed a norm for linear operators, and for Y = R for linear funcionals.It satisfies

‖L‖ = sup‖x‖X=1

‖Lx‖Y , ‖L2 ◦ L1‖ ≤ ‖L2‖ ‖L1‖ ∀ L,L1 ∈ L(X ,Y), L2 ∈ L(Y,Z).

Analogously for bilinear forms b : X × Y → R we use ‖b(x, y)‖R ≤ Cb‖x‖X · ‖y‖Y tointroduce ‖b‖ := inf {Cb}. The kernel, N (L), is a closed subspace, the range, R(L),is usually only a subspace, see Theorem 1.19.

Example 1.10. Special linear operators

1. Linear operators for dimX <∞: Obviously, in this case all linear operators arebounded and have closed kernel and range.


2. Rod with axial load: For small ϕ we have sinϕ ≈ ϕ, hence, see Example 1.2 andits linear counterpart in (1.14), we study

L : C2[0, l] → C[0, l], Lϕ := ϕ + λϕ, (1.30)

with the norms

‖ϕ‖∞ := sup0≤s≤l

|ϕ(s)| and ‖ϕ‖2,∞ := ‖ϕ‖∞ + ‖ϕ‖∞ + ‖ϕ‖∞

in C[0, l] and C2[0, l], respectively. Then L is a bounded linear operator, inde-pendent of the boundary conditions, since

‖Lϕ‖ ≤ |λ|‖ϕ‖∞ + ‖ϕ‖∞ ≤ ‖ϕ‖2,∞ max{1, |λ|} with ‖L‖ ≤ max{1, |λ|}.

�

Above we have introduced the linear functionals on X yielding the space X ′. Ina totally analogous way it is possible to define the linear functionals on X ′, thusintroducing X ′′ := (X ′)′. Since 〈l, x〉 is linear in both arguments,

〈·, x〉 ∈ (X ′)′ =: X ′′ for every x ∈ X

it is possible to identify X with a subset of X ′′, sometimes even with X ′′ itself.

Definition 1.11. The mapping

〈·, ·〉 =: 〈·, ·〉X ′∗×X : X ′ ×X → R, {x′, x} �→ 〈x′, x〉 ∈ R (or ∈ C)

is called a pairing (for X ′ and X ). X is called reflexive if there exists a bijectionb : X ′′ → X such that for the pairing 〈·, ·〉′ : X ′′ ×X ′ → R,

〈x′′, l〉′X ′′×X ′ = 〈l, b(x′′)〉X ′×X = 〈l, x〉X ′×X =: 〈l, x〉 =: l(x)

∀ l ∈ X ′, x = b(x′′) implying ‖x′′‖ = ‖b(x′′)‖ = ‖x‖ ∀ x′′ ∈ X ′′. (1.31)

‖x′′‖ is indeed a norm, and we use the notation X ′′ = X for reflexive X . Sometimes,the analogous

‖x‖d := sup{|〈l, x〉| : ∀ l with ‖l‖X ′ = 1} (1.32)

is called the dual norm of x and we have ‖x‖d = ‖x‖.

The compactness of operators and their products is a powerful property. Weintroduce

Definition 1.12. Weak convergence: A sequence {xn}∞n=1 ⊂ X , a Banach space, isweakly convergent towards x0 ∈ X iff ∀l ∈ X ′ we have limn→∞ l(xn) = l(x0).

Theorem 1.13. Properties of compact operators:

1. For Banach spaces X ,Y,Z let L1 ∈ L(X ,Y), L2 ∈ L(Y,Z). If one of the L1, L2 iscompact, then L2 ◦ L1 is compact as well. Furthermore, L1 ∈ L(X ,Y) is compactif and only if (L1)d ∈ L(Y ′,X ′) is compact; for Ld see (1.33) below.


2. For compact L1, and equibounded Lh2 ∈ L(Y,Z), Lh

2 ◦ L1 is compact.1

3. A compact L1 ∈ L(X ,Y) maps every weakly convergent convergent sequence intoa strongly convergent sequence.

Among the many important theorems on linear operators we particularly needtheorems on extensions, open mappings and closed ranges and some consequences.We summarize the existence of extensions of bounded linear and bilinear forms andoperators, extending the Hahn–Banach extension theorem, and their inverses:

Theorem 1.14. Extension theorem for bounded forms and linear operators:

1. Let X0 ⊂ X ,Y0 ⊂ Y be linear subspaces of the Banach spaces X ,Y. Let l0 : X0 →R, L0 : X0 → Y and b0 : X0 × Y0 → R be bounded linear functionals, operatorsand bilinear forms with respect to the norms in X ,Y, respectively. Then l0, L0

and b0 can be extended as l, L, b to X and X × Y with the same norms. Theyare well defined bounded linear functionals, operators and bilinear forms on X ,Ywith ‖l‖X = ‖l0‖X , ‖L‖Y←X = ‖L0‖Y←X , ‖b‖X×Y = ‖b0‖X×Y , respectively.

2. If X0,Y0 are dense subsets of X ,Y these l, L, b are defined by the correspondinglimits: For x ∈ X , y ∈ Y, choose sequences

xn ∈ X0, x = limn→∞ xn lx := limxn→x l0xn

and define Lx := limxn→x L0xn

yn ∈ Y0, y = limn→∞ yn b(x, y) := limxn→x,yn→y b0(xn, yn).

Then the l, L, b are independent of the choice of the convergent sequences{xn}n∈N, {yn}n∈N. If L0 : X0 ⊂ X → Y is compact, then so is L0 : X → Y.

Theorem 1.15. Banach open mapping theorem: Let L ∈ L(X ,Y) be surjective:L(X ) = Y. Then the image, L(U), of every open subset, U ⊂ X , is open in Y.

Corollary 1.16. Bijective linear operators are invertible: L ∈ L(X ,Y) is bijective,often called an isomorphism, iff it is surjective and injective, that is Lx1 = Lx2 onlyfor x1 = x2. If L is bijective, then L−1 exists and L−1 ∈ L(Y,X ).

For the next theorem we need the concepts of dual operators, generalizing transposematrices, and generalized orthogonal complements. For L ∈ L(X ,Y) the dual operatorLd : Y ′ → X ′ (to distinguish later derivatives, L′, we use Ld for the dual operator), isuniquely defined for every l ∈ Y ′ by

Ld ∈ L(Y ′,X ′) with l ∈ Y ′ �→ Ldl ∈ X ′ :〈l, Lx〉Y′×Y = 〈Ldl, x〉X ′×X ∀ x ∈ X (1.33)

with the pairings in Y ′,Y and X ′,X , often denoted by the same 〈·, ·〉. Only ifthe specific combination is emphasized do we use the index of the pairing as in

1 For the later applications to discretizations with parameter h, we allow Lh2 , cf. Hackbusch [387],

Lemma 6.4.5, and Wloka [666], Satz 2.


(1.31), (1.33). The above Ld is unique, linear and continuous. Furthermore, forL,L1 ∈ L(X ,Y), L2 ∈ L(Y,Z), α ∈ C,

‖Ld‖ = ‖L‖, (L1 ◦ L2)d = Ld2 ◦ Ld

1, (αL)d = αLd;Ldd := (Ld)d = L; (1.34)

the last Ldd = L is valid only in reflexive Banach spaces X = X ′′,Y = Y ′′.We denote the dual space of a Hilbert space as X ∗ := X ′. We obtain the following

representation and natural embedding:

Theorem 1.17. Riesz representation theorem: Let X be a Hilbert space and l ∈ X ′ =X ∗. Then there exists a unique xl ∈ X such that

l(x) = (x, xl) ∀ x ∈ X and ‖xl‖X = ‖l‖X ′ . (1.35)

Corollary 1.18. There exists a unique (Riesz-)isomorphism JX ∈ L(X ,X ∗ = X ′),see Corollary 1.16, such that

JXxl = l, J−1X l = xl, ‖JX ‖ =

∥∥J−1X∥∥ = 1.

X ∗ = X ′ is again a Hilbert space with scalar product (x′, y′)X∗ := (J−1X x′, J−1

X y′)X and‖l‖X∗ = ‖xl‖X . X and (X ∗)∗ = X ′′ can be identified by xl = J−1

X l.

For two Hilbert spaces X ,Y with scalar products (·, ·)X , (·, ·)Y , let L ∈ L(X ,Y).Besides the dual operator, the adjoint operator L∗ : Y → X is defined by

L∗ ∈ L(Y,X ) and ∀ y ∈ Y : (y, Lx)Y = (L∗y, x)X ∀ x ∈ X . (1.36)

The dual and adjoint operators are related via the Riesz isomorphism as

L∗ = (JX )−1LdJY ∈ L(Y,X ). (1.37)

Only if X ,Y are identified with X ′,Y ′, do we find L∗ = Ld. In particular, we introduceself-adjoint or symmetric operators and orthogonal projectors as

L ∈ L(X ,X ) is called a self-adjoint or symmetric operator andan orthogonal projector, if, L = L∗ and L = L2, L = L∗, respectively.

Pairings may be used to generalize orthogonal complements: let subsets or subspacesU ⊂ X and V ′ ⊂ X ′ be given. Then

⊥U := {x′ ∈ X ′ : 〈x′, u〉X ′×X = 0 ∀ u ∈ U} ⊂ X ′, (1.38)

V ′⊥ := {x ∈ X : 〈v′, x〉X ′×X = 0 ∀ v′ ∈ V ′} ⊂ X ,

⊥(V ′) := (V ′⊥)′′ := {x′′ ∈ X ′′ : 〈x′′, v′〉′X ′′×X ′ = 0 ∀ v′ ∈ V ′} ⊂ X ′′,

with X ′′ = X for reflexive Banach spaces. The last space (V ′⊥)′′ is used only veryrarely. For continuous L (and 〈·, ·〉), the subspaces ⊥U ,V ′⊥ and N (L) are closed. ForR(L) we have


Theorem 1.19. Closed range theorem: For L ∈ L(X ,Y) the following fourstatements are equivalent (⇐⇒):

R(L) is closed ⇐⇒ R(Ld) is closed⇐⇒R(L) = N (Ld)⊥ ⇐⇒ R(Ld) =⊥N (L).

Definition 1.20. For L ∈ L (X ,Y), let

1. N (L) = {x ∈ X : Lx = 0}, dimN (L) <∞, and (1.39)

2. R(L) be closed, hence, a subspace N ⊂ Y exists, with

Y = N ⊕R(L) and let codim R(L) := dimN <∞.

Then L is called a Fredholm operator and the number i(L) := dimN (L)− codim R(L)its Fredholm index.

Definition 1.21. Complexification of Banach spaces and operators: We extend a realX to a complex Banach space Xc cf. R to C, as, cf. [676]

Xc := {x + iy : x, y ∈ X}, i2 = −1, (a + ib)(x + iy) := (ax− by) + i(bx + ay),

(x1 + iy1)(x2 + iy2) = (x1x2 − y1y2) + i(y1x2 + x2y1), ∀a, b ∈ R, x1, x2, y1, y2 ∈ X ,

and ‖x + iy‖Xc:= max

0≤φ≤2π‖x(cosφ) + y(sinφ)‖. (1.40)

Similarly we extend an A ∈ L(X ,Y) to

Ac(x + iy) := Ax + iAy,∀x, y ∈ X , thus Ac ∈ L(Xc,Yc). (1.41)

Theorem 1.22. Complexified Banach space L(X ,Y) and linear operator: AssumeBanach spaces X , Y and A ∈ L(X ,Y). Define Xc, ‖ · ‖Xc

, Ac as in (1.40), (1.41). ThenXc, Yc are Banach spaces and Ac ∈ L(Xc,Yc) is linear and bounded with ‖A‖Y←↩X ≤‖Ac‖Yc←↩Xc

≤ 2‖A‖Y←↩X .

1.4.2 Inequalities and Lp(Ω) spaces

We start with some inequalities for finite sums. The discrete Chebyshev inequalityholds for ordered sums. Then we obtain, e.g. [396], p 99, for the two sets of f1 ≥ f2 ≥. . . ≥ fN ∈ R and g1 ≥ g2 ≥ . . . ≥ gN ∈ R,

N∑j=1

fj/NN∑

j=1

gj/N ≤N∑

j=1

fjgj/N. (1.42)

This implies for arbitrary f1, f2, . . . , fN ∈ R that

N∑j=1

fj ≤

∣∣∣∣∣∣N∑

j=1

fj

∣∣∣∣∣∣ ≤√√√√N

N∑j=1

f2j . (1.43)


Often we need the discrete Holder inequalities. We always choose a pair of 1 ≤ p, q ≤∞ with 1/p + 1/q = 1. Then we find, with

N∑j=1

|gj |q|1/q :=N

maxj=1

|gj | for q =∞ [396,397]

for 1 < p, q <∞, and for 1 = p < q =∞ as in (1.46),∣∣∣∣∣∣N∑

j=1

fjgj

∣∣∣∣∣∣ ≤⎛⎝ N∑

j=1

|fj |p⎞⎠1/p⎛⎝ N∑

j=1

|gj |q⎞⎠1/q

for 1 ≤ p, q ≤ ∞, 1/p + 1/q = 1.

(1.44)

For cuboids Q :=∏n

j=1[aj , bj ] ∈ Rn, aj ≤ bj we have μ(Q) :=∏n

j=1(bj − aj). For theset Ω, we choose two sets Ai and Ao, the union of countably many inner and outercuboids Qi

l and Qok with disjoint interiors. They have to satisfy Ao := ∪Qo

k ⊃ Ω andAi := ∪Qi

l ⊂ Ω. For μ(Ao) :=∑

k μ(Qok) and μ(Ai) :=

∑l μ(Qi

l), let infAo{μ(Ao)} =supAi{μ(Ai)}. Then Ω is called μ-measurable, and μ(Ω) := infAo{μ(Ao)}. For Ωmeasurable, a function, f : Ω → R ∪∞∪−∞, is called measurable, if for every α ∈ Rthe set {x ∈ Ω : f(x) ≥ α} is measurable.

For any (measurable) Ω ⊂ Rn we define and estimate

‖f‖Lp(Ω) :=

⎛⎝∫Ω

|f(x)|pdx

⎞⎠1/p

, 1 ≤ p <∞, and ‖f‖L∞(Ω) := ess supx∈Ω

|f(x)|

Lp(Ω) := {f : Ω → R : ‖f‖Lp(Ω) <∞},Ω ⊂ Rn, (1.45)

‖fg‖L1(Ω) =∫Ω

|f(x)g(x)|dx ≤

⎛⎝∫Ω

|f(x)|pdx

⎞⎠1/p⎛⎝∫Ω

|g(x)|qdx

⎞⎠1/q

= ‖f‖Lp(Ω)‖g‖Lq(Ω) ∀ f ∈ Lp(Ω), g ∈ Lq(Ω)

∀ 1 ≤ p, q ≤ ∞ with 1/p + 1/q = 1, the Holder inequalities.

The usually missing case 1 = p, q = ∞ in (1.44), (1.45) is directly obtainable: f ∈L1(Ω), g ∈ L∞(Ω) imply fg ∈ L1(Ω), and similarly for (1.44),

‖fg‖L1(Ω) =∫Ω

|f(x)g(x)|dx (1.46)

≤ ‖g‖L∞(Ω)

⎛⎝∫Ω

|f(x)|dx

⎞⎠= ‖f‖L1(Ω)‖g‖L∞(Ω). (1.47)

In particular we obtain for the case p = q = 2 the Cauchy–Schwarz inequality

|(u, v)L2(Ω)| ≤ ‖u‖L2(Ω) · ‖v‖L2(Ω). (1.48)


For later discussions of nonlinearities we need the extension∥∥∥∥∥∥d∏

j=1

fj

∥∥∥∥∥∥L1(Ω)=∫Ω

∣∣∣∣∣∣d∏

j=1

fj

∣∣∣∣∣∣ dx ≤d∏

j=1

⎛⎝∫Ω

|fj(x)|pjdx

⎞⎠1/pj

(1.49)

=d∏

j=1

‖fj‖Lpj (Ω) ∀ fj ∈ Lpj (Ω)∀ 1 ≤ pj ≤ ∞ withd∑

j=1

1/pj = 1.

1.4.3 Holder and Sobolev spaces and more

For a function u ∈ C∞0 (Ω,R) the following partial derivatives are well defined, with

α := (α1, . . . , αn) ∈ Nn0 , |α| := α1 + · · ·+ αn, α ! := (α1)! · · · (αn)! (1.50)

∂iu :=∂u

∂xi, ∂u := (∂1u, . . . , ∂nu), ∂αu := ∂α1

1 . . . ∂αnn u :=

∂|α|

∂xα11 . . . ∂xαn

nu.

The Ck, Ck,1 and Ck+μ indicate the usual classical k-times continuously differentiablefunctions u : D → R, with appropriate D ⊂ Rn. For u ∈ Ck,1 [and u ∈ Ck+μ], respec-tively, all derivatives ∂αu for |α| ≤ k are Lipschitz [and Holder] continuous, hence|∂αu(x)− ∂αu(y)| ≤ C|x− y|μ for all x, y ∈ D for μ = 1 [and 0 < μ < 1].

For Sobolev spaces we have to generalize this concept:

Definition 1.23. For a function u ∈ Lp(Ω), 1 ≤ p ≤ ∞, a function v := ∂αu ∈ Lp(Ω)is called a weak α derivative of u, iff

(w, v)Lp(Ω) = (−1)|α|(wα, u)Lp(Ω) for all w ∈ C∞0 (Ω). (1.51)

The Sobolev space W k,p(Ω), k ∈ N0, 1 ≤ p ≤ ∞, is the set of functions with weakderivatives up to order k:

W k,p(Ω) := {u ∈ Lp(Ω) : ∀α with |α| ≤ k : ∂αu ∈ Lp(Ω)}. (1.52)

For u ∈ Lp(Ω) let the weak derivative ∂αu ∈ Lp(Ω), and in Ω′ ⊂ Ω let the classicalderivative ∂αu ∈ Lp(Ω′) exist. Then the weak and the classical derivative coincidealmost everywhere (a.e.) in Ω′.

Theorem 1.24. Sobolev spaces are Banach spaces: The Sobolev spaces W k,p(Ω),k ∈ N0, 1 ≤ p ≤ ∞, are Banach spaces with respect to the norm

‖u‖W k,p(Ω) := ‖u‖W k,p := ‖u‖k,p :=

⎛⎝∑|α|≤k

‖∂αu‖pLp(Ω)

⎞⎠1/p

. (1.53)

For the special case p = 2 we have the Hilbert space Hk(Ω) with

(u, v)k := (u, v)Hk(Ω) :=∑|α|≤k

(∂αu, ∂αv)L2(Ω), ‖u‖Hk(Ω) := (u, v)1/2

Hk(Ω); (1.54)


sometimes we use ‖u‖k := ‖u‖Hk(Ω). The spaces W k,p0 (Ω) and Hk

0 (Ω), defined asclosure of C∞

0 (Ω) with respect to the corresponding norm, e.g. ‖ · ‖Hk(Ω),

W k,p0 (Ω) := C∞

0 (Ω)‖·‖k,p

,Hk0 (Ω) := C∞

0 (Ω)‖·‖

Hk(Ω) , (1.55)

are again Banach and Hilbert spaces with the same norm (1.53) and (1.54), respec-tively. Sometimes, we need the following seminorms

|u|W k,p(Ω) :=

⎛⎝∑|α|=k

‖∂αu‖pLp(Ω)

⎞⎠1/p

, |u|Hk(Ω) := |u|W k,p=2(Ω). (1.56)

If Ω is bounded, then ‖ · ‖W k,p0 (Ω) := | · |W k,p(Ω)

∣∣W k,p

0 (Ω)and ‖ · ‖W k,p(Ω) are equivalent

norms on W k,p0 (Ω); similarly, for p = 2, on Hk

0 (Ω). In particular, we obtain withPoincare’s constant CP (Ω)

|u|W k,p(Ω) ≤ ‖u‖W k,p(Ω) ≤ CP (Ω)|u|W k,p(Ω)∀u ∈W k,p0 (Ω). (1.57)

Proposition 1.25. cf. Zeidler, [678], p. 1032: For Ω ∈ C0,1 the standard norm‖u‖W k,p(Ω) for the Sobolev spaces W k,p(Ω) with k ∈ N is equivalent to the norms⎛⎝∫

Ω

∑|α|=k

|∂αu|pdx +∣∣∣∣∫

Ω

udx

∣∣∣∣p⎞⎠1/p

(1.58)

⎛⎝∫Ω

∑|α|=k

|∂αu|pdx +∣∣∣∣∫

∂Ω

u ds

∣∣∣∣p⎞⎠1/p

⎛⎝∫Ω

∑|α|=k

|∂αu|pdx +∫

∂Ω

|u|p ds

⎞⎠1/p

,

where ds indicates the surface element on ∂Ω. For n = 1 and Ω = (a, b) we define∫∂Ω

g ds := g(b)− g(a).

For X ⊂ Y we denote the mapping

�I : X → Y, x �→ �I x := x ∈ Y for X ⊂ Y

as an inclusion or embedding, often abbreviated as X ↪→ Y. If �I is a continuous andcompact operator, then X is called continuously and compactly embedded into Y,respectively, so ‖x‖Y ≤ C‖x‖X∀x ∈ X . For the compact embedding, additionally eachbounded sequence {xn} in X has a subsequence {xnk

} converging in Y, cf. [677],pp. 1026 ff., [676], p. 248, [387], Example 6.4.7:


Theorem 1.26. Embedding theorem for Sobolev spaces: Let Ω be a boundedLipschitz domain, j ∈ N0, k ∈ N, 0 ≤ j < k, 1 ≤ p, q <∞, 0 < α < 1 and defined := 1/p− (k − j)/n. Then the following embedding results are valid.2

1. W k,p(Ω) ↪→W j,q(Ω) and W k,p0 (Ω) ↪→W j,q

0 (Ω) are continuous for d ≤ 1/q, j ≤ k,

and compact for d < 1/q, j < k. These results remain valid for W k,p0 (Ω) and

arbitrary open Ω.2. In particular, W k,p(Ω) ↪→W k,q(Ω) is continuous for 1 ≤ q ≤ p ≤ ∞, 0 ≤ k.3. W k,p(Ω) ↪→ Cj(Ω) are compact for d < 0, i.e. k − j > n/p, and W k,p(Ω) ↪→ Cj,α

(Ω) ↪→ Cj(Ω) for d + α/n < 0 or d < 0. This inclusion might require changingu ∈W k,p(Ω) on a subset of Ω of n-dimensional measure 0.

4. If we replace the above bounded Ω by Rn we find: W k,p(Rn) ↪→ Cj(Rn) ford < 0, i.e. k − j > n/p, and W k,p(Rn) ↪→W j,q(Rn) for d ≤ 1/q and j < k arecontinuous.

5. Cα(Ω) ↪→ Lp(Ω) and C0,1(Ω) ↪→ Lp(Ω) are continuous for 0 ≤ α ≤ 1 and 1 ≤p ≤ ∞.

6. For 0 < α ≤ 1, j < k, the embeddings Ck,α(Ω) ↪→ Ck(Ω) ↪→ Cj(Ω) are compact.

The previous compact embedding result implies, e.g.

Proposition 1.27. Under the conditions of Theorem 1.26, for every ε > 0 a Cε > 0exists such that

‖x‖W j,q(Ω) ≤ ε‖x‖W k,p(Ω) + Cε‖x‖Ls(Ω) for 0 ≤ j < k,

1 ≤ p, q, s <∞, 1/p− (k − j)/n < 1/q, 1/q − k/n ≤ 1/s.

In particular, we find for p = q = s and 0 < j < k:

‖x‖W j.p(Ω) ≤ ε‖x‖W k,p(Ω) + Cε‖x‖Lp(Ω).

Theorem 1.28. Sobolev–Stein extension theorem [141], p. 31: Let Ω ⊂ Rn have aLipschitz boundary, let k ∈ N0 and p ∈ R with 1 ≤ p ≤ ∞. Then there exists a boundedoperator E : W k,p(Ω) → W k,p(Rn) and a constant, C, independent of v, such that

Ev|Ω = v and ‖Ev‖W k,p(Rn) ≤ C‖v‖W k,p(Ω) ∀v ∈W k,p(Ω). (1.59)

Further bounded extension operators for an open set with Ω ⊂ Ω′ of the formED : Ck,α

D (Ω) → Ck,α0 (Ω′), with 0 and D indicating trivial and nontrivial Dirichlet

boundary conditions are discussed in [344], p. 136, [677], pp. 305–306. In Chapter 5,for example, we need Theorem 1.28.

Theorem 1.29. Extension of a boundary function, [344], p 137: Let Ω ⊂ Rn be aCk,α domain, let k ∈ N and let Ω′ be an open set with Ω ⊂ Ω′. Let v ∈ Ck,α(∂Ω) Thenthere exists a function w ∈ Ck,α

0 (Ω′), such that v = w on ∂Ω.

Trace operators for Sobolev spaces of fractional order, s ∈ R+, are necessary indifferent contexts, in particular for nonconforming, adaptive FE and discontinuousGalerkin methods. We formulate the following results in Hs(Ω), and W s,p(Ω) form,

2 Note that Lq(Ω) = W 0,q(Ω) and d is independent of n for k = j.


cf. Zeidler [678], pp. 1031 ff., Hackbusch [387], Subsections 6.2.4 and 6.2.5. Theyrepresent an integral variant of the Holder spaces Ck,μ(Ω). We define for these

Hs(Ω),W s,p(Ω) for ∂Ω ∈ C0,1, s = k + μ, k = 0, 1, . . . , 0 < μ < 1, 1 < p <∞, (1.60)

the corresponding scalar products and Sobolev–Slobodeckij norms as

(u, v)Hs(Ω) := (u, v)Hk(Ω) +∑|α|≤k

f(∂αu, ∂αv), ‖u‖Hs(Ω) :=((u, u)Hs(Ω)

)1/2, (1.61)

with

f(u, v) :=∫

Ω×Ω

[u(x)− u(y)][v(x)− v(y)]|x− y|n+2μ

dxdy

and for 1 ≤ p ≤ ∞, fp(u) :=∫

Ω×Ω

|u(x)− u(y)|p|x− y|n+pμ

dxdy,

‖u‖W s,p(Ω) := (‖u‖pW k,p(Ω)

+∑|α|≤k

fp(∂αu))1/p.

For Hs(∂Ω),W s,p(∂Ω) replace in (1.61) the Ω by ∂Ω and dxdy by dsds′, and n byn− 1.

Theorem 1.30. Sobolev spaces of fractional order: The Sobolev spaces Hs(Ω), Hs

(∂Ω), and W s,p(Ω),W s,p(∂Ω), s=k + μ, k=0, 1, . . . , 0 < μ < 1, 1 < p <∞, in (1.60)are Hilbert and reflexive Banach spaces, respectively, with respect to the scalar productand norms in (1.61).

For considering the boundary and functions defined there, we introduce the conceptof an atlas. We achieve that by introducing a parametrization, hence, local bijectionsbetween neighborhoods of boundary points and the unit sphere along the boundary.This allows formulating the necessary differentiability conditions for ∂Ω and a para-metrization for the boundary.

Definition 1.31. Let k ∈ N0 [and t with 0 < t ≤ ∞, t = k + λ, k ∈ N0, 0 ≤ λ < 1] begiven. [Here and below we indicate by [...] this case 0 < t ≤ ∞,. . . and its conse-quences.] For each x ∈ ∂Ω we require a neighborhood U = U(x) ⊂ Rn and a bijectionφ : U → B1(0) := Bn

1 (0) = {x ∈ Rn : |x| ≤ 1}, the unit ball in Rn, such that

1. φ ∈ Ck,1(U), φ−1 ∈ Ck,1( B1(0) )[and φ ∈ Ct ¯(U), φ−1 ∈ Ct( B1(0) )],

2. φ(U ∩ ∂Ω) = {ξ ∈ B1(0) : ξn = 0},3. φ(U ∩ Ω) = {ξ ∈ B1(0) : ξn > 0},4. φ(U ∩ (Rn\Ω)) = {ξ ∈ B1(0) : ξn < 0}.

(1.62)

Then we denote Ω as (k, 1) [and t] smooth, indicated by Ω ∈ Ck,1 [and ∈ Ct]. Ω ∈C(0,1) is called a Lipschitz domain. These (U , φ) are denoted as an atlas.


U 2

U 0U 3

U 1

Figure 1.5 Atlas for a circle.

Proposition 1.32. Let Ω ∈ Ck,1, k ∈ N0 [and Ω ∈ Ct, 0 < t ≤ ∞] be open andbounded. Then we need only finitely many neighborhoods, U i, bijections, φi, 0 ≤ i ≤N, and parametrizations of the boundary, αi, 1 ≤ i ≤ N , in (1.62), such that cf.Figures 1.5 and 1.6,

1. U i is open, bounded, 0 ≤ i ≤ N,N∪

i=0U i ⊃ Ω, U0 ⊂ Ω,

2. φi : U i → Bn1 (0) is bijective, φi ∈ Ck,1

(U i)

[and ∈ Ct(U i)], 0 ≤ i ≤ N,

3. Ui := U i ∩ ∂Ω, αi := φi|Ui, 1 ≤ i ≤ N,

N∪i=1Ui = ∂Ω, (1.63)

4. αi : Ui → αi(Ui) = Bn−11 (0) ⊂ Rn−1 is bijective, 1 ≤ i ≤ N,

5. αi ◦ α−1j ∈ Ck,1

(αj(Ui ∩ Uj)

)[and αi ◦ α−1

j ∈ Ct(αj(Ui ∩ Uj)

)].

U 0

U 3

U 4

U 1

U 2

Figure 1.6 Atlas for a square.


These U i, φi, 0 ≤ i ≤ N , define an atlas, hence satisfy (1.62). The system (Ui, αi), 1 ≤i ≤ N , is denoted as a parametrization or coordinate system for ∂Ω. In particular, Ωis sometimes called a smooth manifold, if in (1.63) the φi, αj and their inverses are∈ C∞.

The above boundary operators have to be applied to functions u ∈Wmp (Ω) and will

yield functions u|∂Ω ∈Wm−1/pp (∂Ω) . Using the above φi, αi, it is possible to reduce

the discussion to functions defined on the U i,Ui or the Bn1 (0),Bn−1

1 (0) or even toRn,Rn−1. We use the following partition of unity:

Proposition 1.33. Let U i, 0 ≤ i ≤ N, satisfy (1.63). Then functions σi ∈ C∞0 (Rn),

0 ≤ i ≤ N, exist such that

carr (σi) := closure {x ∈ Rn : σi(x) �= 0}

= supp (σi) ⊂ U i,

N∑i=0

σ2i (x) = 1 for all x ∈ Ω. (1.64)

This representation is called a partition of unity.

Since, by (1.64),∑N

i=0 σ2i (x)u(x) =

∑Ni=0 u(x)σ2

i (x) = u(x), we may consider eachof the σiu separately. In particular, we will use the above bijections αi : Ui → Bn−1

1 (0).Then we study the local parametrizations (σiu) ◦ (αi)−1 : Bn−1

1 (0) → R and sum overall appropriate terms. With carr (σi) ⊂ U i, we extend (σiu) ◦ (αi)−1 to (σiu) ◦ (αi)−1 :Rn−1 → R.

Definition 1.34. Let Ω be bounded and choose for Ω ∈ Ck,1 [and Ω ∈ Ct], ans with s ≤ k + 1 [and s < t for t /∈ N, t > 1 or s ≤ t ∈ N]. Furthermore, let theU i, φi, Ui, αi and σi, 0 ≤ i ≤ N, satisfy (1.63), (1.64). Then the Sobolev spaceW s,p(∂Ω) is defined as

W s,p(∂Ω) :={u : ∂Ω → R : (σiu) ◦ α−1

i ∈W sp,0

(Bn−1

1 (0)), 1 ≤ i ≤ N

}.

By U0 ⊂ Ω we may restrict the discussion essentially to 1 ≤ i ≤ N.

Theorem 1.35. Sobolev spaces W s,p(∂Ω) are Banach spaces: For bounded Ω ∈ Ck,1

[and Ct] and s as in Definition 1.34, let the U i, φi, Ui, αi and σi, 0 ≤ i ≤ N, satisfy(1.63), (1.64). Then the above Sobolev space W s,p(∂Ω) is a Banach and a Hilbert space,for 1 ≤ p <∞, p �= 2 and p = 2, respectively, with the obvious modification of thenorms in (1.61), and replacing Ω by ∂Ω. The W s,p(∂Ω) are reflexive for 1 < p <∞.For p = 2 the scalar product is

(u, v)Hs(∂Ω) :=N∑

i=1

((σiu) ◦ α−1

i , (σiv) ◦ α−1i

)Hs(Rn−1)

. (1.65)

A different choice of atlas and partition of unity yields the same Sobolev spaces.More precisely, replace the Ui, αi, σi chosen above by Ui, αi, σi, 0 ≤ i ≤ N , againsatisfying (1.63), (1.64). This defines another Ck,1 [and Ct] coordinate system andanother partition of unity. The norms, as in Therem 1.24 and the modified (1.61),


are equivalent and the two Sobolev spaces W s,p(∂Ω) and W s,p(∂Ω) contain the sameelements, so can be identified.

Now we are able to define the restriction operators for functions defined on Ω to theboundary ∂Ω. Let Ω ⊂ Ω0. Then a restriction to the boundary is uniquely determinedfor sufficiently smooth functions. So

γ0 : C∞0 (Ω0) → Cτ (∂Ω), (γ0u)(x) := u(x)∀x ∈ ∂Ω, (1.66)

for appropriate τ ∈ R, is well defined.

Definition 1.36. Trace or restriction operator: Let an extension γ of γ0 in (1.66)exist such that γp = γ ∈ L(W s,p(Ω),W t,p(∂Ω)) for appropriate s, t, with

γ|C∞0 (Ω) = γ0, and denote it as u|∂Ω := γu.

Then this γ is called a trace operator or a restriction operator.

The following theorem shows that these trace operators exist, if Ω ∈ C0,1 and thesmoothness in W s,p(Ω) is better than in W t,p(∂Ω), cf. [677], p. 1030.

Theorem 1.37. Properties of the trace operator: Let Ω be open and bounded, Ω ∈C0,1, n ≥ 2, 1 ≤ p <∞, and γ as in Definition 1.36. Then

1. For p <∞ the γ = γp : W 1,p(Ω) →W 1−1/p,p(∂Ω) is linear, surjective, and con-tinuous: ‖γpu‖W 1−1/p(∂Ω) ≤ C‖u‖W 1,p(Ω) ∀u ∈W 1,p(Ω).

2. u ∈W 1,p0 (Ω) is equivalent to u ∈W 1,p(Ω) and γ u = 0 on ∂Ω.

3. For Ω ∈ Cm−1,1 the u ∈Wm,p0 (Ω) is equivalent to u ∈Wm,p(Ω) and Dα u ∈

W 1,p0 (Ω) or γ Dα u = 0 ∀ |α| ≤ m− 1.

4. For p <∞ there exists a linear continuous extension operator

Ep : W 1−1/p,p(∂Ω) →W 1,p(Ω)

such that the following diagram is commutative with the identity I : W 1,p(Ω) →W 1,p(Ω)

W 1,p(Ω)γp−→ W 1−1/p,p(∂Ω)

I ↖ ↙ Ep

W 1,p(Ω)

Theorem 1.38. Trace theorem: Let Ω ⊂ Rn have a Lipschitz boundary, Ω ∈ C0,1,and let 1 ≤ p ≤ ∞. Then there exist constants, C, such that

‖v‖Lp(∂Ω) ≤ C‖v‖1−1/pLp(Ω) · ‖v‖

1/pW 1,p(Ω) ∀v ∈W 1,p(Ω) and (1.67)

‖v‖Lp(∂Ω) ≤ C‖v‖1/2L2(Ω)‖v‖

1/2H1(Ω) ≤ C‖v‖H1(Ω) for p = 2.

The ‖v‖W k,p(Ω) satisfy a trace estimate as well:

‖v‖W k,p(∂Ω) ≤ C‖v‖1−1/p

W k,p(Ω)· ‖v‖1/p

W k+1p (Ω)

∀v ∈W k+1p (Ω). (1.68)


Theorem 1.39. Trace operators for Sobolev spaces of fractional order: For Ω ∈ Ck,1,choose s such that 1/2 < s = k + 1 ∈ N [and for Ω ∈ Ct choose s such that 1/2 <s ≤ t ∈ N or 1/2 < s < t �∈ N] and s > |α|+ 1/2, |α| ≥ 0. We formulate separately for|α| = 0 and |α| > 0. Then the following trace and extension statements are correct: thetrace operators γ∂α : Hs(Ω) → Hs−|α|−1/2(∂Ω) exist.

1. γ ∈ L(Hs(Ω),Hs−1/2(∂Ω)), so u ∈ Hs(Ω) implies γu = u|∂Ω ∈ Hs−1/2(∂Ω).2. γ∂α ∈ L(Hs(Ω),Hs−|α|−1/2(∂Ω)), hence, for u ∈ Hs(Ω) the restriction implies

γ∂αu = ∂αu|∂Ω ∈ Hs−|α|−1/2(∂Ω) and ‖∂αu|∂Ω‖Hs−|α|−1/2(∂Ω) ≤ C‖u‖Hs(Ω). Inparticular the normal derivatives ∂lu/∂nl exist, e.g. for t > s > l + 1/2 and Ω ∈Ct, and u ∈ Hs

0(Ω) with s > l + 1/2 implies γ∂αu = ∂αu|∂Ω = 0, if 0 ≤ |α| ≤ l.3. For each w ∈ Hs−1/2(∂Ω) there exists u ∈ Hs(Ω) such that w = γu and‖u‖Hs(Ω) ≤ Cs‖w‖Hs−1/2(∂Ω).

For the Gauss integral theorem, we use the notation F = (F1, · · · , Fn) ∈W 1,1(Ω)

divF =n∑

i=1

∂Fi/∂xi.

Theorem 1.40. Gauss integral theorem: Let Ω ⊂ Rn be a C0,1 compact domain withouter normal vector ν = (ν1, · · · , νn) along ∂Ω, and F ∈W 1,1(Ω,Rn). Then∫

Ω

divFdx =∫

∂Ω

〈F, ν〉nds and∫

Ω

∂Fi/∂xidx =∫

∂Ω

Fiνids, i = 1, . . . , n. (1.69)

1.4.4 Derivatives in Banach spaces

We introduce two types of derivatives in Banach spaces. We motivate it by thenonlinear operator, cf. (1.13), for the loaded rod.

G : C2[0, l] → C[0, l], G(ϕ) = d2ϕ/ds2 + λ sinϕ. (1.70)

(1) Choosing functions ϕ,ψ ∈ C2[0, l] with small

‖ψ‖2,∞ := ‖ψ‖∞ + ‖ψ‖∞ + ‖ψ‖∞,

we obtain the relation, with fixed parameter λ,

G(ϕ + ψ)−G(ϕ) = d2ψ/ds2 + λ(sin(ϕ + ψ)− sinϕ)

= d2ψ/ds2 + (λ cosϕ)ψ + o(ψ)

=: L(ϕ)ψ + o(ψ)

=: G′(ϕ)ψ + o(ψ).

The notation L(ϕ) = G′(ϕ), called the Frechet derivative, is formalized in (1.71) below:

G′(ϕ) : C2[0, l] → C[0, l], G′(ϕ)ψ := d2ψ/ds2 + (λ cosϕ)ψ := ψ + (λ cosϕ)ψ.


This operator G′(ϕ) is obviously linear and it is bounded, see Example 1.2,

‖G′(ϕ)ψ‖ ≤ |λ cosϕ|‖ψ‖∞ + ‖ψ‖∞ ≤ max{1, |λ|}‖ψ‖2,∞ or ‖G′(ϕ)‖ ≤ max{1, |λ|}.For appropriate ϕ and ψ we even have equality.

(2) To determine the directional derivative, we choose fixed ϕ,ψ ∈ C2[0, l], τ ∈ [0, l].We introduce a function of a real variable t ∈ [0, l], as

g : R → R, g(t) := G(ϕ(τ) + tψ(τ)) = (d2(ϕ + tψ)/ds2)(τ) + λ sin(ϕ(τ) + tψ(τ)).

We can use the classical differentiation rules, e.g. the chain rule for a function of areal variable to obtain

dg/dt∣∣t=0

= (d2ψ/ds2)(τ) + (λ cosϕ(τ))ψ(τ).

By considering this relation for variable τ ∈ [0, l], and fixed functions ϕ,ψ, we couldequally well have defined the corresponding

g : R → C[0, l], g(t) := d2(ϕ + tψ)/ds2 + λ sin(ϕ + tψ).

Similarly to the above, we can linearize

g(t)− g(0) = td2ψ/ds2 + λ(sin(ϕ + tψ)− sinϕ)

= td2ψ/ds2 + t(λ cosϕ)ψ + t o(ψ)

= t(L(ϕ)ψ) + o(tψ)

= t G′(ϕ)ψ + o(tψ).

This L(ϕ)ψ = G′(ϕ)ψ will be called the Gateaux derivative in (1.72). Obviously,by fixing the functions and variables, e.g. the ϕ,ψ, τ above, the computation ofthese directional derivatives only uses classical calculus. The linearization is morecomplicated, but it has advantages for the bifurcation in [120]. We need a simplecriterion which guarantees a directional derivative to be a Frechet derivative, seeDefinition 1.41 and Theorem 1.42.

Definition 1.41. Frechet (F) and Gateaux (G) derivative:

1. Let H : D ⊂ X → Y, x0 ∈ U := U(x0) ⊂ D and U be open. Then F is calledFrechet (F) differentiable in x0 if there is an LF ∈ L(X ,Y), depending on x0

but not on h, such that

H(x0 + h)−H(x0) = LFh + o(h) for h→ 0, x0 + h ∈ U . (1.71)

Then H ′F (x0) := H ′(x0) := LF is called the Frechet derivative of H in x0.

2. H is called Gateaux (G) differentiable in x0 ∈ U , if for every h (such that ‖h‖ =1) there is a H ′

G(x0, h) ∈ Y such that

H(x0 + th)−H(x0) = tH ′G(x0, h) + o(t) for t→ 0, x0 + th ∈ U ; (1.72)

this o(t) = oh(t) depends on h. If (1.72) should be valid only for a fixed h (with‖h‖ = 1), then H is called Gateaux differentiable in x0 in the direction of h.


Furthermore, the H ′G(x0, h) in (1.72) is called the Gateaux (G) derivative of H

in x0 in the direction of h.

These Frechet and Gateaux derivatives generalize the total and directional deriv-atives for F : D ⊂ Rn → Rm, respectively. We will mainly apply Frechet derivatives.However, by G(t) := H(x0 + th) we have reduced H : D ⊂ X → Y to a G : I ⊂ R → Ymapping for a real variable t. Thus, all the rules, e.g. sum, product and chain rule,obtained by limits of divided differences, remain correct for Gateaux derivatives aswell, if they are interpreted as limits. This requires that x0 and h are kept fixedduring this differentiation. This allows an easier and more straightforward approach.Under appropriate conditions, which essentially cover all our applications, Gateauxderivatives yield a simple tool to determine Frechet derivatives as well.

F and G derivatives are uniquely determined. Sometimes, “one-sided” derivatives areappropriate, in particular, if U(x0) is not open and, hence, not all x0 + h and x0 + thare in U(x0), for sufficiently small h and t. An operator which is F differentiable inx0 is continuous in x0. That is not true, if it is only G differentiable. This situation iswell known for real valued functions of two variables.

It is obvious that both derivatives are generated by linearizing the difference ofoperator values: In (1.71) we have linearized with respect to h ∈ X , in (1.72) withrespect to t ∈ R. In particular we have as a consequence of (1.72)

H ′G(x0, h) = lim

t→0

H(x0 + th)−H(x0)t

=d

dtH(x0 + th)|t=0.

For some examples we have the situation that LG ∈ L(X ,Y) exists such that for all‖h‖ = 1,H ′

G(x0, h) = LGh. Sometimes this is even part of the definition of a Gateauxderivative, see [677]. The o(t) term in (1.72) does depends on h. We have

Theorem 1.42. Frechet (F) and Gateaux (G) derivatives and their relations:

1. The Frechet and Gateaux derivatives are uniquely determined.2. If H is Frechet differentiable in x0 ∈ U(x0) ⊂ D, it is continuous in x0, Gateaux

differentiable and H ′G(x0, h) = LFh.

3. Let H be Gateaux differentiable in x0, such that

LG = LG(x0) = H ′G(x0) ∈ L(X ,Y) exists, such that

H(x0 + th)−H(x0) = tH ′G(x0)h + oh(t) and (1.73)

oh(t) = o(t)C(h) with ‖C(h)‖ ≤ C for ‖h‖ = 1.

Then H is Frechet differentiable in x0 and for all h with ‖h‖ = 1 we haveH ′

G(x0, h) = H ′G(x0)h = LGh = LFh, hence LF = H ′

G(x0).4. Let H be Gateaux differentiable in a neighborhood U(x0) of x0 and such that

H(x+ t h)−H(x) = t H ′G(x)h + oh(t) s.t. (1.74)

LG(x) = H ′G(x) ∈ L(X ,Y) is continuous w.r.t x for all x ∈ U(x0).

Then H is Frechet differentiable in U(x0) and LF (x) = LG(x).


Theorem 1.43. Mean value theorem: Let F : D ⊂ X → Y and a, b ∈ D given withab = {a + t(b− a); 0 ≤ t ≤ 1} ⊂ D, such that F is F differentiable on ab (or in D) and

‖F ′(a + t(b− a))‖ ≤ g′(t), 0 ≤ t ≤ 1,

were g′ and f ′(t) := F ′(a + t(b− a))(b− a) are integrable in [0, 1]. Then the followingrelations are valid, prefeably with x0 near ab,

F (b)− F (a) =∫ 1

0

F ′(a + t(b− a))dt(b− a), (1.75)

‖F (b)− F (a)‖ ≤ (g(1)− g(0))‖b− a‖, (1.76)

‖F (b)− F (a)‖ ≤ ‖b− a‖ supx∈ab

‖F ′(x)‖, (1.77)

‖F (b)− F (a)− F ′(x0)(b− a)‖ ≤ ‖b− a‖ supx∈ab

‖F ′(x)− F ′(x0)‖. (1.78)

Modifications for G derivatives into the direction of b− a are straightforward.

Partial derivatives are defined by splitting the variable into components. Let

F := (F1(x), . . . , Fm(x)) : D ⊂ X =n∏

i=1

Xi → Y =m∏

j=1

Yj (1.79)

with different arguments x = (x, . . . , xu) ∈ X and components, (F1(x), . . . , Fm(x)).For Banach spaces Xi and Yj and the norm ||x||X := maxn

i=1{||xi||Xi}, the X ,Y are

Banach spaces as well. We define partially differentiable F :

Definition 1.44. Choose x0 ∈ U(x0) ⊂ D,U(x0) a neighborhood of x0. Assume thereexists a Li ∈ L(Xi,Y) and a Lj

i ∈ L(Xi,Yj), respectively, such that for all small hi ∈Xi,

F (x0 + hi)− F (x0)− Lihi = o(hi) and Fj(x0 + hi)− Fj(x0)− Ljihi = o(hi). (1.80)

Then the operator F and its components Fj , are called partially (Frechet) differentia-ble in x0 with respect to xi and the (∂F/∂xi)(x0) := (∂xi

F )(x0) := Li and Lji are

called its partial derivatives. By replacing hi by thi and t→ 0 we obtain the partialGateaux differentiability. For Xi = Yj = R and hi = tei with the i-th unit vector, ei,and small t ∈ R, we obtain the classical partial derivatives.

Theorem 1.45. Continuously partially differentiable operators are (Frechet) differ-entiable:

1. Let F in (1.79) be continuously partially differentiable in U(x0), that is ∂F/∂xi ∈C(U(x0)). Then F is (Frechet) differentiable in x0.

2. Let F in (1.79) be differentiable in x0. Then

F ′(x0) =(∂F

∂xi(x0)

)n

i=1

=(∂Fj

∂xi(x0)

)n m

i=1,j=1

=(Lj

i

)n m

i=1,j=1.


With h = (h1, . . . , hn) this yields the result

F (x0 + h)− F (x0) + o(h) = F ′(x0)h =n∑

i=1

∂F

∂xi(x0)hi =

(n∑

i=1

∂Fj

∂xi(x0)hi

)m

j=1

.

Theorem 1.46. Implicit function theorem: Let U ⊂ X , and V ⊂ Y be open sets, F :U × V → Z, F ∈ C1(U × V), and let (x0, y0) ∈ U × V be a solution of F (x0, y0) = 0.Furthermore, let ∂xF (x0, y0) ∈ L(X ,Z) be boundedly invertible. Then there existneighborhoods U(x0) ⊂ U(x0) ⊂ U , V(y0) ⊂ V for x0, y0, and an operator, g : V(y0) →U(x0) such that

F (g(y), y) = 0 ∀y ∈ V(y0). (1.81)

Furthermore, g ∈ C1(V(y0)) and g is unique in the following sense: For every (x, y) ∈U(x0)× V(y0) such that F (x, y) = 0 this x = g(y), and limy→y0 g(y) = g(y0).

Definition 1.47. C1 diffeomorphism: Let M,N be arbitrary subsets of Banachspaces X ,Y and let F : M→N .

1. F : M→N is called a C1 diffeomorphism iff F is bijective and F and F−1 areC1 mappings.

2. F is called a C1 local diffeomorphism at x0 ∈M iff U(x0) ⊂M and F (U(x0))⊂ N are neighborhoods of x0 and F (x0), respectively, and F : U(x0) → F (U(x0))is a C1 diffeomorphism.

Theorem 1.48. Local inverse function theorem [675], Theorem 4.F: Let the map-ping F : U(x0) ⊂M→ N be C1 with Banach spaces X ,Y. Then F is a local C1

diffeomorphism at x0 iff F ′(x0) : U(x0) → F ′(x0)(U(x0)) is bijective.

Corollary 1.49. Derivative of inverse functions [676], Corollary 4.37: Let F :U(x0) ⊂M→ N be in C1, and let F ′(x0) :M→N be bijective. Then F is a localC1 diffeomorphism at x0 and, in a small neighborhood U(x0) of x0 :

(F−1)′(y) = (F ′(x))−1 with y = F (x), x ∈ U(x0). (1.82)

Higher order derivatives of order l are obtained by applying the Gateaux definitionfor l independent h1, . . . , hl. Then results are valid analogously to Theorem 1.42. Sincethey are hardly used in this book, but intensely in [120]. Starting with Theorem 1.46,we could then replace the C1 by Cr, r ≥ 1.

2

Elements of analysis for linear andnonlinear partial elliptic differentialequations and systems

2.1 Introduction

We attempt to give a necessarily incomplete survey of this huge area. The presentationis restricted to facts which seem to be most important for our context. In particular,it is totally impossible to refer to all the many highly interesting results availablein this area. Nonlinear elliptic differential equations or systems and their numericalsolutions, including bifurcation and center manifolds in Bohmer [120] are our maingoal. Accordingly, we select the references essentially from among textbooks. Thestudy of numerical solutions and bifurcations is based upon linearization and theFredholm alternative for the linearized problem.3 So we study linear and nonlinearelliptic differential equations and systems and their linearization of order 2 and 2m.The Fredholm alternative for the linearized problem shows that, in some sense, thesesystems behave similarly to finite-dimensional systems: the kernel and co-range of thelinearized operator, A, have the same finite dimension unless parameters extend thekernel. Solutions for Au = f exist if and only if f is perpendicular to the kernel of thedual operator Ad. We summarize the existence and uniqueness results for solutions,and the Fredholm alternative for the bifurcation context. Regularity results for thesolutions are necessary for good convergence of numerical methods. We only formulateproofs for results that are straightforward generalizations of the reviewed books.However, we omit the results related to the different types of maximum principles andoscillations and the concept of viscosity solutions. Monotony and (nonlinear) coercivitytechniques yield a lot of results, summarized here for our context. Good references forthese important results are the books, by e.g. Gilbarg and Trudinger [346], Dong [299],Chen and Wu [170], Evans [310], Kreiss and Lorenz [454], Showalter [589], Taylor[618–620], and Zeidler [675–678].

The transition from order 2 to 2m, from one to several space variables, from linearto nonlinear, and from one elliptic differential equation to systems complicates thesituation. Correspondingly, the results become less strong. In particular, not manyregularity results for quasilinear and nonlinear equations and systems of higher order

3 Linearized problem always means the derivative of a problem and differentiable is always Frechetdifferentiable, unless stated otherwise.

2.1. Introduction 33

seem to be available in the textbooks and monographs used in this survey. For afascinating survey of the historic development of the analysis of partial differentialequations, see Brezis and Browder [143].

As motivation we consider the following simple semilinear model problem

Au = −Δu + f(u) = g ∈ C(Ω), u|∂Ω = 0, (2.1)

with the Laplacian Δ and continuous functions f, g. A straightforward way to solve itwould be to determine a “classical solution” u0 ∈ C2(Ω) satisfying (2.1). In fact thisis still one of the (necessary) solution concepts for nonlinear problems, which we willsummarize in Section 2.5. For linear problems with f(u) = cu, c ∈ R, and nonlinearproblems other important concepts have been developed. In particular, the nonlinearf(u) must not grow too fast for |u| → ∞, so growth conditions have to be imposed forf(·). Often a solution u0 ∈ C2(Ω) does not exist. So we have to generalize the “classicalsolution” u0 ∈ C2(Ω), Au0 ∈ C(Ω) in two steps into u0 ∈ H2(Ω), Au0 ∈ L2(Ω), the“strong solution”, and u0 ∈ H1(Ω), Au0 ∈ H−1(Ω), the “weak solution”. Accordingly,the boundary condition u0|∂Ω = 0 has to be replaced by u0 ∈ H1

0 (Ω), cf. Theorem1.24 and Remark 2.3. For linear problems the bilinear form a(·, ·) induced by A isan essential tool. Sobolev spaces H1(Ω),H1

0 (Ω),H−1(Ω), and their generalizationsWm,p(Ω),Wm,p

0 (Ω), W−m,q(Ω), 1/p + 1/q = 1, cf. Subsection 1.4.3, the necessaryboundary analysis for ∂Ω, and the trace operators are summarized in Section 1.4.

We start in Sections 2.2–2.4 with linear problems. In Section 2.2, motivated bythe linear version (2.2) of our above model problem (2.1). We determine appropriateconcepts for generalized solutions in Hilbert spaces for a second order elliptic operatorand the corresponding bilinear form. In this motivating section, we sometimes referto definitions and properties introduced later on. Section 2.3 studies, in the Banachspace setting, bilinear forms a(·, ·) and the induced linear operators A and vice versa.The norms, duals and invertibility of A are characterized by properties of a(·, ·). Theconcepts of Gelfand triples, coercive and elliptic bilinear forms a(·, ·) are introducedand relations to extremal problems are discussed. Finally the Riesz–Schauder theory,the Fredholm alternative and conditions for elliptic bilinear forms are studied. Theseresults provide a solid foundation for the later linearization. General 2m-th orderelliptic operators and the corresponding bilinear forms are the topic of Section 2.4.We formulate Fredholm alternatives, estimate the solutions, summarize the essentialregularity results in Holder and Sobolev (Banach) spaces for 2m-th order ellipticoperators with C∞ coefficients. Ck coefficients are allowed for order 2. Finally, weaksolutions in Sobolev (Hilbert) spaces are studied.

In Section 2.5 we introduce nonlinear elliptic problems. Most of them live inBanach spaces, often in Holder or Sobolev spaces. Usually, growth conditions for thecoefficients have to be imposed. We start with the different definitions and types ofsolutions and study semilinear, quasilinear and fully nonlinear problems. Examplesand some existence, uniqueness, regularity and Fredholm-type results are added. InSection 2.6 we turn to linear and nonlinear systems of elliptic equations. Again wedistinguish between elliptic and coercive systems, their compact perturbations, andthe corresponding bilinear forms.

34 2. Analysis for linear and nonlinear elliptic problems

We do know that the linearized operators play a dominant role for the stability ofnumerical methods and for bifurcation. So we study in Section 2.7 the linearized formsof the nonlinear operators presented in Sections 2.5 and 2.6. It turns out that, underslightly stronger conditions, Fredholm-type results are available for all of them. Oneof the most challenging examples of nonlinear systems of elliptic equations are theNavier- Stokes equations, studied in Section 2.8. We start with the Stokes operator. Itrepresents a (noncoercive) saddle point problem. Its extension to the nonlinear versionrequires trilinear forms and bilinear operators.

Linearization of the nonlinear problems in this book yields linear operators, A,inducing bilinear forms, a(u, v), sums of

∫Ωa∂αu∂βvdx, |α|, |β| ≤ m. For a ∈ L∞(Ω),

these are well defined only for functions u, v ∈Wm,p(Ω) with 2 ≤ p ≤ ∞, but not for1 ≤ p < 2, cf. Sections 2.2, 2.3, and Subsection 2.4.4. The A are compact perturbationsE.g. the linearized Navier-Stokes operator compactly perturbs the invertible Stokesoperator (with stable discretization). Similarly operators B with Hm(Ω)-coercivebilinear forms are induced by their principal parts. For 1 ≤ p < 2, we would obtainWm,p(Ω)-coercive bilinear forms; unfortunately they are defined no longer. We willshow in Chapters 3 ff. that operators with Hm(Ω)-coercive bilinear forms are stablewith respect to discrete Hm(Ω) norms. Furthermore, linear boundedly invertibleoperators with stable consistent discretization essentially inherit this stability to theircompactly perturbed invertible operators. Consistency results will be available forthis large class of linear and nonlinear equations and systems as well. Stability of thelinearized problem implies stability of the nonlinear problem as well.

Therefore, linear and nonlinear operators A : Wm,p(Ω) →W−m,q(Ω), 1/p + 1/q =1, are particularly important, cf. Corollary 2.44. These results yield the key fora complete convergence theory for the following equations, systems and the maindiscretization methods for Wm,p(Ω), 2 ≤ p ≤ ∞ w.r.t the Hm(Ω) norm for 2 < p. Thisallows in many areas strong extensions of the state of the art, e.g. for wavelet andradial basis or mesh-free methods. The excluded case 1 ≤ p < 2 allows a completeconvergence theory as well, however based upon monotone operators, cf. Section 4.5.

Many problems in science, e.g. in mechanics, are related to variational problemsdefined via integrals. We want to touch on some aspects. In many cases (systemsof) elliptic equations are the so-called Euler systems of a variational problem, e.g.in Theorem 2.97. This is a main topic in the calculus of variations, see, e.g. themonographs by Giaquinta and Hildebrandt [341, 342]. The first variation of theseintegrals applied to a test function yields the weak form, the so-called Euler system,and weak solutions of a PDE. A transformation of this equation into the strongform yields a PDE in divergence form, sometimes known as a divergent PDE.This transition from a variational problem to its first variation or the weak, oftendivergence form is discussed in Subsection 2.6.4. Many of these problems are relatedto monotone operators. They are systematically discussed by Zeidler [677, 678] andpartially summarized in Subsections 2.5.5, 2.5.6 and 2.6.4, Theorems 2.97, 2.98. Forlinear problems the divergence form and the general form can easily be exchanged,if only appropriate regularity properties for the coefficients are guaranteed. Thereforein most of the literature nonlinear problems are studied in divergence form. Wetoo concentrate our presentation mainly on this form. However, the exceptions in


Subsections 2.5.3, 2.5.7 and 2.6.8 are important. For the general case we refer, e.g. toLadyzenskaja and Uralceva [463,464,466]. For fully nonlinear elliptic problems, thereis no divergence form corresponding to the strong form, and, hence, no weak form orsolution of a PDE in strong form. This is relevant for their dicretizations, formulatedfor the first time for the general case in this book, mainly as an FEM. In the prefacewe have already mentioned the viscosity solutions. We do not discuss them here.

Most of the following results are valid only for uniformly elliptic equations. Thecase of nonuniformly elliptic equations is indicated in Subsection 2.5.3, and in, e.g.Ivanov [415].

For this chapter we consulted only a few of the original papers, but mainly referto the textbooks and monographs cited in the text. We begin with a short list, notaiming for completeness. Elliptic PDEs of second order, including quasilinear andfully nonlinear equations are studied by Ladyzenskaja and Uralceva [464], Gilbarg andTrudinger [346], and Chen and Wu [170]. More general PDEs essentially for the linearcase, are discussed by Garabedian [336], Hormander [402], Necas [509,510] and Wloka[667]. PDEs, now including nonlinear problems, are further studied by Evans [310] andDong [299]. Evans [310] extensively discusses different representations for solutions forfirst and second order equations. His, and some of the other, theoretically orientedbooks extend the theory for linear equations to nonlinear problems. Usually the Holder,Sobolev and Schauder results are complemented with maximum principles. Dong [299]concentrates on second order equations and presents several important examplesmainly of parabolic type. Two impressive “multi-graphs” on linear and nonlinearproblems are due to Zeidler [675–678], and Taylor [618–620]. Zeidler’s main interestis nonlinear functional analysis with applications to PDEs. He gives a large numberof results for linear and nonlinear PDEs. Taylor more directly aims for equations andsystems of PDEs. Both combine many of the preceding techniques for elliptic equationsand systems, Taylor even on manifolds. Triebel [634,635] initiated and obtained a newhierarchy of spaces, the so-called Besov–Triebel–Lizorkin spaces and many resultsfor nonlinear equations. Based upon these results, Runst and Sickel [561] develop avery general theory for nonlinear PDEs. Polyanin and Zaitsev [535] collect the exactsolutions for many nonlinear elliptic (parabolic and hyperbolic) equations. Necessarily,in our context, the maximum principle, lower and upper solutions and oscillationare neglected compared to the preceding monographs. Morrey [500] and Giaquintaand Hildebrandt [341, 342] study mainly quasilinear differential equations related tovariational problems, a main topic in the calculus of variations. For higher orderPDEs nonhomogeneous boundary conditions are far more complicated, in contrastto second order; in fact they are a serious problem. They are presented by Lions andMagenes [478–480]. For more literature see the following sections.

In this book, we mainly consider isolated solutions u0 with boundedly invertiblelinearized operator G′(u0). Extensions are described by Bohmer [120]. For isolatedsolutions, Theorem 1.46 and the results from Section 2.4 usually allow us the restrictionto homogeneous boundary conditions. But for fully nonlinear problems, we discuss theseparate role of differential and boundary operators. So, in general, it is possible todefine an exact extension of the boundary function and use the standard substitutiontrick. Alternatively, an approximating function, extended to the full domain, yields asmall error at the boundary, causing only a small change in the exact solution.


2.2 Linear elliptic differential operators of second order,bilinear forms and solution concepts

In this section we motivate different solution concepts for the easiest case of secondorder linear equations. The linear form of our above model problem (2.1) is

Au = −Δu + cu = f ∈ C(Ω), u|∂Ω = 0. (2.2)

But even for c > 0, f ∈ C(Ω), and a smooth enough ∂Ω, a “classical solution” u0 ∈C2(Ω) with ‖u0‖C2(Ω) ≤ C‖f‖C(Ω) does not always exist, hence A : C2(Ω) → C(Ω)is not boundedly invertible. For a counterexample, see Hackbusch [387] and forappropriate modifications see, e.g. our Theorem 2.37. Linear functionals allow us away out of this dilemma. From Definition 1.8 we do know that for a Hilbert space Vand its dual V ′,

f ∈ V ′ with 〈f, v〉V′×V = 0 ∀ v ∈ V is equivalent to f = 0. (2.3)

So we use linear functionals for testing Au = −Δu + cu = f . We relax Au ∈ C(Ω) intwo steps into Au ∈ L2(Ω) and then Au ∈ H−1(Ω).

We have to relate operators, bilinear forms and boundary terms. We use the standardnotations for partial derivatives

∂iu :=∂u

∂xi, ∂i∂ju =

∂2u

∂xi∂xj, ∂0u := u, Δu :=

n∑i=1

(∂i)2u. (2.4)

Unless stated otherwise, we require throughout this chapter:

Ω ⊂ Rn is open, bounded and ∂Ω is Lipschitz-continuous. (2.5)

For this Ω the corresponding spaces are defined with respect to their Sobolev (Hilbert)space inner products, norms, and seminorms, with H0(Ω) = L2(Ω),

(u, v)H1(Ω) :=n∑

i=0

(∂iu, ∂iv)L2(Ω), (u, v)H2(Ω) :=n∑

i,j=0

(∂j∂iu, ∂j∂iv)L2(Ω), (2.6)

‖u‖Hi(Ω) := (u, u)1/2Hi(Ω), i = 0, 1, 2, |u|H1(Ω) :=

( n∑i=1

‖∂iu‖2L2(Ω)

)1/2

.

As usual, H−i(Ω), i = 1, 2, denote the dual spaces of bounded linear functionals onHi(Ω), i = 1, 2.

We start testing (2.2) with v ∈ L2(Ω) in the sense of (2.3) and for u ∈ H2(Ω) andintroduce classical, strong and weak operators and bilinear forms. In contrast to thestandard literature we use in this section different notations Ac, for the strong formsdefined on C2 and As, as(·, ·), on H2, sometimes called the Friedrichs extension, and

2.2. Linear elliptic differential operators of second order 37

A, a(·, ·) for the weak forms in (2.11):

Ac : C2(Ω) → C(Ω), Ac u = −Δu + cu, (2.7)

As : H2(Ω) → L2(Ω), As u = −Δu + cu, and

as(·, ·) : H2(Ω)× L2(Ω) → R defined by (2.8)

as(u, v) := (Asu, v)L2(Ω) =∫

Ω

(−Δu + cu) vdx ∀v ∈ L2(Ω).

The condition u ∈ H2(Ω) still is too restrictive. We apply (2.9). Zeidler [677], pp. 19and 20, calls this one of the cornerstones of modern analysis. For v, w ∈ H1(Ω) wegeneralize the classical integration by parts formula,

∫ b

av′wdx = vw|ba −

∫ b

avw′dx.

With the outer normal ν = (ν1, . . . , νn)T and a Lipschitz-continuous manifold ∂Ω,we obtain the fundamental Green’s formula, cf. Ciarlet [174], p. 14,∫

Ω

w∂v/∂xidx +∫

Ω

v∂w/∂xidx =∫

Ω

∂(vw)/∂xidx =∫

∂Ω

vwνids, (2.9)

∂Ω ∈ CL,∀w, v ∈ H1(Ω).

We apply this to the last equation in (2.8) to get the first Green’s formula∫Ω

(−Δu + cu) vdx = as(u, v) =∫Ω

(∇u,∇ v)n + cuvdx−∫

∂Ω

∂u

∂νvds, (2.10)

with the Euclidean product and norm (·, ·)n and | · |n in Rn. Note that∫

∂Ω∂u/∂ν vds

= 0 if ∀∂u/∂ν ∈ L2(∂Ω) and ∀v ∈ H1(Ω) with v = 0 on ∂Ω. By Theorems 1.37,1.38, we denote this as ∀v ∈ H1

0 (Ω) in the trace sense. This allows us to reduce thedifferentiability conditions for u and the solution u0. We introduce the weak bilinearform and operator, a(·, ·), A sometimes called the energy extension, as

a(·, ·) : H1(Ω)×H1(Ω) → R and A : H1(Ω) → H−1(Ω) defined by (2.11)

a(u, v) :=∫Ω

∇u∇ v + cuvdx =: 〈Au, v〉H−1(Ω)×H1(Ω), thus, for “regular” u,

∀u ∈ H2(Ω), v ∈ H1(Ω) : a(u, v) = as(u, v) +∫

∂Ω

∂u

∂νvds and = as(u, v),

if ∂Ω1 ∪ ∂Ω2 = Ω and∫

∂Ω

∂u

∂νvds = 0 for ∀v|∂Ω1 = 0 and ∀ ∂u

∂ν

∣∣∣∣∂Ω2

= 0,

often with ∂Ω1 = ∂Ω or ∂Ω2 = ∂Ω. The restriction v|∂Ω1 for v ∈ H1(Ω) yields v|∂Ω ∈H1/2(Ω) in the trace sense. For our Ω in (2.5), this is an immediate consequence ofTheorems 1.37, 1.39.

Remark 2.1.

1. For the weak form a(u, v) we need u, v ∈ H1(Ω), and for the strong form as(u, v)we need u ∈ H2(Ω), v ∈ L2(Ω). The relation a(u, v) = as(u, v) +

∫∂Ω

∂u/∂ν vds,


shows that a(u, v) = as(u, v) if∫

∂Ω∂u/∂ν vds = 0. In (2.11), u ∈ H2(Ω) implies

∂u/∂ν|∂Ω2 ∈ H1/2(Ω), and a well-defined∫

∂Ω∂u/∂ν vds. Under appropriate con-

ditions even for u ∈ H1(Ω) a ∂u/∂ν|∂Ω ∈ H−1/2(Ω) is possible, cf. Costabel[211]. With v ∈ H1(Ω), hence v|∂Ω ∈ H1/2(Ω) the

∫∂Ω

∂u/∂ν vds is still defined.Under those conditions, the previous Neumann boundary value, but not thestrong form as(u, v), in (2.11) is well defined for u ∈ H1(Ω). Other results forthis problem of boundary traces are discussed, e.g. in Taylor [618], Chapter4, Proposition 4.5, and Jonsson and Wallin [425], Jerison and Kenig [419],Schwab [575] and Grisvard [373]. We do not consider these extensions here inmore detail.

2. For the later nonconforming discretization methods in Chapters 5 and 7, andadaptive FEMs in Chapter 6, we will consider (2.11) on subtriangles T ⊂ Ω,indicated as aT (u, v) = as,T (u, v) +

∫∂T

∂u/∂ν vds. This will turn out to beone of the deciding tools for estimating the consistency errors in the discretegeneralization of the a(u, v). The piecewise functions, uh|T , usually polynomials,in FEMs and DCGMs always satisfy uh ∈ Hk(T ), k ≥ 2, hence aT (uh, vh) =as,T (uh, vh) +

∫∂T

∂uh/∂ν vhds. For the corresponding consistency estimates,we will require u as close as possible to the u ∈ H1(Ω) in the weak form. Thereforemotivated by Theorems 1.37, 1.39 and 2.48, we replace the above u ∈ H1(Ω)either by u ∈ H2(Ω) as in (2.11), or require, in DCGMs, u ∈ H3/2+ε(Ω) withany ε > 0. Then ∂u/∂ν|∂Ω2 ∈ Hε(∂Ω) ⊂ L2(∂Ω) is valid and

∫∂Ω

∂u/∂ν vds iswell defined, cf. Remarks 5.51 and 7.1.

3. For simplicity, we assume, here and for analogous cases, unless stated otherwise:For natural or Neumann boundary conditions let u ∈ H2(Ω) or u ∈ H3/2+ε(Ω),hence ∂u/∂ν|∂Ω2 ∈ L2(∂Ω).

Obviously the two operators and bilinear forms defined in (2.8) and (2.11) aredifferent. We could even go one step further: By an additional partial integration,again using (2.9), one would end up with the distributional bilinear form

ad(u, v) :=∫

Ω

u(−Δv + cv)dx ∀ v ∈ C∞0 (Ω), ∀ u ∈ (C∞

0 (Ω))′ . (2.12)

This allows us to formulate solutions u0 ∈ (C∞0 (Ω))′, the space of distributions. We

do not study this possibility in this book; see, e.g. Lions and Magenes [478], Runstand Sickel [561] Showalter [589].

Based upon (2.8) and (2.11), we are now able to introduce the concepts ofclassical and strong and weak solutions u0 = uc,0 and u0 = us,0 and u0 = uw,0,

respectively, of Au = f. We use uc,0, us,0 or uw,0 and Ac, As, A = Aw only if thecontext does not directly show the chosen solution concept. The existence, uniquenessand regularity of the solutions are studied in the following sections. These regularityconditions will show that the type of solution uc,0, us,0 or uw,0 is essentially aconsequence of the smoothness of the problem. Note that the definition of the linearoperators and bilinear forms does not require boundary conditions, but the followingdefinition of the solutions does require boundary conditions, e.g. u0 ∈ H1

0 (Ω). The


classical solution u0 = uc,0 and the strong solution us,0 are then defined, e.g. forDirichlet boundary conditions, by

u0 = uc,0 ∈ C2(Ω), u0|∂Ω = 0 : As u0 = −Δu0 + cu0 = f ∈ C(Ω), (2.13)

u0 = us,0 ∈ H2(Ω) ∩H10 (Ω) : As u0 = −Δu0 + cu0 = f ∈ L2(Ω) ⇔ (2.14)∫

Ω

fvdx = (Asu0, v)L2(Ω) =∫

Ω

(−Δu0 + cu0)vdx = as(u0, v) ∀ v ∈ L2(Ω).

The weak solution u0 is defined, with f ∈ H−1(Ω) instead of f ∈ L2(Ω), by

u0 = uw,0 ∈ H10 (Ω) ⇔ Au0 = f ∈ H−1(Ω) ⇔ (2.15)

〈Au0, v〉H−1(Ω)×H10 (Ω) =

∫Ω

(∇u0 ,∇ v)n + c u0 v dx = a(u0, v)

=∫

Ω

f0v +n∑

j=1

fj∂jvdx

= 〈f, v〉H−1(Ω)×H10 (Ω) ∀ v ∈ H1

0 (Ω), cf. (2.107).

As a special case of Theorem 2.43 and Corollary 2.44, we obtain

Theorem 2.2. For c ≥ 0 and f ∈ H−1(Ω) or f = f0 ∈ L2(Ω) the problem (2.15) hasa unique solution u0 ∈ H1

0 (Ω).

In (2.15) we required f ∈ H−1(Ω) and tested with v ∈ H10 (Ω), indicating the need for

Au0 = f ∈ H−1(Ω). In fact, there are reasons for distinguishing and for identifyingH−1(Ω) and H−1

0 (Ω). Since we need this discussion for all the following more generalcases, we formulate it here.

Remark 2.3. We assume a bounded Ω as in (2.5). Then, according to Proposi-tion 2.34, cf. Adams [1], pp. 49 ff. Wm,p

0 (Ω) ⊂Wm,p(Ω) �= Wm,p0 (Ω). On the other

hand, Taylor [618], Chapter 4, Proposition 5.1, proves, for a smooth Ω, hence ∂Ω ∈C∞, cf. Proposition 1.32, and m ∈ N0, that there is a natural and norm-preservingisomorphism

H−m0 (Ω) = dual space of Hm

0 (Ω) ≈ H−m(Ω).

This fact is one of the reasons why sometimes in the literature H−m(Ω) and H−m0 (Ω),

and W−m,q(Ω), 1/p + 1/q = 1, and W−m,q0 (Ω) are not distinguished. On the other

hand, often only f ∈W−m,q(Ω) are used for the f ∈W−m,q0 (Ω), e.g. in Proposition

2.34. That H−m(Ω) and H−m0 (Ω) are in fact different is demonstrated by (2.107)

versus (2.110).

Now we extend (2.15) to more general boundary and elliptic operators of secondorder. The strong operator As can be directly defined and is related to its strong


bilinear form as(·, ·) , with real valued coefficients as

As : H2(Ω) → L2(Ω), as(·, ·) : H2(Ω)× L2(Ω) → R and (2.16)

Asu :=n∑

i,j=0

(−1)j>0∂j(aij∂

i u) with (−1)j>0 := 1 for j = 0 else := −1,

as(u, v) := (Asu, v)L2(Ω),∀u ∈ H2(Ω), v ∈ L2(Ω), aij ∈W 1,∞(Ω) and (2.17)

Ac : C2(Ω) → C(Ω), Ac := As| C2(Ω), aij ∈ C1(Ω).

Even for Ac : C2(Ω) → C(Ω) test spaces can be defined as functions of boundedvariation defining a bilinear form for uc,0 ∈ C2(Ω). We do not consider this here,nor generalize the above distributional bilinear form ad(·, ·) in (2.12).

For the (standard) weak problem we directly define the weak bilinear form a(·, ·) andthe induced A by partial integration of (2.16) and the Green’s formula with vanishingboundary terms as

a(·, ·) : H1(Ω)×H1(Ω) → R and A : H10 (Ω) → H−1(Ω), with (2.18)

a(u, v) :=∫

Ω

n∑i,j=0

aij∂iu∂jvdx

=: 〈Au, v〉H−1(Ω)×H10 (Ω), ∀u, v ∈ H1

0 (Ω), aij ∈ L∞(Ω).

Definition 2.4. Principal part and ellipticity: The principal parts Ap and ap(u, v)of the operator, A, and its bilinear form, a(u, v), in (2.18) are

Apu :=n∑

i,j=1

(aij∂i u)∂j ∈ H−1(Ω) and

ap(u, v) :=∫

Ω

⎛⎝ n∑i,j=1

aij∂i u ∂j v

⎞⎠ dx =∫

Ω

(Apu) ◦ vdx, (2.19)

and similarly for As, Ac and as(·, ·). For ϑ = (ϑ1, . . . , ϑn)T ∈ Rn let the characteristicpolynomial of Ap satisfy; 4

n∑i,j=1

aij(x)ϑiϑj ≥ 0 and ≥ ε′|ϑ|2n , for an ε′ > 0, ∀x ∈ Ω, ϑi ∈ R, respectively,

(2.20)

with the Euclidean norm |ϑ| = |ϑ|n in Rn. Then the operator A (or As, Ac) is calledelliptic (≥ 0), and strongly elliptic

(≥ ε′|ϑ|2n

). For aij ∈ L∞(Ω) (or aij ∈W 1,∞(Ω))

this condition is only required almost everywhere (a.e.) in Ω.

4 It is always possible to use ≥ instead of ≤ in (2.20) by multiplying A by −1.


Obviously, the above bilinear forms and linear operators in (2.16), (2.18) arecontinuous or bounded, that is, positive constants, C, exist, such that

|a(u, v)| < C‖u‖H1(Ω)‖v‖H1(Ω) ∀u, v ∈ H1(Ω), (2.21)

|as(u, v)| < C‖u‖H2(Ω)‖v‖L2(Ω) ∀u ∈ H2(Ω), v ∈ L2(Ω),

‖Au‖H−1(Ω) < C‖u‖H1(Ω) ∀u ∈ H1(Ω),

‖Asu‖L2(Ω) < C‖u‖H2(Ω) ∀u ∈ H2(Ω) and

‖Acu‖C(Ω) < C‖u‖C2(Ω) ∀u ∈ C2(Ω).

As an immediate consequence of (2.19), (2.20), we obtain, for a strongly ellipticdifferential operator of second order, the so-called H1

0 (Ω) coercivity of the principalpart:

for aij ∈ L∞(Ω), (2.20) implies ap(u, u) ≥ ε′|u|2H1(Ω). (2.22)

For u ∈ H10 (Ω) the seminorm |u|H1(Ω) is equivalent to the standard Sobolev norm

‖u‖H1(Ω), see (2.6). In fact, (2.21), (2.22) show the equivalence of ‖ · ‖H1(Ω) and(ap(·, ·))1/2 in H1

0 (Ω), for As = −Δ, in general, called the energy norm.A generalization of (2.21) to A : W 1,p(Ω) →W−1,q(Ω), 1 ≤ p ≤ ∞, 1/p + 1/q = 1,

with test spaces Vb = W 1,p(Ω) and to As : W 2,p(Ω) → Lq(Ω) with test spaces Lp(Ω),is possible as well; see Sections 2.4 and 2.5, and, e.g. [561,618–620,634,635].

For higher order problems we will consider A : Wm,p(Ω) →W−m,q(Ω). However,generalizations to A : C2(Ω) → C(Ω), and a corresponding theory is not appropriate,but only for A : C2,γ(Ω) → Cγ(Ω), 0 < γ < 1, see Subsections 2.4.2 and 2.4.3.

Again the relation between the weak and strong bilinear forms is based upon theso-called natural boundary conditions. With the outer normal ν = (ν1, . . . , νn)T on aC1 manifold ∂Ω, we obtain from (2.9) or the Gauss integral theorem,∫

Ω

v∂j(aij∂iu)dx = −

∫Ω

(∂jv)aij∂iu +

∫∂Ω

vaij∂iuνjds. (2.23)

This is applied to a(·, ·) in (2.18) yielding a generalized first Green’s formula

a(u, v) =∫

Ω

(As u)vdx +∫

∂Ω

(Ba u)vds = as(u, v) +∫

∂Ω

(Bau)vds (2.24)

with

Ba u :=j=1,...,n∑i=0,...,n

νjaij∂i u, the natural boundary operator. (2.25)

Sometimes, instead of the boundary operator, Ba, a more general form is used:

Bu :=n∑

i=1

bi∂iu + b0u with b = (b1, . . . , bn) ∈ Rn, (ν, b)n �= 0. (2.26)


Bau|∂Ω and Bu|∂Ω ∈ H−1/2(∂Ω) or Bu|∂Ω ∈ L2(∂Ω) are well defined for u ∈H3/2+ε(Ω), cf. Remark 2.1 and Theorem 1.38. In contrast to B we call Ba the natural,or induced boundary operator, since it is “induced” by a(·, ·). For the special case of(2.2) we obtain Neumann boundary conditions

Asu = −Δu + cu induces Ba u = ∂u/∂ν. (2.27)

The last condition, (ν, b)n �= 0, in (2.26) excludes tangential derivatives∑n

i=1 bi∂iu of u

along ∂Ω. By rescaling Bu we can obtain (ν, b)n = νTb = νTAν �= 0 for a strongly ellip-tic operator A, see Proposition 2.5, Remark 2.6, and Hackbusch [387], Remark 7.4.10.

The differential operator Au = f and on ∂Ω Dirichlet and natural boundary condi-tions Bu := u = ϕ and Bau = ϕ are tested by Vb, and are realized for u0 ∈ H1(Ω)with

Vb = H10 (Ω) for Dirichlet boundary conditions: H1/2(Ω) � u0|∂Ω = 0 and

Vb = V = H1(Ω) for natural boundary conditions: we require (2.28)

u ∈ H2(Ω), implying H1/2(∂Ω) � Bau0|∂Ω = ϕ

in the trace sense, cf. Remark 2.1 and Theorems 1.38, 1.37 and 2.43. Since v|∂Ω ∈H1/2(∂Ω) for v ∈ H1(Ω), the

∫∂Ω

(Bau)vds is again well defined in this case, cf.textbooks on PDEs, e.g. [135,141,387].

By the standard trick u := u− u with u = ϕ on ∂Ω we obtain homogeneous Dirichletboundary conditions. In fact we may use Theorem 1.29 for extending the boundaryfunction, ϕ to Ω, yielding ϕ and thus u := ϕ, see Remark 2.7.

An extension to parts of ∂Ω with Dirichlet and natural boundary conditions as in(2.11) is presented in most textbooks, e.g. [141]. Therefore we only indicate it here;compare Remark 2.7 below. For u = ϕ or Bau = ϕ on an open subset ∂Ω1 of theboundary ∂Ω we first extend ϕ from ∂Ω1 to ∂Ω and then to u on Ω and again use theabove trick, applicable in later sections as well. Then we use as test space

Vb := {u ∈ H2(Ω) : u|∂Ω1 = 0}. (2.29)

We determine the weak and strong solution u0 = uw,0 and u0 = us,0 of the Dirichletboundary value problem with the bilinear forms in (2.18) and (2.16) by

uw,0 = u0 ∈ H10 (Ω) : a(u0, v) = 〈f, v〉H−1(Ω)×H1(Ω) ∀v ∈ H1

0 (Ω), and (2.30)

us,0 = u0 ∈ H10 (Ω) ∩H2(Ω) : as(u0, v) = (f, v)L2(Ω) ∀v ∈ L2(Ω),

hence Au0 = Auw,0 = f ∈ H−1(Ω) and

Asu0 = Asus,0 = f ∈ L2(Ω), respectively

In Section 2.4 we will return to classical solutions, Asus,0 = f ∈ C(Ω).For a smoother u0 ∈ H2(Ω) ∩H1

0 (Ω) and u0|∂Ω = 0, (2.24) and (2.25) imply uw,0 =u0 = us,0, since H1(Ω) is dense in L2(Ω), so

u0 ∈ H2 ∩H10 (Ω) : a(u0, v) =

∫Ω

(Asu0)vdx = as(u0, v) =∫

Ω

fvdx ∀v ∈ H10 . (2.31)


The natural boundary operator Ba was “induced” by a(·, ·). By partial integrationof as(·, ·) we obtained a(·, ·) and a boundary term defined via Ba. But a(·, ·) does notnecessarily uniquely determine As and Ba and vice versa. So the question naturallyarises whether a boundary value problem with a general boundary operator B as in(2.26) uniquely determines a bilinear form a(·, ·) and vice versa. Let, cf. (2.16), (, )n

be the Euclidean scalar product in Rn,

Asu :=n∑

i,j=0

(−1)j>0∂j(aij∂

i u) and with b(x) := (b1(x), · · · , bn(x)) ∈ Rn, (2.32)

B u :=n∑

i=0

bi∂i u assume (b(x), ν(x))n �= 0 ∀x ∈ ∂Ω. (2.33)

If (b(x), ν(x))n = 0 for a specific x ∈ ∂Ω were allowed,∑n

i=1 bi(x)∂i u would be atangential derivative of u in x. This sometimes does not make sense, however seeWloka [667] p. 161 ff., where b0 ≡ 0, (b(x), ν(x)) = 0 is discussed.

Hackbusch’s [387] Theorem 7.4.11 and Remark 7.4.12 show

Proposition 2.5. Let As and B be given as in (2.32), (2.33). Then there exists amodified bilinear form a1(·, ·) : H1(Ω)×H1(Ω) as in (2.18), such that the variationalformulation as

a1(u0, v) =∫

Ω

fvdx for u0 ∈ H2(Ω) ∀v ∈ H1(Ω), Ba1u0 = Bu0|∂Ω = ϕ (2.34)

corresponds to the classical formulation Asu = f in Ω and Bu = ϕ on ∂Ω.If a(·, ·) in (2.18), corresponding to As in (2.16), is H1(Ω) elliptic, cf. Definition

2.17, and the coefficients of As and Ω are C1, this5 modified a1(·, ·) is again H1(Ω)elliptic.

Remark 2.6.

1. This proposition allows us to restrict the discussion for m = 1 to the Dirichletor the natural boundary condition, or a combination, see below.

2. It is important to realize the impact of the boundary conditions, cf. Treves[632], Chapter 37. In fact, for f ∈ L2(Ω), we obtain identical strong andweak solutions, u0 = us,0 = uw,0, in (2.31) if one of the following conditionsis satisfied, see (2.24). Either u0 ∈ H2(Ω) satisfies the natural boundary con-dition Ba(u0)|∂Ω = 0, for Ba see (2.25). Then the boundary term drops out

5 Hackbusch indicates that it might be advantageous to employ additional boundary integral terms.This would avoid a smooth extension of a function, defined on ∂Ω, to Ω. Theorem 1.29 and its proofin Gilbarg and Trudinger [346] presents a constructive way to solve this problem. Thus a1(·, ·) canbe constructed in the form (2.18).


∀v ∈ H1(Ω). Or u0 satisfies the Dirichlet boundary condition u0|∂Ω = 0. Thenthe test functions are chosen as v ∈ H1

0 (Ω) = Vb. Again the boundary term dropsout. Finally, Vb has to be appropriately modified, cf. (2.143), for a combinationas in (2.11).

For natural boundary conditions Bau = ϕ on ∂Ω the reduction to Bau = 0 as forDirichlet conditions, is usually not considered. Instead, we choose

u0 ∈ U = V = H1(Ω) and solve Au0 = f ∈ H−1(Ω). (2.35)

Note that for u0 ∈ H2(Ω) and (2.5), the strong forms Asu0 ∈ L2(Ω) and Bau0 ∈H1/2(∂Ω) are well defined. Even for u0 ∈ H1(Ω), Bau0 ∈ H−1/2(∂Ω), the dual ofH1/2(∂Ω), might make sense, cf. Remark 2.1 and Theorem 1.37. So for v ∈ H1/2(∂Ω),the∫

∂ΩBau v ds are well defined.

So the strong solution u0 = us,0 of the natural boundary value problem solves

us,0 = u0 ∈ H2(Ω) : Asu0 = f ∈ L2(Ω), Bau0 = ϕ ∈ H1/2(Ω). (2.36)

Returning to (2.24), we obtain instead of (2.31), with the smoother u0 ∈ H2(Ω),

a(u0, v) =∫

Ω

(Asu0)vdx +∫

∂Ω

Bau0 v ds = as(u0, v) +∫

∂Ω

Bau0 v ds (2.37)

=∫

Ω

fvdx +∫

∂Ω

ϕ v ds or, more generally, = 〈f, v〉V′b×Vb

+∫

∂Ω

ϕ v ds,

with Vb = H10 (Ω), cf. Proposition 2.34 and Remark 7.1.

Remark 2.7.

1. Equation (2.37) is used in two steps: Firstly, we insert v ∈ H10 (Ω) and delete the

boundary integral term. This yields Au0 − f = 0 ∈ H−1(Ω) or even Au0 − f =0 ∈ L2(Ω). For the previous Dirichlet conditions with u0, v ∈ H1

0 (Ω) = Vb = Ub

we would be done. Natural boundary conditions require a second step: By theprevious a(u0, v) = 〈f, v〉V′×V or =

∫Ωfvdx we are left with

∫∂Ω

Bau0 v ds =∫∂Ω

ϕ v ds ∀v ∈ H1(Ω), yielding Bau0|∂Ω = ϕ.2. An extension to parts of ∂Ω with Dirichlet and natural boundary conditions

as in (2.11) is presented in most textbooks, e.g. Treves [632], Chapter 37, andHackbusch [386, 387], Section 7.4 and Brenner and Scott [141]. Therefore weonly indicate it here. For Dirichlet and natural boundary conditions let ∂Ω =∂Ω1 ∪ ∂Ω2 be the union of two disjoint “regular” subsets. For u = ϕ on ∂Ω1 ⊂⊂∂Ω �= ∂Ω1 we first extend ϕ from ∂Ω1 to ∂Ω and then to u on Ω and again usethe above trick for enforcing homogeneous Dirichlet conditions for u0 − u. Thenwe use as test and solution space

Vb := {v ∈ H1(Ω) : v|∂Ω1 = 0} (2.38)

and, cf. (2.11), (2.24), (2.31), define

f(v) :=∫

Ω

fvdx +∫

∂Ω2

ϕ v ds or = 〈f, v〉V′b×Vb

+∫

∂Ω2

ϕ v ds.

2.3. Bilinear forms and induced linear operators 45

2.3 Bilinear forms and induced linear operators

This presentation is strongly influenced by Hackbusch [386, 387]. His proofs for theHilbert space situation can usually be simply modified to get the corresponding resultsfor Banach spaces, see e.g. Proposition 2.18. However, motivated by the discussion atthe beginning of this chapter, Hilbert spaces are still the most important for this book.We use the notation of many authors in numerical analysis, and exchange, in contrastto [387] ellipticity to coercivity and vice versa. Hence coercive bilinear forms induce(generalized energy) norms and elliptic bilinear forms are induced by elliptic differen-tial operators. We have seen in the last section that the linear differential operatorswhich we want to consider in this context are closely related to bilinear forms. In factthey essentially determine each other uniquely, however see Proposition 2.5. So we startin this section by studying bilinear forms and the properties of linear operators inducedby them. We will apply and extend this in the following sections to linear and nonlinearelliptic differential operators of order 2m in Sobolev spaces, see Adams, [1]. Weassume

V is a real Banach space, (2.39)

and indicate whenever we need reflexivity. Linear and nonlinear (weak) differentialoperators related to bilinear or nonlinear forms are mostly studied as A : V → V ′,tested with v ∈ V. Often forms As : V → Y are used, e.g. for Holder spaces in thenext section. The discussion of bilinear forms in this section allows us to apply theseresults to linear and nonlinear operators and systems in Sections 2.4–2.6 and theirlinearization in Section 2.7. In the numerical context, they are often appropriatelydefined in Banach spaces with different spaces, U , for solutions and, V, for testfunctions. For our theoretical goals in this section, this is hardly ever relevant. Sowe mainly discuss the case U = V.

For many of the following results, e.g. coercivity and the inf–sup condition, boundaryconditions are mandatory. If we want to emphasize this we use the notation Vb for V,where the index b indicates the boundary conditions. These Vb are closed subspacesof the original V, hence again are Banach spaces. So we formulate most results in thissection with respect to V.

Definition 2.8. Properties of bilinear forms: Let a bilinear form a(·, ·) be definedon U × V → R, e.g. V = H1(Ω) or V = Wm,p

0 (Ω). This a(·, ·) is called continuous orbounded, if there exists a constant Cb ∈ R+ such that

|a(u, v)| ≤ Cb‖u‖U‖v‖V ∀ u ∈ U , v ∈ V. (2.40)

If in (2.41)–(2.43), U = V, this a(·, ·) is called V-coercive, if it is bounded and thereexists α ∈ R+ such that

a(u, u) > α‖u‖2V ∀u ∈ V. (2.41)

The dual bilinear form ad(·, ·) : V × V → R for a(·, ·) is defined as

ad(u, v) := a(v, u) ∀ u, v ∈ V. (2.42)


The a(·, ·) is called symmetric if

a(u, v) = ad(u, v) = a(v, u) ∀ u, v ∈ V. (2.43)

(For complex Banach spaces, a(u, u) and a(v, u) in (2.41) and (2.42), (2.43) have tobe replaced by � a(u, u) and a(v, u), respectively.)

Thus a V-coercive bounded bilinear form introduces a norm, ‖ · ‖a. Indeed, see(2.40), (2.41), this ‖ · ‖a is equivalent to the norm ‖ · ‖V on V. So, positive constants,γ1 = (α)1/2, γ2 = (Cb)1/2, exist such that 0 < γ1 < γ2 and

γ1‖u‖V < ‖u‖a := (a(u, u))1/2 < γ2‖u‖V∀u ∈ V. (2.44)

For the special case Asu = −Δu or aij = δij , a00 = 0 the corresponding ‖u‖a =((∇u,∇u)L2(Ω))1/2 = ‖∇u‖L2(Ω) is denoted as the energy norm.

Remark 2.9. Often the bilinear form a(·, ·) is defined, as in Section 2.2 and Sub-section 2.4.4, via an elliptic differential operator. Then the standard choice for Vis Hm(Ω). An extension to V = Wm,p(Ω) is possible for p ≥ 1. For Ω in (2.5),Wm,p(Ω) ⊂Wm,q(Ω), p ≥ q, is densely and continuously embedded, cf. Theorem 1.26.For the linearization of quasilinear and fully nonlinear equations, see e.g. Section 2.7.3,Theorem 2.122, 1 ≤ p ≤ ∞, is required, cf. the beginning of Subsection 2.4.4. Only for2 ≤ p ≤ ∞ do we obtain the full convergence results for all discretizations with respectto the discrete Hm(Ω)norm.

We use, for linear operators, A : U → V ′, with U ,V, Banach spaces, the notation

L(U ,V ′) := {A : U → V ′} for bounded A, (2.45)

N (A) := {u ∈ U : Au = 0} = ker(A),

R(A) := {v ∈ V ′ : ∃u ∈ U : v = Au} = range (A).

The notation of a dual operator in (1.33) for a linear operator A ∈ L(U ,V ′) in areflexive Banach space, V = V ′′, is slightly updated here into

Ad ∈ L(V,U ′) : 〈u,Adv〉U×U ′ = 〈Au, v〉V′×V ∀ u, v ∈ V. (2.46)

Proposition 2.10. Let a(·, ·) be a continuous bilinear form, see (2.40).

1. Then there exists a unique operator A ∈ L(U ,V ′) such that

a(u, v) = 〈Au, v〉V′×V ∀ u ∈ U , v ∈ V, ‖A‖V′←U ≤ Cb. (2.47)

A is called the operator induced by the bilinear form. In particular,

‖A‖V′←U = supu∈U,‖u‖U=1

{ supv∈V,‖v‖V=1

{|a(u, v)|} } (2.48)

= sup{|a(u, v)| : u ∈ U , v ∈ V, ‖u‖U = ‖v‖V = 1}.


2. Similarly, for U = V, the ad(u, v) = a(v, u) induces the special dual operator Ad ∈L(V,V ′) by

〈Adu, v〉V′×V = ad(u, v) = a(v, u) = 〈Av, u〉V′×V (2.49)

∀ u, v ∈ V ⇒ Ad ∈ L(V,V ′).

This Ad is the dual operator to A, see (1.33), and ‖Ad‖V′←V = ‖A‖V′←V .3. Let V1 be a dense subspace of V and let a(·, ·) be defined on V1 × V1, satisfying

(2.40) for u, v ∈ V1 with the norm ‖ · ‖V . Then there is a unique extension �a(·, ·)of a(·, ·) to V × V such that (2.40) is still satisfied with the same Cb and norms∀ u, v ∈ V.

4. If the (real) Hilbert space V is identified with V ′, then A ∈ L(V,V) induces theadjoint operator A∗ ∈ L(V,V), in the sense of (1.36) by replacing (2.49) by

(A∗u, v)V = ad(u, v) = a(v, u) = (u,Av)V ∀ v, u ∈ V. (2.50)

Finally, see (1.37), ‖A∗‖V←V = ‖A‖V←V and A∗ = A for a symmetric bilinearform a(·, ·).

5. An a(·, ·) : U × V → R, with U = Wm,p1 ,V = Wm,p2 , with coefficients ai,j ∈Wm,p3 , has to satisfy 1/p1 + 1/p2 + 1/p3 = 1. This case plays a certain role inthe linearization of nonlinear elliptic equations.

Remark 2.11. We have restricted the discussion here to bilinear forms a(·, ·) andtheir uniquely induced operator A ∈ L(U ,V ′), and its dual operator Ad ∈ L(V =V ′′,U ′). If, more generally, A ∈ L(X ,Y) for two normed spaces X ,Y, then Ad ∈L(Y ′,X ′) is, for every y′ ∈ Y ′, defined by the following unique x′ ∈ X ′, suchthat

〈Ax, y′〉Y×Y′ = 〈x, x′〉X×X ′∀ x ∈ X ⇒ Ad ∈ L(Y ′,X ′), Ady′ := x′ ∀ y′ ∈ Y ′.

In many situations it is important to characterize the invertibility of an induced A byappropriate properties of the bilinear form a(·, ·), in particular, the following inf–supcondition. This and the invertibility of A usually require boundary conditions to beincorporated into U ,V, see below and [386], p. 127, [135], p. 117. The following theoremis often applied to numerical methods. So it is sometimes important to distinguish theu ∈ U �= V � v. It is straightforward to generalize Hackbusch’s proof for his Lemma6.5.3 [387], from his A ∈ L(U ,U ′) form for Hilbert spaces U ,U ′ to our Banach spacesetting A ∈ L(U ,V ′).

Theorem 2.12. Brezzi–Babuska condition: Let U ,V be reflexive Banach spaces andA ∈ L(U ,V ′) be induced by the bounded a(·, ·) : U × V → R, see (2.47). Then the four


statements (2.51), (2.52), (2.53) and (2.54) are equivalent:

A−1 ∈ L(V ′,U) exists, hence C exists, s.t. ‖A−1‖U←V′ ≤ C, (2.51)

∃ ε, ε′ > 0 s.t. sup0�=v∈V

|a(u, v)|/‖v‖V ≥ ε‖u‖U ∀u ∈ U (2.52)

and sup0�=u∈U

|a(u, v)|/‖u‖U ≥ ε′‖v‖V ∀v ∈ V,

∃ ε > 0 s.t sup0�=v∈V

|a(u, v)|/‖v‖V ≥ ε‖u‖U ∀u ∈ U (2.53)

and sup0�=u∈U

|a(u, v)|/‖u‖U > 0 ∀v ∈ V, 0 �= v

∃ ε > 0 s.t sup0�=v∈V

|a(u, v)|/‖v‖V ≥ ε‖u‖U ∀u ∈ U (2.54)

and ∀ v ∈ V, v �= 0 ∃ u ∈ U : a(u, v) �= 0.

Each of the conditions (2.51)–(2.54) implies ‖A−1‖U←V′ ≤ 1/ε. If ε, ε′ are chosen asthe suprema in (2.52)–(2.54), then ‖A−1‖U←V′ = 1/ε.

If one of these (2.51)–(2.54) holds, then ∀ f ∈ V ′ ∃1 u0 ∈ U such that

Au0 = f ⇔ a(u0, v) = 〈Au0, v〉V′×V = 〈f, v〉V′×V ∀ v ∈ V. (2.55)

We do not give a proof, see [386], p. 127, [135], p. 117. But it is straightforward toshow that (2.52) or (2.53) are necessary for (2.51): If 0 �= w ∈ R(A)⊥ exists, choose0 �= v ⊥ w. Then a(w, v) = 0, contradicting (2.52), (2.53).

Remark 2.13. Injective A and the inf–sup condition:

1. The notation inf-sup condition is due to the equivalence of, e.g. the first line of(2.52) with

inf0�=u∈U

{sup

0�=v∈V{|a(u, v)|/‖v‖V}/‖u‖U

}≥ ε.

Some authors replace the above ≥ ε by = ε.2. Sometimes we refer to the second line of (2.53) as

sup{|a(u, v)| : u ∈ U , ‖u‖U = 1} > 0 for all 0 �= v ∈ V. (2.56)

3. The first line of the above condition (2.52) is directly related to the dual normof Au as

ε‖u‖U ≤ sup{|〈Au, v〉V′×V | : v ∈ V, ‖v‖V = 1} = ‖Au‖V′ . (2.57)

This inequality is necessary for an injective A, and for the existence of A−1.A combination with the second condition of the second line of (2.52) yields thebijectivity.


4. Often Theorem 2.12 is applied to a discrete Ah : Uh → V ′h. Then the lastestimate in (2.51) applied to ‖(Ah)−1‖Uh←V′h ≤ C yields the stability of Ah.

5. Under appropriate conditions, the first and second line of (2.52) are equivalent,see Proposition 2.19.

Sometimes a corollary of Theorem 2.12 is useful, obtained simply by rescaling.

Corollary 2.14. Let U ,V be reflexive Banach spaces and A ∈ L(U ,V ′) be induced bythe bounded a(·, ·) : U × V → R, see (2.47). Then the two statements (2.58) and (2.59)are equivalent:

A−1 ∈ L(V ′,U) exists, hence C exists, s.t. ‖A−1‖V′←U ≤ C, (2.58)

∃ ε, ε′ > 0 s.t. supv∈V,‖v‖V=1

|a(u, v)| ≥ ε ∀u ∈ U , ‖u‖U = 1, (2.59)

and supu∈U,‖u‖U=1

|a(u, v)| ≥ ε′ ∀v ∈ V, ‖v‖V = 1.

Obviously a V-coercive, hence bounded bilinear form a(·, ·) satisfies each of (2.52)–(2.54). Hence a combination of (1.33) and (2.52) shows:

Theorem 2.15. Coercivity implies invertibility; invertible A,Ad:

1. Let V be reflexive and the bounded a(·, ·) be either V-coercive, see (2.40), (2.41),or satisfy one of (2.52)–(2.54), thus A−1 ∈ L(V ′,V) exists, see Theorem 2.12.Then A and Ad are simultaneously invertible and

A,Ad ∈ L(V,V ′), ‖A‖V′←V = ‖Ad‖V′←V ≤ C ≤ Cb in (2.40), (2.60)

A−1, (Ad)−1 ∈ L(V ′,V), ‖A−1‖V←V′ = ‖(Ad)−1‖V←V′ ≤ C ′.

2. If a(·, ·) is V-coercive with α as in (2.41), then the previous C ′ ≤ 1/α, or if a(·, ·)satisfies one of (2.52)–(2.54), then C ′ ≤ 1/ε.

3. In particular, Au0 = f in (2.55) and its dual counterpart are uniquely solvable.

In the remainder of this section we study V-coercive and V-elliptic bilinear forms.Sometimes, Equation (2.55) is equivalent to a minimum problem:

Theorem 2.16. Extremal problem:

1. Let a(·, ·) be V-coercive, hence bounded, and symmetric, and f ∈ V ′ be given.Then there is a unique minimum u0 ∈ U = V for the extremal problem

J(u0) = inf{J(u) := a(u, u)− 2f(u) : ∀ u ∈ V}. (2.61)

2. This minimum is attained for the (unique) solution u0 of (2.55).

The elliptic differential operators in Sections 2.2 and 2.4 do not correspond tocoercive, but to more general, so-called elliptic bilinear forms. So we need:


Definition 2.17. Embeddings, Gelfand triple, elliptic bilinear forms:

1. Let V ⊂ W be Banach spaces and I : V ↪→W, Iv := v ∈ W ∀v ∈ V. This embed-ding I : V ↪→W is called dense and continuous or compact if V is dense in Wand I is continuous or compact, respectively

2. Let V be a Banach and W a Hilbert space, with6 W =W ′ identified. Then V ⊂W ⊂ V ′ is called a Gelfand triple if

V ↪→W =W ′ ↪→ V ′, V ↪→W is dense and continuous. (2.62)

3. Under this condition, the bounded bilinear form a(·, ·) : V × V → R is calledV-elliptic if there exist constants Cc, α ∈ R such that

a(u, u) ≥ α‖u‖2V − Cc‖u‖2W ∀ u ∈ V with α > 0. (2.63)

The second inequality is a so-called Garding inequality. For Cc = 0 we obtainagain a V-coercive bilinear form a(·, ·).

We will show in Subsection 2.4.4 that elliptic operators induce, under standardconditions, Hm(Ω)-elliptic bilinear forms.

In (2.62) only embedding properties of V ⊂ W are listed. Relation (2.62) impliesW ′ to be continuously embedded into V ′ for Banach spaces V,W, but V,W denselyand continuously into V ′ for Hilbert spaces, see [387], Corollary 6.3.10. Therefore, see(2.62), with V ↪→W =W ′ dense and continuous, we obtain for

Hilbert spacesV,W : (2.62) ⇒ V ↪→ W =W ′ ↪→ V ′ dense and continuous. (2.64)

For applications of partial differential equations the most important Gelfand triplesare those with compact embeddings. To apply the following Riesz–Schauder theorywe need criteria for compact operators, cf. Hackbusch [387], Theorem 6.4.10.

Proposition 2.18. In the Gelfand triple (2.62) let the embedding V ↪→W be compactand T ∈ L(V ′,V) be given. Then the dense embeddings in (2.62) allow us to modifythis T ∈ L(V ′,V), e.g. A−1 = T , yielding compact

T1 ∈ L(V ′,V ′), T2 ∈ L(V ′,W), T3 ∈ L(V,V), T4 ∈ L(W,V), T5 ∈ L(W,W).

Proof. The compact embedding I : V ↪→W implies the embedding I1 : W = W ′ ↪→ V ′

to be compact as well. If in a product of continuous operators at least one factor iscompact, the product is compact as well, see [387], Lemma 6.4.5. Now T1 = I1IT :V ′ → V →W ′ → V ′, similarly T2 = IT : V ′ → V →W =W ′, T3 = IT I1 : W =W ′ →V ′ → V →W =W ′, T4 = T I1 : W =W ′ → V ′ → V, T5 = T I1I : V → W = W ′ → V ′

→ V. �

It is interesting that for the most important Gelfand triples we can reduce theconditions in Theorem 2.12, cf. [387], Lemma 6.5.17:

Proposition 2.19. In a Gelfand triple, see (2.62), with reflexive Banach space Vlet the embedding V ↪→W be compact. Then in Theorem 2.12 the two conditions in

6 A Hilbert space W can always be identified with its dual W ′.


(2.52) are equivalent. In particular, A−1 ∈ L(V ′,V) is guaranteed by one of the twoequivalent lines of (2.52).

Later we will apply the Liapunov–Schmidt method. It requires results concern-ing kernels and ranges of linear(ized) operators. Important candidates are compactperturbations of boundedly invertible operators and operators induced by ellipticbilinear forms. This includes essentially all the following problems, e.g. Navier–Stokesoperators. The following Riesz–Schauder theory and its modification show that theseoperators behave in some sense similarly to finite dimensional linear operators. Infact, the Fredholm alternative is7 valid. We formulate it for the two cases indicatedabove. The form in Theorem 2.20 for the Riesz–Schauder theory is a combination ofHackbusch’s Theorem 6.4.12 and the Closed Range Theorem 1.19. For the many casesof operators induced by elliptic bilinear forms we modify this result in the followingTheorem 2.21.

Theorem 2.20. Riesz–Schauder theory: Let V be a Banach space, I ∈ L(V,V) denotethe identity and let T ∈ L(V,V) be compact, for T d ∈ L(V ′,V ′) cf. Remark 2.11.

1. For every λ ∈ C \ {0}, one of the following alternatives is valid:(i) (T − λI)−1 ∈ L(V,V);(ii) λ is an eigenvalue for T ⇔ ∃x �= 0 : Tx = λx.For (i) the (T − λI)x = y is uniquely solvable in V for every y ∈ V and

(T − λI)−1 ∈ L(V,V) ⇐⇒ (T d − λI)−1 ∈ L(V ′,V ′),

that is, both are simultaneously bijective and boundedly invertible, so

R(T − λI) = V, R(T d − λI) = V ′,

N (T − λI) = {0} ⊂ V, N (T d − λI) = {0} ⊂ V ′. (2.65)

For (ii) we find: λ is an eigenvalue for T ⇐⇒ λ is an eigenvalue for T d. Thenthere exist finite dimensional eigenspaces

E(T, λ) = N (T − λI) �= {0}, and E(T d, λ) = N (T d − λI) �= {0}with dim E(T, λ) = dim E(T d, λ) <∞ and Tx = λx ∀x ∈ E(T, λ). Then Tx−λx = y has at least one, and hence infinitely many solutions x ∈ V if and only if〈y, x′〉V×V′ = 0 ∀ x′ ∈ E(T d, λ).

2. The spectrum of T is defined as the set

σ(T ) := {λ eigenvalue for T} ∪ {0, if T−1 �∈ L(V,V)}.This σ(T ) is at most countable and the only possible limit point in C is 0. Thisis the reason for excepting λ = 0 here and in 1. We find σ(T d) = σ(T ) := {λ :λ ∈ σ(T )}.

7 A Fredholm operator, compare Definition 1.20, is a bounded linear operator A ∈ L(V,Y) withfinite dimensional kernel and co-range. The following A − λI and Ad − λI for λ ∈ C and its inversesusually requires the complexification of V,V′ and the operators A, Ad, see Theorem 1.22.


3. For (i), (ii) and λ �= 0, the closed ranges, R(T − λI),R(T d − λI), and kernels,N (T − λI) = E(T, λ), N (T d − λI) = E(T d, λ), are subspaces of the same finiteco-dimension and dimension, respectively. They are related as

R(T − λI) = (N (T d − λI))⊥, and R((T − λI)d) =⊥ (N (T − λI))(2.66)

equivalent to 〈y, x′〉V×V′ = 0 ∀ y ∈ R(T − λI), x′ ∈ E(T d, λ),

and 〈x, y′〉V×V′ = 0 ∀ x ∈ E(T, λ), y′ ∈ R((T − λI)d),

respectively. Hence, T − λI are Fredholm operators of index 0.

The last part of this theorem is a special case of the Closed Range Theorem 1.19and usually not included in the Riesz–Schauder theory. The relation between thisand the next theorem is mainly based upon the compact embedding V ↪→W. Thisimplies a compact embedding I : V ↪→ V ′ as well. Let A be induced by the abovebounded elliptic bilinear form a(·, ·). Then for large enough CK > 0 the A + CKI isboundedly invertible. So, T := (A + CKI)−1I : V → V is compact by Theorem 1.13.Thus Theorem 2.20 can be applied to this T − μI : V → V and some parts only haveto be slightly changed to yield, cf. [388], Theorem 6.5.15:

Theorem 2.21. Fredholm alternative: Theorem 2.20 remains correct with the fol-lowing modifications: Let V ⊂ W =W ′ ⊂ V ′ be a Gelfand triple, see (2.62), with aHilbert space W, a compact embedding V ↪→W (hence V ↪→ V ′ is compact as well)and with a reflexive Banach space V = V ′′ . Let a(·, ·) be bounded elliptic, and letA ∈ L(V,V ′) and I ∈ L(V,V ′) be the induced operator and the embedding (hence I iscompact), respectively, and Ad ∈ L(V,V ′) be the dual for A.

1. For every λ ∈ C (including λ = 0!) one of the following alternatives is valid:(i) (A− λI)−1 ∈ L(V ′,V);(ii) λ is an eigenvalue for A.For (i), (A− λI)u0 = f is uniquely solvable in V for every f ∈ V ′ with

(A− λI)−1 ∈ L(V ′,V) ⇐⇒ (Ad − λI)−1 ∈ L(V ′,V),

that is, both are simultaneously bijective and boundedly invertible, so

R(A− λI) = R(Ad − λI) = V ′,N (A− λI) = N (Ad − λI) = {0} ⊂ V. (2.67)

For (ii), Au0 − λu0 = f has at least one, and in fact dimE(Ad, λ) many solutionu0 ∈ V,, if and only if 〈f, ud〉V′×V = 0 ∀ ud ∈ E(Ad, λ) ⊂ V, and (2.67) is updatedaccording to (2.66).

2. The spectrum of A is at most countable and admits no limit points in C, and isdefined, differently from above, as

σ(A) := {λ eigenvalue for A}, and λ ∈ σ(A)⇔ λ ∈ σ(Ad).

3. For (i) and (ii) the closed ranges and kernels again satisfy (2.66) and A− λIand Ad − λI are Fredholm operators of index 0.


Remark 2.22.

1. With A− λI, I ∈ L(V,V ′), I compact and (A− λI)−1 ∈ L(V ′,V) the operator(A− λI)−1I ∈ L(V,V) is compact by Theorem 1.13.

2. Solving Au0 = f and (A− λI)u0 = f the above two cases have to be distinguished.The unique existence of a solution u0 for the problem Au0 = f and (A− λI)u0 =f is correct if and only if λ = 0 and λ is not an eigenvalue of the operatorA. If (A− λI)u0 = f has to be solved for an eigenvalue λ, finitely many, n′ :=dimN (A− λI), conditions have to be imposed for existence and uniqueness, cf.Theorem 1.19:

f ∈ R(A− λI) = (N (Ad − λI))⊥ ⇐⇒ 〈f, yj〉V′×V = 0, j = 1, . . . , n′, with

span {y1, . . . , yn′} = N (Ad − λI) ⊂ V, u0 ∈ V⊥,V = N (A− λI)⊕ V⊥.

If a special solution us for (A− λI)u0 = f is known, the set of general solutionsu0 is u0 ∈ {us +N (A− λI)}, similarly for (Ad − λI)u0 = f and N (Ad − λI).

3. Not all linearizations of the quasilinear equations, see Section 2.7, satisfy (2.63),particularly in Wm,p(Ω) for p > 2. Then Theorem 2.21 is applicable with appro-priate modifications.

The following two propositions are often useful, cf. [387], Lemmas 6.4.13 and 6.5.18:

Proposition 2.23. Let X ↪→ Y ↪→ Z be continuously embedded Banach spaces, andX ↪→ Y be compactly embedded. Then for every ε > 0 there exists Cε such that

‖u‖Y ≤ ε‖u‖X + Cε‖u‖Z ∀ u ∈ X . (2.68)

Proposition 2.24. Let a(·, ·), b(·, ·) be bilinear forms and a(·, ·) be V-elliptic, with theGelfand triple V ⊂ W =W ′ ⊂ V ′ and a reflexive Banach space V, see (2.63). Thena(·, ·) + b(·, ·) is again V-elliptic if b(·, ·) satisfies one of the following conditions:

1. For every ε > 0 there exists Cε such that, compare Proposition 2.23,

|b(u, u)| ≤ ε‖u‖2V + Cε‖u‖2W ∀ u ∈ V. (2.69)

2. Let the embeddings V ↪→ X ,V ↪→ Y be continuous and at least one of them becompact and let

|b(u, u)| ≤ Cb‖u‖X · ‖u‖Y ∀ u ∈ V. (2.70)

3. Let the embeddings V ↪→ X ,V ↪→ Y be continuous and (2.70) be satisfied. For‖ · ‖X or ‖ · ‖Y , we assume: For every ε > 0 a C

′ε exists such that, cf. (2.68),

‖u‖X ≤ ε‖u‖V + C ′ε‖u‖W or ‖u‖Y ≤ ε‖u‖V + C ′

ε‖u‖W ∀u ∈ V. (2.71)

This condition is guaranteed by Proposition 1.27.

We will come back to these bilinear forms in the case of Hilbert spaces, and linearoperators A of order 2m, see the following (2.133) ff., in Section 2.4.4, and to the


general case of Banach spaces in Section 2.7, particularly in Subsection 2.7.4, wherethe linearization of nonlinear differential operators is studied.

2.4 Linear elliptic differential operators, Fredholm alternativeand regular solutions

2.4.1 Introduction

The aim of this section is the definition of sufficiently general concepts of linear ellipticdifferential operators of order 2m in the Holder and Sobolev space settings. We needexistence, uniqueness, regularity results and Fredholm alternatives for the solutions,which can be combined with the nonlinear problems in Sections 2.5 ff. and numericalmethods in the following chapters. A linear differential operator A of order 2m, directlygeneralizing Section 2.2, is

A : V → V ′ with V = Hm(Ω) ⊂ W = L2(Ω) = W ′ ⊂ V ′ = H−m(Ω). (2.72)

But it makes sense to consider competing very successful approaches in Holderand general Sobolev spaces. We will present them in this section. In Section 2.2we introduced the classical, strong and weak operators Ac : C2(Ω) → C(Ω), As :H2(Ω) → L2(Ω) and A : H1(Ω) → H−1(Ω). We will generalize them to Holder andSobolev spaces and study elliptic operators Ac = As : C2m+k,γ(Ω) → Ck,γ(Ω) andAs : W 2m+k,p(Ω) →W k,p(Ω) in Subsection 2.4.2 under C∞-conditions for the coef-ficients of the differential and boundary operators and for Ω. In contrast to Section2.2 we do no longer use these nonstandard indices in Ac, As, A. The regularity resultsin this section show that the operators are essentially the same; only the spaces andthe smoothness conditions vary. We use the As, A only if necessary, e.g. when partialintegration and the Green’s formula require this notation. It is worthwhile realizingthat for second order equations the C∞-conditions can be relaxed to Ck-conditions andstill get A : C2+k,γ(Ω) → Ck,γ(Ω) results, summarized in Subsection 2.4.3. Finally, inSubsection 2.4.4 we study the direct generalization of Section 2.2 to elliptic operatorsof order 2m and formulate the necessary results for A : Hm(Ω) → H−m(Ω). In Lionsand Magenes [478] and Oden and Reddy [518], further generalizations are discussed.An essential difference between elliptic operators of order 2 and 2m is caused by thefact that only for order 2 is it always possible to transform the problem to homogeneousboundary conditions. Lemma 2.26 shows that for order 2m problems, m > 1, this isonly possible for Dirichlet boundary systems. General boundary operators for 2m aremore complicated and induce surprising effects, see Lions and Magenes [478], Odenand Reddy [518], and Runst and Sickel [561].

For defining linear elliptic operators of order 2m we use the standard notation formulti-indices, α, reals or vectors ϑα or ϑ and (partial) derivatives ∂αu. The notationfor the wide range of problems in the present chapter is a nightmare. Many differentpossibilities have been suggested in the literature. We choose relatively similarsymbols, ∂iu, ∂αu, ∂u, for partials, and ϑi, ϑα, ϑ, for reals or vectors. Let

2.4. Linear elliptic differential operators 55

α := (α1, . . . , αn) ∈ Nn0 , |α| := α1 + · · ·+ αn, α ! := (α1)! · · · (αn)! (2.73)

∂iu :=∂u

∂xi, ∂u := (∂1u, . . . , ∂nu), ∂αu := ∂α1

1 . . . ∂αnn u :=

∂|α|

∂xα11 . . . ∂xαn

nu,

∇ku := (∂αu)|α|=k,∇≤ku := (∂αu)|α|≤k, ∂0u := ∇0u := u,∇u := ∇1u, with

u(x), ∂iu(x), ∂αu(x) ∈ R, ∂u(x) ∈ Rn, ∇ku(x) ∈ Rnk ,∇≤ku(x) ∈ RNk , where

nk :=∑|α|=k

1, Nk :=k∑

j=0

nj , and Θk ∈ Rnk , Θ≤k ∈ RNk ,Θ0 ∈ R,Θ := Θ1,

ϑ := (ϑ1, . . . , ϑn) ∈ Rn, ϑα := (ϑ1)α1 · · · (ϑn)αn , ϑi ∈ R, i ≥ 0, ϑ0 = 1,

and sometimes Dku := ∇ku,D≤ku := ∇≤ku,D0u := ∇0u := D0u := u,Du := ∇1u.

Only in Subsections 2.5.4 and 2.5.7, 2.6.4 do we appropriately replace our notation(x,Θ0,Θ1) and (x,Θ0,Θ1,Θ2) by (x, z, p) and (x, z, p, r), respectively.

The standard Sobolev inner products, norms, and seminorms are defined as

(u, v)Hm(Ω) :=∑

|α|≤m

(∂αu, ∂αv)L2(Ω), and ‖v‖Hm(Ω) := ((u, u)Hm(Ω))1/2,

|v|Hm(Ω) :=

⎛⎝ ∑|α|=m

‖∂αv‖2L2(Ω)

⎞⎠1/2

, ‖v‖W m,p(Ω) :=

⎛⎝ ∑|α|≤m

‖∂αv‖pLp(Ω)

⎞⎠1/p

.(2.74)

As a consequence of the polynomial formula the inequalities8

1 ≤(mα

):=(

mα1 · · ·αn

):=

m!α1! · · ·αn!

≤ Cm,n∀m ∈ N, α ∈ Nn0 , ϑ ∈ Rn,

imply |ϑ|2mn /Cm,n ≤ |ϑ|2m

n,α :=∑

|α|=m

(ϑ)2α

=∑

|α|=m

(ϑ1)2α1 · · · (ϑn)2αn (2.75)

≤∑

|α|=m

(m

α1 · · ·αn

)(ϑ1)2α1 · · · (ϑn)2αn

=

(n∑

i=1

(ϑi)2)m

= |ϑ|2m

= |ϑ|2mn =

(n∑

i=1

(ϑi)2)m

= ≤ Cm,n

∑|α|=m

(ϑ)2α, |ϑ|n Euclidean norm in Rn.

8 As a consequence of (2.75), we can use, in (2.79), here and throughout, either |ϑ|n,α or |ϑ|n withappropriately modified λ(x), Λ(x).


Definition 2.25. Linear elliptic differential operators of order 2m and differentboundary conditions:

1. A linear differential operator of order 2m has the strong form

Au :=∑

|α|≤2m

aα∂αu with real valued coefficients aα; (2.76)

for the weak form A we prefer (2.102). The symbol S for A

SA(x, ϑ) := S(x, ϑ) :=∑

|α|≤2m

aα(x)ϑα, (2.77)

is often called the characteristic polynomial for A. The principal part Ap is

Apu :=∑

|α|=2m

aα(x)∂αu. (2.78)

2. Generalizing (2.20), the principal part of the symbol of an elliptic A (or of −A)has to satisfy, in D ⊂ Ω and ∀ϑ ∈ Rn, x ∈ D,

λ(x)|ϑ|2mn ≤ (−1)m

∑|α|=2m

aα(x)ϑα ≤ Λ(x)|ϑ|2mn ∀ϑ ∈ Rn, x ∈ D. (2.79)

3. This A is called elliptic and strongly elliptic and uniformly elliptic, in D,if 0 ≤ λ(x) and 0 < ε ≤ λ(x) and 0 < ε ≤ λ(x),Λ(x)/λ(x) bounded ∀ x ∈ D,respectively.For aα ∈ L∞(D) or aα,β ∈W |β|,∞(D) in (2.102), and aα,β ∈ L∞(D) in (2.110)this condition is only required almost everywhere in D.

4. The most important types of boundary operators, in the trace sense, see Remark2.1 and Theorem 1.37, are imposed ∀x ∈ ∂Ω or in complementing subsets ∂Ω =∂Ω1 ∪ ∂Ω2, and are denoted as

Dirichlet operator BDu := (∂iu/∂νi)m−1i=0 (2.80)

Neumann operator ∂iu/∂νi ,m ≤ i ≤ 2m− 1, and (2.81)

General operator Bju :=∑

|α|≤mj

bj,α∂αu of order mj < 2m, j = 1, · · · ,m,

(2.82)

with real bj,α, e.g. the induced Ba or, for m = 1 : u,∂u

∂ν, b0u +

∂u

∂ν. (2.83)

5. The system (2.82) is called a Dirichlet system if mj = j − 1, j = 1, . . . ,m. It iscalled a normal system if 0 ≤ m1 < · · · < mm and SBj

, the symbol of Bj , satisfiesSBj

(x, ν) �= 0, ν = ν(x) the normal vector ∀x ∈ ∂Ω, j = 1, . . . ,m. A boundaryvalue problem is defined by the corresponding pair (A,B = (Bj)m

j=1), with one ofthe previous B = (Bj)m

j=1.


Here and for our numerical methods we essentially require the case of uniformlyelliptic equations, hence

0 < ε ≤ λ(x), Λ(x)/λ(x) ≤ ∞ ∀x ∈ Ω. (2.84)

If (2.84) is violated, then Tychonov regularization, here with (−1)mδ∑

|α|=2m ∂αu,

δ > 0, may be used for δ → 0. Inequality (2.84) is violated usually at boundary points.Then for numerical methods, these violating boundary points may be cut off andapproximate boundary conditions may be imposed. The case of nonuniformly ellipticequations is indicated in Subsection 2.5.3, and systematically studied by, e.g. Ivanov[415].

By a standard trick, inhomogeneous boundary conditions can be transformed intohomogeneous boundary conditions. If u1 and u2 satisfy the prescribed inhomogeneouslinear conditions, then the difference u1 − u2 satisfies the homogeneous conditions.For a Dirichlet system this transformation is possible for m ≥ 1 due to the followinglemma. In fact, the proof for Lemma 5.1 in Oden and Reddy [518] can be slightlygeneralized and extended, yielding in particular, the estimate in (2.86):

Lemma 2.26. Transformation into homogeneous boundary conditions: Let (2.82)be a Dirichlet system with bj,α ∈ Cm−j+1(Ω), ∂Ω ∈ Cm and (gj)m

j=1 with gj ∈Wm−j+1−1/p,p(∂Ω), j = 1, . . . ,m. Then there exists u1 ∈Wm,p(Ω) such that in thetrace sense, see Remark 2.1 and Theorem 1.37 ,

Bju1 − gj = 0 in ∂Ω, j = 1, · · · ,m, and (2.85)

‖u1‖W m,p(Ω) ≤ C‖(g1, . . . , gm)‖Πmj=1W m−j+1−1/p,p(∂Ω). (2.86)

With any u2 ∈Wm,p(Ω), such that Bju2 = gj in ∂Ω, j = 1, · · · ,m, we obtainBj(u1 − u2) = 0 in ∂Ω, j = 1, . . . ,m.

As a consequence of Theorem 1.37, we can extend g0 ∈W k−1/p,p(∂Ω) to g0 ∈W k,p(Ω). This allows us, for the most important special case, m = 1, to relax theprevious condition ∂Ω ∈ C1.

Lemma 2.27. Transformation to homogeneous conditions for m = 1: Choose m = 1,a Lipschitz boundary ∂Ω ∈ C0,1, and the Dirichlet condition with u = g0 on ∂Ω, withp <∞. Then for g0 ∈W k−1/p,p(∂Ω), a u1 ∈W 1,p(Ω) exists, such that in the tracesense, u1 − g0 = 0 in ∂Ω. Then again (u1 − u2)− g0 = 0 in ∂Ω.

Corresponding lemmas for Dirichlet systems for systems of equations are valid aswell, so we study mainly trivial Dirichlet boundary conditions.

For operators A of higher order m > 1, the problem of defining induced and wellfitting boundary operators Ba is much more complicated than for m = 1. We will comeback to this in Subsection 2.4.2 (2.92) ff. and Subsection 2.4.4. In particular, for m > 1the transformation of inhomogeneous into homogeneous boundary conditions is onlypossible in exceptional cases. So nonhomogeneous boundary conditions are studied byLions and Magenes, [478–480].


2.4.2 Linear operators of order 2m with C∞ coefficients

The topic of this subsection are strong operators of the form (2.76) with C∞ coefficientsand boundary. An adequate existence, uniqueness and regularity theory is not knownfor A : C2(Ω) → C(Ω), but only for A : C2+s,γ(Ω) → Cs,γ(Ω), 0 < γ < 1, s ∈ N0. Weessentially formulate the results in Holder and Sobolev spaces, C2m+s,γ(Ω) andW 2m+s,p(Ω). For extensions we refer to Runst and Sickel [561]. Instead of thestandard C2m(Ω) spaces we use the corresponding Holder spaces Ck,γ(Ω), wherek = 2m + s, with the norms

‖u‖Ck(Ω) :=∑|α|≤k

supx∈Ω

|∂αu(x)| with k ∈ N0, γ ∈ R, 0 < γ < 1, and

‖u‖Ck,γ(Ω) := ‖u‖Ck(Ω) +∑|α|=k

supy �=x,y∈Ω

|∂αu(x)− ∂αu(y)||x− y|γn

with (2.87)

Ck(Ω) := {u : Ω → R : ‖u‖Ck(Ω) <∞}, Ck,γ(Ω) := {u s.t. ‖u‖Ck,γ(Ω) <∞}.

We study BVPs of the form (A,B), see (2.76)–(2.82), with the solution u0

A : C2m,γ(Ω) → Cγ(Ω) � f, Bj : C2m,γ(Ω) → C2m−mj ,γ(∂Ω) � gj , (2.88)

Au :=∑

|α|≤2m

aα∂αu, B := (Bj)m

j=1, Bju :=∑

|α|≤mj

bjα∂αu, mj < 2m,

Au0(x) = f(x)∀x ∈ Ω, Bu0(x) = 0 or = g(x) = (gj(x))mj=1∀x ∈ ∂Ω, (2.89)

for u, u0 ∈ C2m,γ(Ω) or u, u0 ∈ C2m+k,γ(Ω), compare (2.93).Instead of this complicated situation one certainly would prefer a simpler setting.

For the case 2m = 2 and Dirichlet boundary conditions we would aim for a solutionu0 ∈ C2(Ω) ∩ C(Ω), u0|∂Ω = 0 for, e.g.

2m = 2 : A := −Δ : C2(Ω) → C(Ω), −Δu0(x) = f(x)∀x ∈ Ω. (2.90)

It is known that this solution u0 is unique and it exists, if the Green functionexists for Ω, see Hackbusch [387], Theorems 3.1.2, 3.2.11, 3.2.13. This u0 does notdepend continuously upon f with respect to the sup norms in C2(Ω) and C(Ω). So‖u0‖C2,γ(Ω) ≤ C‖f‖Cγ(Ω), 0 < γ < 1 is correct in the Holder norms, see e.g. Theorem2.30 below, but it is wrong for the norms ‖u0‖C2(Ω), ‖f‖C(Ω), compare Hackbusch’sTheorem 3.2.15. We get for (2.90) an estimate with the “unusual” norm ‖u0‖C(Ω)

instead of ‖u0‖C2(Ω). More generally, let A be strongly elliptic on a bounded Ω withcoefficients aα ∈ C(Ω), a0 ≤ 0 in Ω. Then there exists a C ∈ R+, see [387], Theorem5.1.9 and Exercise 5.1.10d, such that

‖u0‖C(Ω) ≤ C‖f‖C(Ω)∀f ∈ C(Ω). (2.91)

These are two reasons for studying and returning to (2.88) and (2.89).For the following results we require A to be a uniformly elliptic differential operator

in Ω. The B have to “fit” to A, a condition much more complicated for m > 1 than for


m = 1. The coefficients of A,B and the boundary have to be in C∞, more precisely,

A,B in (2.88) satisfy aα ∈ C∞(Ω), bjα ∈ C∞(∂Ω) ∀|α| ≤ 2m, (2.92)

j = 1 . . . ,m, ∂Ω ∈ C∞, A is strongly (so uniformly) elliptic,

(Bj)mj=1 satisfies the complementary condition, e.g. Dirichlet boundary

conditions for m ≥ 1, see (2.80), or for m = 1, let b0(x) �= 0, see (2.83),

compare Theorem 2.50 below. These complementary conditions and the relatedconcept of normal boundary value problems9 are intensely studied, e.g. by Lionsand Magenes [478], pp. 113 ff., and Oden and Reddy [518], pp. 152 ff. Equivalent(nonalgebraic) formulations via initial value problems of ODEs are described byWloka [667], pp. 150 ff. and Taylor [620], pp. 379 ff. They are often handled moreeasily. As an example Wloka [667] shows that for a strongly elliptic operator A as in(2.76) with m = 1 the Dirichlet and natural boundary conditions are complementaryand normal. The same is correct for the Laplacian −Δ combined with (2.83). Forthe biharmonic operator (Δ)2, hence m = 2, several cases are considered, see [667]and (2.88). So m1 = m2 = 0 is neither normal nor complementary. m1 = 0,m2 = 1 isnormal and complementary if b10 × ((b1α)|α|=1, ν)n(x) �= 0∀x ∈ ∂Ω. This includes theDirichlet boundary conditions u|∂Ω = g1, (∂u/∂ν)|∂Ω = g2.

For the following regularity results we choose smoother Banach spaces

X := C2m+k,γ(Ω), Y := Ck,γ(Ω), Z := C(Ω), with (2.93)

X0 := {u ∈ X : Bu|∂Ω = 0} for k ∈ N0, γ ∈ R, 0 < γ < 1.

So this X0 with homogeneous boundary conditions corresponds to our V above. Com-pared to (2.88) we change the spaces in A : X → Y, Bj : X → Yj := C2m+k−mj ,γ(Ω),but use the same notation A,B. The following results in Theorems 2.28–2.33 and thelast statement in 2.35 are due to Zeidler [676], Chapter 6.3, and Problems 6.8, p. 259.They remain valid for the A,B in (2.98), (2.101).

Theorem 2.28. Fredholm alternative, regularity:

1. Under the conditions (2.92), (2.93) the elliptic operator A ∈ L(X0,Y) is aFredholm operator of index i = d− c := dimN (A)− codimR(A) <∞. This i,determined by the pair A,B, is independent of k. This remains unchangedif in A and Bj terms of order < 2m and < mj are changed, respectively.This remains correct for all the combinations in this section. For many

9 Complementary condition: Let SA(x, ϑ) and SBj(x, ϑ), see (2.77), denote the principal parts of

the symbols for A and Bj , respectively. These polynomials correspond to the terms with the highestderivatives. For all x ∈ ∂Ω let ν = ν(x) denote the outer normal in x ∈ ∂Ω, let ϑ �= 0 lie in the tangentplane of ∂Ω in x and let A be an elliptic operator with real coefficients. Then SA(x, ϑ + zν(x))is a polynomial in z ∈ C of degree 2m. It can be shown that it admits exactly m solutions zk ∈C with positive imaginary part ∀x ∈ ∂Ω. Let P (z) := Πm

k=1(z − zk). Then the system A, Bj , j =1, . . . , m satisfies the complementary condition if the polynomials SBj

(x, ϑ + zν(x)) in z are linearly

independent modulo P (z), i.e. divide the SBj(x, ϑ + zν(x)) by P (z). Then the remainders Sr

Bj(x, ϑ +

zν(x)) of degree < m of this division are linearly independent.


important cases i vanishes, e.g. compare the conditions in Theorem 2.50below.

2. If index(A) = i = 0 and dimN (A) = 0 then A ∈ L(X0,Y) is boundedly invertible,hence ∃1u0 for Au0 = f and ‖u0‖X0 ≤ C‖f‖Y .

3. Finally, the Fredholm alternative in Theorem 2.21 for Au = f has to be modified:There exist c = codimR(A) and d = dimN (A) linearly independent elementsf∗

i ∈ Y∗ ⊥ R(A), i = 1, . . . , c and ui ∈ N (A) ⊂ X0, i = 1, . . . , d such that: Eitherf∗

i (f) �= 0 for at least one i ∈ {1, . . . , c}. Then Au = f is not solvable. Other-wise Au = f is solvable. If us is a special solution, the general solution u0 isobtained as

u0 = us +d∑

i=1

ciui with ci ∈ R.

Remark 2.29.

1. Compared to Theorem 2.21 the eigenvalue formulation does not make sensehere. For d > 0 we have N (A) �= 0 ∀ λ. So the unique existence of an exactsolution u0 for any f ∈ Y, g ≡ 0 ∈ Πm

j=1Yj in (2.89) is correct if and only ifN (A|X0) = N (A) = {0} and index i = 0.

2. The above Remark 2.22 has to be updated for the new situation. The linear spaceV in Theorem 2.21 usually corresponds to our X0 with homogeneous boundaryconditions, see above and Lions and Magenes [478], Oden and Reddy [518],Wloka [667], and Runst and Sickel [561], Section 3.5.3; cf. Section 8.6 in thisbook as well.

3. Some results for the index i can be found in these books as well.

The next theorems summarize a priori estimates and regularity results for arbitraryfunctions u ∈ X0 and for solutions u0 ∈ C2m(Ω). In part this is related to the last lineof Theorem 2.21 or to Theorem 2.45:

Theorem 2.30. Estimates in X0: For (2.92), (2.93) any u ∈ X0 satisfies

‖u‖X ≤ C(‖Au‖Y + ‖u‖Z) and ‖u‖X ≤ C‖Au‖Y for N (A) = {0}. (2.94)

For N (A) = {0} the A : X0 = C2m+k,γ(Ω)) → R(A) is boundedly invertible. If addi-tionally index i = 0 then R(A) = Y = Ck,γ(Ω) with A−1 ∈ L(Y,X0) and A−1 ∈L(Y,Y) is compact.

Theorem 2.31. Regularity in X0: For (2.92), (2.93) with g = 0 in (2.89) anysolution u0 for Au0 = 0 satisfies u0 ∈ C∞(Ω). For f ∈ Y in (2.93), let the classicalsolution u0 of (2.89) be in C2m(Ω). Then again u0 ∈ X0.


Now we turn to problems with inhomogeneous boundary conditions. Choose

X , Y, Z, as in (2.88), (2.93), Yj := Ck+2m−mj ,γ(∂Ω), and (2.95)

Au(x) = f(x)∀x ∈ Ω, Bu(x) = g(x) = (gj(x))mj=1 ∀ x ∈ ∂Ω. (2.96)

Theorem 2.32. Estimates, regularity and Fredholm operator in X :

1. Let (A,B) ∈ L(X ,Y ×Πm

j=1Yj

)be a normal boundary value problem and satisfy

(2.92), (2.95), (2.96). Then it defines a Fredholm operator of index i. It allowsus the a priori estimate

‖u‖X ≤ C

⎛⎝‖Au‖Y + ‖u‖Z +m∑

j=1

‖Bju‖Yj

⎞⎠ ∀u ∈ X and

‖u‖X ≤ C

⎛⎝‖Au‖Y +m∑

j=1

‖Bju‖Yj

⎞⎠ for N (A,B) = {0}. (2.97)

For N (A,B) = {0} and i = 0, the (A,B) : X → R(A,B) is boundedly invertibleand we find R(A,B) = Y ×Πm

j=1Yj .

2. If the classical solution u0 of (2.96) is in C2m,γ(Ω), and f ∈ Y, gj ∈ Yj , j =1, . . . ,m with k > 0 in (2.95), then u0 ∈ X with k in (2.93),(2.95).

Now we replace the operators A := A,B := (Bj)mj=1 in (2.88), with the Holder spaces

in (2.93), (2.95), by the following Sobolev spaces:

A : W 2m,p(Ω) → Lp(Ω), Bj : W 2m,p(Ω) →W 2m−mj−1/p,p(∂Ω), 1 < p <∞, (2.98)

and the corresponding spaces for homogeneous boundary conditions

X := W k+2m,p(Ω),X0 := {u ∈ X : Bu = 0},Y := W k,p(Ω),Z := L1(Ω)

and for inhomogeneous boundary conditions: (2.99)

X ,Y as in (2.99), Z := Lp(Ω), Yj := W k+2m−mj−1/p,p(∂Ω). (2.100)

We define solutions, for details see Subsections 2.4.4, 2.5.6 and Zeidler [676],Problems 6.8, p. 261, as

u0 ∈W 2m,p(Ω), Au0 = f in Ω, Bu0 = 0 or g = (gj)mj=1 on ∂Ω. (2.101)

Theorem 2.33. Results for the Sobolev case: In Theorems 2.28–2.32 and the oper-ators in (2.88), replace the Holder spaces (2.93), (2.95) by the Sobolev spaces (2.98),(2.99), (2.100). Correspondingly, replace the homogeneous and inhomogeneous bound-ary value problems (2.89), (2.96) by (2.101), require (2.92) and a normal boundary


value problem. Finally, in Theorems 2.30–2.32 replace the conditions for Cj,γ(Ω) in(2.93), (2.95) by W j,p(Ω) in (2.99), (2.100). Then Theorems 2.28–2.32 remain validin this modification and the index remains unchanged.

This astonishing similarity of the results is due to the fact that Holder and Sobolevspaces are special cases of the so-called Besov–Triebel–Lizorkin spaces. For these spacesthe above results are correct, see Runst and Sickel [561].

For our later applications to numerical methods we reformulate the original lineardifferential operator in (2.76) into the so-called divergence form.10 The aη ∈ C∞(Ω)and ∂η in (2.88) and (2.73) define appropriate aα,β ∈ C∞(Ω) with

Au :=∑

|η|≤2m

aη∂ηu =

∑|α|,|β|≤m

(−1)|β|∂β(aα,β∂αu), (2.102)

e.g. with aη = (−1)maα,β for α + β = η and |α| = |β| = m.In Subsection 2.4.3 we discuss, for 2m = 2, general boundary operators, see (2.82).

However here, to avoid too many technicalities, we restrict the discussion to specialDirichlet systems, see Theorem 2.50 and (2.159) below, satisfying

on ∂Ω for j = 1, . . . ,m : Bju :=∑

|α|≤j−1

bj,α(x)∂αu = gj ⇐⇒ ∂j−1u/∂νj−1 = gj

⇐⇒ (∂αu)|α|≤m−1 = (gα)|α|≤m−1 for ∂Ω ∈ Cm−1 and gj ∈ Cm−j . (2.103)

For more general cases combine Oden and Reddy [518], pp. 184ff. and Zeidler [677],pp. 331 ff. By Lemma 2.26 we use the standard trick for modifying f := f in (2.101)for obtaining homogeneous boundary conditions

Au(x) = f(x) ∀ x ∈ Ω, (∂j−1u/∂νj−1)mj=1 = 0 ∀ x ∈ ∂Ω, (2.104)

or abbreviated into u ∈Wm,p0 (Ω). So (2.88), (2.89), (2.92) has the simplified form for

the aη or aα,β ,

A in (2.102) or (2.102) satisfies aη or aα,β ∈ C∞(Ω)∀|η| ≤ 2m or |α| = |β| = m,

A is strongly (so uniformly) elliptic and ∂Ω ∈ C∞. (2.105)

With (2.79), (2.102), uniform ellipticity means, noting that ϑα ∈ R, ϑ ∈ Rn, |ϑ|n ∈ R,

γ|ϑ|2mn ≤

∑|α|,|β|=m

aα,β(x)ϑαϑβ ≤ Γ|ϑ|2mn for 0 < γ,Γ <∞ ∀ x ∈ D. (2.106)

The special form of f in (2.107) is proved by Adams [1], pp. 47 ff.

10 As in the literature, we use different notations for the standard and the divergence form asindicated, e.g. in (2.102) systematically with consequences for the different ellipticity conditions;compare, e.g. (−1)m in (2.79) versus (2.106).


Proposition 2.34. Continuous forms, f ∈ V ′b = W−m,p′

0 (Ω) : For 1 < p <∞, 1/p +1/p′ = 1, we obtain a special unique representation:

〈f, v〉V′b×Vb

:=∫

Ω

∑|β|≤m

fβ∂βvdx ∀ v ∈ Vb := Wm,p

0 (Ω) with (2.107)

fβ ∈ Lp′(Ω) and ‖f‖V′ =

⎛⎝ ∑|β|≤m

‖fβ‖p′

Lp′ (Ω)

⎞⎠1/p′

. (2.108)

For p = 1 or p =∞ (2.108) has to be replaced by ‖f‖V′ := minfβ

(∑|β|≤m . . .

)1/p′

.

For f ∈ V ′ = W−m,p′(Ω) we obtain more generally

〈f, v〉V′×V :=∫

Ω

∑|β|≤m

fβ∂βvdx +

∫∂Ω

∑|β|<m

ϕβ∂βvds ∀ v ∈ V := Wm,p(Ω) with

ϕβ ∈ Lp′(∂Ω) and ‖f‖V′ =

⎛⎝ ∑|β|≤m

‖fβ‖p′

Lp′ (Ω)+∑

|β|<m

‖ϕβ‖p′

Lp′ (∂Ω)

⎞⎠1/p′

. (2.109)

Note that for f ∈ V ′ = W−m,p′(Ω) the f0 := f |

W−m,p′0 (Ω)

is reduced to an f0 ∈W−m,p′

0 (Ω)as in (2.107). This is the reason that the following

a(u0, v) = 〈Au0, v〉V′×V = 〈f, v〉V′×V ∀v ∈ V = Hm0 (Ω) � u0

tested ∀v ∈ Hm0 (Ω) makes sense for f ∈W−m,p′

(Ω) in (2.109) as well.

We combine the standard partial integration with Green’s formula:

Theorem 2.35. Solutions for the divergence form:

1. Assume the form (2.102), the conditions (2.103)–(2.106) and define the bilinearform a(u, v), the principal part ap(u, v), and the linear f as in (2.107)

a(u, v) := 〈Au, v〉V′×V :=∫

Ω

∑|α|,|β|≤m

aα,β∂αu∂βvdx, (2.110)

ap(u, v) := 〈Apu, v〉V′×V :=∫

Ω

∑|α|=|β|=m

aα,β∂αu∂βvdx,∀u, v ∈ V = Hm

0 (Ω).

These a(u, v), ap(u, v), 〈f, v〉V′×V are continuous.2. Any solution u0 ∈ Ck+2m,γ(Ω) or u0 ∈W k+2m,p(Ω) ⊂ Hm(Ω), k + m ≥ n(1/p−

1/2) of (2.104) necessarily satisfies the standard equation

u0 ∈ Hm0 (Ω) : a(u0, v) = 〈f, v〉V′×V∀v ∈ V = Hm

0 (Ω). (2.111)


Vice versa, under the conditions (2.92) in (2.93) and (2.99), (2.100), everysuch solution u0 ∈ Hm

0 (Ω) even satisfies u0 ∈ Ck+2m,γ(Ω) and u0 ∈W k+2m,p(Ω),respectively.

Theorem 2.36. Ellipticity, and Fredholm alternative for the divergence form: Inaddition to the conditions in Theorem 2.35 let the coefficients aα,β with |α| = |β| = min (2.102), (2.106), be continuous in Ω. Then for a uniformly elliptic A in (2.105) theap(u, v) is Hm

0 (Ω)-coercive. For an Hm0 (Ω)-coercive principal part ap(u, v), the above

a(u, v) is an Hm0 (Ω)-elliptic bilinear form. So the Fredholm alternative in Theorem

2.21 applies, with index (A,B) = 0. The unique solution of the variational problem(2.61) for ap(u, v) satisfies (2.111) for ap(u, v).

Proof. a(u, v)− ap(u, v) are lower order terms. So a combination of (2.63) andProposition 2.24 shows that since for aα,β ∈ C(Ω) with |α| = |β| = m the ap(u, v)by (2.105) is Hm

0 (Ω)-coercive, see Theorem 2.43 below, the a(u, v) is Hm0 (Ω)-elliptic

and the claim is proved, compare Theorems 2.89, 2.104. �

2.4.3 Linear operators of order 2 under Ck conditions

In Subsection 2.4.2 we studied results under strong C∞-conditions for coefficients andthe boundary of the differential and boundary operators. For the most importantspecial case of second order equations it is possible to relax this C∞-condition andstill get very similar results. We list the essential regularity results and the Fredholmalternative in Gilbarg and Trudinger [346]. It is appropriate to arrange, see (2.76), theaα and ∂α as ai,j , ∂

0u = u and ϑ0 := 1, ϑi, ϑj ∈ R, i, j = 0, . . . , n. With ν, the outernormal unit vector for ∂Ω, the differential and one of the three boundary operators(2.88) and (2.89), the boundary value problem has the form

Au :=n∑

i,j=0

aij∂i∂ju : C2(Ω) → C(Ω), (2.112)

Bu = B1u = u, or B2u = ∂u(x)/∂ν, or B3u = b0u +n∑

i=1

ci∂iu on ∂Ω, (2.113)

determine u0 ∈ C2,γ(Ω) such thatAu0 = f ∀x ∈ Ω, Bu0 = ϕ∀ x ∈ ∂Ω. (2.114)

Then the λ(x) and Λ(x) in (2.79) are the minimal and the maximal (real) eigenvaluesof the (symmetric) matrix Ap defining the different types of ellipticity:11

Ap(x) := (ai,j(x))ni,j=1 ∀ x ∈ Ω and (2.79) has the form (2.115)

λ(x)|ϑ|2n ≤ −n∑

i,j=1

aij(x)ϑiϑj ≤ Λ(x)|ϑ|2n ∀ϑ ∈ Rn. (2.116)

11 Note the − sign in (2.106) compared to [346]; formula (6.2) yields a00(x) ≥ 0 in (2.118).


For the following results we require A to be a uniformly elliptic differential operatorwith smooth coefficients, boundary and inhomogeneities; more precisely, let

A,B in (2.112), (2.113) satisfy f, aij ∈ Cγ(Ω), b0, cj , ϕ ∈ C1,γ(Ω), 0 < γ < 1,

∀i, j = 0, . . . , n, ∂Ω ∈ C2,γ , for B1 let ϕ ∈ C2,γ(Ω), for B3and c = (c1, . . . , cn) let

|(c(x), ν(x))n| > κ > 0 ∀ x ∈ ∂Ω, A is strongly (so uniformly) elliptic, (2.117)

cf. (2.26), (2.33). The following results are proved by Gilbarg and Trudinger [346],see Theorems 6.14, 6.15, 6.17, 6.19, 6.30, 6.31 and the last lines in Section 6.7. Wesummarize the existence, uniqueness and regularity results in the interior Ω, its closureΩ and the Fredholm alternative:

Theorem 2.37. Existence, (non-)uniqueness and estimates: Let the operators A,Bin (2.112), (2.113) satisfy (2.116), (2.117), and

a00(x) ≥ 0 ∀ x ∈ Ω for B1u and for B3u with b0(x) > 0 ∀ x ∈ ∂Ω,

a00(x) > 0 ∀ x ∈ Ω for B2u. (2.118)

1. Then (A,B) : C2,γ(Ω) → Cγ(Ω)× C1,γ(∂Ω) for B = B2, B3 and (A,B) :C2,γ(Ω) → Cγ(Ω)× C2,γ(∂Ω) for B = B1.

2. If λ = 0 is not an eigenvalue, the (A,B)u = (f, ϕ) is boundedly invertible, hencethere exists a unique solution u0 ∈ C2,γ(Ω) for (2.114).

3. If λ = 0 is not an eigenvalue, e.g. for violated (2.118), (A,B) is not boundedlyinvertible, any solution u0 ∈ C2,γ(Ω) for (2.114) can be estimated as, compare(2.94),

‖u‖C2,γ(Ω) ≤ C(‖u‖C(Ω) + ‖f‖Cγ(Ω) + ‖ϕ‖C1,γ(Ω)) for B = B2, B3, but (2.119)

‖u‖C2,γ(Ω) ≤ C(‖f‖Cγ(Ω) + ‖ϕ‖C1,γ(Ω)) for a boundedly invertible (A,B).

For B = B1 the ‖ϕ‖C1,γ(Ω) in (2.119) has to be replaced by ‖ϕ‖C2,γ(Ω).

Theorem 2.38. Regularity in Ω: For the A,B1 in (2.112), (2.113) modify (2.117)by replacing Ω by Ω, Cγ by Ck,γ-conditions for the coefficients, no conditions forthe boundary and ϕ, and assume that u0 ∈ C2(Ω) solves the Dirichlet boundary valueproblem (2.114). Then u0 ∈ Ck+2,γ(Ω); note in Ω. Here k = ∞ is allowed.

Corresponding results for weak solutions u0 ∈ H10 (Ω) are formulated in Subsection

2.4.4. Regularity in Ω requires additional conditions for the boundary.

Theorem 2.39. Regularity in Ω: For the operator A,B1 in (2.112), (2.113) modify(2.117) by replacing only the Cγ , C2,γ-conditions by Ck,γ , Ck+2,γ , let ∂Ω be in Ck+2,γ

and assume that u0 ∈ C0(Ω) ∩ C2(Ω) solves the Dirichlet boundary value problem(2.114), see Theorem 2.37. Then u0 ∈ Ck+2,γ(Ω).

Note that (2.118) is not imposed in the last two theorems. Corresponding u0 ∈W k+2,p=2(Ω) = Hk+2(Ω) results are formulated in Subsection 2.4.4, cf. [346],Chapter 8.


Theorem 2.40. Fredholm alternative: Let the operators A,B in (2.112), (2.113)satisfy (2.117). Then

A ∈ L(X0,Y) with X0 := {u ∈ C2,γ(Ω) : Bu = 0}, Y := Cγ(Ω), (2.120)

is a Fredholm operator of index i = 0, satisfying the Fredholm alternative inTheorem 2.21.

As in the last subsection we want to relate the original boundary value problem(2.112), (2.113) to the corresponding strong divergence form. In particular, we aimfor existence and uniqueness results and the Fredholm alternative for different com-binations of second order equations (2.112), and boundary conditions (2.113). Withaij := aij in (2.112), and appropriate new aij , let, see (2.16),

Au := Asu :=n∑

i,j=0

aij∂i∂ju =

n∑i,j=0

(−1)j>0∂j(aij∂

iu) : C2(Ω) → C(Ω),

aij ∈ C1,γ(Ω), ai0 ∈ Cγ(Ω) i, j = 0, . . . , n, aij = −aij , i, j ≥ 1. (2.121)

In this Subsection we have studied u ∈ Ck+2,γ(Ω), so u ∈ H2(Ω). We modify (2.117)by (2.121) and get instead of (2.116) for the uniform ellipticity for (2.121)

λ′|ϑ|2n ≤n∑

i,j=1

aij(x)ϑiϑj ≤ Λ′|ϑ|2n ∀ϑ ∈ Rn, x ∈ Ω for 0 < λ′. (2.122)

As in (2.24) the strong and weak bilinear forms are related, with A = As in (2.121):

a(u, v) =∫

Ω

n∑i,j=0

aij∂iu∂jvdx =

∫Ω

(As u)vdx +∫

∂Ω

(Ba u)vds (2.123)

with Ba u :=j=1,...,n∑i=0,...,n

νjaij∂i u and the principal parts

ap(u, v) =∫

Ω

n∑i,j=1

aij∂iu∂jvdx =

∫Ω

(As,p u)vdx +∫

∂Ω

(Ba,p u)vds (2.124)

with Ba,p u :=n∑

i,j=1

νjaij∂i u so Ba −Ba,p = ha :=

n∑j=1

νja0j ,

where here Ba and Ba,p are the natural boundary operators induced by A and Ap,respectively The special Au = −Δu + cu induces Ba u = Ba,p u = ∂u/∂ν.

We discuss the mutual dependency of coercivity properties and boundary conditionsfor the first (Dirichlet), the second or natural (generalized Neumann) and the third(generalized Robin) boundary value problem: With a function h ≥ h0 on ∂Ω andVi, f−i, i = 1, 2, 3, below let, cf. (2.123)–(2.124),

Bu = B1u = u,B2u = Ba u,B3u = Ba,p u + hu on ∂Ω and determine (2.125)

u0 ∈ H2(Ω) ∩ Vi such that Au0 = f−i a.e.∀x ∈ Ω, Biu0 = ϕ∀ x ∈ ∂Ω, i = 1 or 2 or 3.


These generalized Neumann and Robin boundary operators, B2 and B3, modify thestandard forms ∂u/∂ν and au + b∂u/∂ν.

For different boundary conditions in (2.124), (2.125), compared to (2.113), we aimfor compact perturbations of the original bilinear forms a(u, v). They will turn out tobe coercive for the above boundary operators Bi and the chosen function spaces Vi, i =1, 2, 3, cf. (2.126) ff. For i = 1, the principal part has this property; for i = 2, 3, it has tobe modified. These Vi-coercive ai

c(u, v) yield unique existence and are combined withthe Fredholm alternative for the Vi-elliptic a(u, v), and the linear forms 〈f−i, v〉V′×V ,cf. Theorems 2.20 and 2.21. The Vi-coercive ai

c(u, v) induce the linear operators Aic as

compact perturbations of A induced by the original a(u, v) in (2.121). We consider:

BVPs: Aicu = f−i ∈ V ′ in Ω, Biu|∂Ω = ϕ, i = 1, 2, 3, and ∀v ∈ V := Vi. Let for

i = 1 : a1c(u, v) := ap(u, v) = 〈f−1, v〉V′×V V := V1 := H1

0 (Ω), (2.126)

with 〈f−1, v〉V′×V :=∫

Ω

(f0v +

∑n

j=1fj∂

jv)dx, B1u = u = 0 for the first BVP,

i = 2 : a2c(u, v) := ap(u, v) + c(u, v)L2 = 〈f−2, v〉V′×V , B2u = Bau, c > 0, (2.127)

〈f−2, v〉V′×V := 〈f−1, v〉V′×V +∫

∂Ω

ϕvds,V := V2 := {v ∈ H1(Ω) : Bav = 0}

i = 3 : a3c(u, v) := ap(u, v) +

∫∂Ω

huvds = 〈f−2, v〉V′×V ,V3 := H1(Ω), (2.128)

B3u = Ba,p u + hu, f−3 := f−2,with h(x)|∂Ω ≥ h0 > 0 for the third BVP.

The case i = 2 and the two alternatives have to be discussed. For Ba 1 = 0 and〈f2, 1〉 = 0 with ha in (2.124), ap(u, v) = 〈f2, v〉∀v ∈ H1(Ω), Ba u = 0 and itsdual problem admit the nontrivial solution u0 ≡ 1, thus the Fredholm alternativeapplies. Enforcing unique existence for Ba 1 = 0 = 〈f2, 1〉 is possible by definingV2 := {v ∈ H1(Ω) : 〈1, v〉V′×V = 0}. This excludes the nontrivial solution u0 ≡ 1.Then the equivalent Sobolev norms in Proposition 1.25 allow the followinggeneralization of Zeidler [677], Sections 22.2 ff. for (2.126)–(2.128).

Theorem 2.41. Unique existence, and Fredholm alternative for (2.126)–(2.128):

1. Assume the form (2.121), the conditions (2.122), (2.125), for Ba, c, h, f−i, anddefine ai

c(u, v),V = Vi, and 〈f−i, v〉V′×V as in (2.126)–(2.128). Then these bilin-ear and linear forms are continuous ∀u, v ∈ V and the ai

c(u, v) are Vi-coercivefor i = 1, 2, 3 with the conditions for Ba, ha, f−i listed in (2.126)–(2.128). Thenunique solutions ui

0 exist for

∃1ui0 = u0 ∈ H1(Ω) : ai

c(u0, v) = 〈f−i, v〉V′×V , Biu0 = ϕ,∀v ∈ Vi = V. (2.129)

The unique solution of the minimal problems (2.61) for these aic(u, v) satisfies

the variational problem (2.129).2. Define the corresponding ai(u, v) by replacing the principal part of a(u, v) by

the above aic(u, v) and impose the previous conditions. Then the ai(u, v) are


Vi-elliptic bilinear forms. So the Fredholm alternative in Theorem 2.21 appliesto (A,Bi), even with index (A,Bi) = 0, i = 1, 2, 3.

3. Any solution u0 ∈ Ck+2,γ(Ω) ⊂ H1(Ω) of (2.125) with the original ai(u, v), andthe previous 〈f−i, v〉V′×V satisfies the standard equation

u0 ∈ H1(Ω) : ai(u0, v) = 〈f−i, v〉V′×V∀v ∈ V = Vi, Biu0 = ϕ, i = 1, 2, 3.

Vice versa, under the conditions (2.117), (2.120), every such solution u0 ∈ Veven satisfies u0 ∈ Ck+2,γ(Ω).

Proof. As in Theorem 2.35 obviously the a(u, v), aic(u, v) are bounded, except for i = 3,

and the 〈f−iv〉, except for i = 2, 3, in H1(Ω). For the exceptional cases, Remark 2.1and the Trace Theorem 1.38 for v ∈ H1(Ω) imply v|∂Ω ∈ L2(∂Ω)

‖v‖2L2(∂Ω) =∫

∂Ω

v2ds ≤ C‖v‖2H1(Ω) =⇒ |∫

∂Ω

vϕds| ≤ C‖ϕ‖L2(∂Ω)‖v‖H1(Ω),

and hence the boundedness of all the 〈f−i, v〉V′×V , i = 1, 2, 3. For i = 3 we estimate

|∫

∂Ω

huvds| ≤ ‖h‖C(∂Ω)

∫∂Ω

|uv|ds (2.130)

≤ ‖h‖C(∂Ω)‖u‖L2(∂Ω)‖v‖L2(∂Ω) ≤ C1‖h‖C(∂Ω)‖u‖H1(Ω)‖v‖H1(Ω).

Finally, by Proposition 1.25, the(ai

c(u, u))1/2 define norms equivalent to ‖u‖Vi

, i =1, 2, 3.

The following proof, that the ac(u, v) := aic(u, v) are Vi-coercive, is a straightforward

generalization of Zeidler [677], pp. 331 ff. Relations (2.125), (2.123), (2.124) imply

0 =∫

Ω

As u0vdx− 〈f−1, v〉V′×V +∫

∂Ω

(Bi u0 − ϕ)vds (2.131)

= a(u0, v)− 〈f−1, v〉V′×V +∫

∂Ω

((Bi −Ba)u0 − ϕ)vds

= ac(u0, v)− 〈f−1, v〉V′×V +∫

∂Ω

((Bi −Ba)u0 − ϕ)vds

+ (a(u0, v)− ac(u0, v)).

Now we drop the lower order terms (a(u0, v)− ac(u0, v)) and are left with ac(u0, v).Then (2.121), (2.122) imply, with ha(x) ≥ 0,

ac(u, u) =∫

Ω

n∑i,j=1

aij∂iu∂ju + hau

2dx ≥ γ

∫Ω

n∑i=1

(∂iu)2dx + minx∈Ω

{ha(x)}‖u‖2L2(Ω)

= γ‖∂u‖2L2(Ω) + minx∈Ω

{ha(x)}‖u‖2L2(Ω) ≥ γi‖u‖2H1(Ω), ∀u ∈ Vi, i = 1, 3.

(2.132)

The last inequality is correct for i = 1, since the norms ‖ac(u, u)‖1/2 and ‖u‖H1(Ω) areequivalent on V1 = H1

0 (Ω). For the different combinations of Vi, i = 2, the Vi-coercivity


of ac(u, u) is a consequence of the discussion preceding Theorem 2.41, based upon thedefinition of the Vi and by, Proposition 1.25, the equivalence of the norms ‖u‖H1(Ω)

and (ac(u, u) + |∫Ωudx|2)1/2. For i = 3 it is correct with γ3 ≥ min{γ, h0}.

Again ai(u, v)− aic(u, v) are only lower order terms. Since ai

c(u, v) is Vi-coercive,a combination of (2.63) and Proposition 2.24 implies a V-elliptic ai(u, v). That thesesolutions satisfy the differential equation and the boundary conditions is shown asindicated in Remarks 2.6 and 2.7. A combination of (2.63) and Proposition 2.24 impliesa V-elliptic ai(u, v), and thus the claim. �

We discuss in more detail the implications of the Fredholm alternative for one ofthe above examples. We consider for i = 2 the special case, cf. Section 8.6,

a(u, v) =∫

Ω

n∑i=1

∂iu∂ivdx =∫

Ω

∇u∇vdx ∀v ∈ V2, inducing Bau = ∂u/∂ν,

with a(1, v) = 0 ∀v ∈ V2, Ba1 = 0 implying N (A) = N (Ad) = R.Consequently, a (nontrivial) solution u0 for

a(u0, v) = 〈f−2, v〉V′×V =∫

Ω

f0vdx +∫

∂Ω

ϕvds ∀v ∈ V2,

exists, by the Fredholm alternative, iff 〈f−2, 1〉V′×V = 0 and every u0 + c, c ∈ R solvesthis problem. Otherwise the trivial solution exists only for the homogeneous problema(u0, v) = 0, f−2 = 0.

As we have seen, enforcing a uniquely existing solution for Ba 1 = 0 = 〈f2, 1〉is possible by defining, according to the Fredholm alternative, V2 := {v ∈ H1(Ω) :∫Ωvdx = 0}. This excludes the nontrivial solution u0 ≡ 1.

2.4.4 Weak elliptic equation of order 2m in Hilbert spaces

In Subsection 2.4.2, we studied strong and weak forms of elliptic operators in generalSobolev spaces Wm,p(Ω), 1 < p <∞, under continuity conditions for the coefficients ofthe differential and boundary operators. The results in this subsection remain correct,in particular with respect to Hm(Ω)-coercivity with appropriate modifications, forWm,p(Ω), 2 ≤ p ≤ ∞ as well, see below. This is most important in our context for thelater linearization of operators and their stability, cf. Section 2.7 and Corollary 2.44.Regularity results in textbooks are usually formulated in the Hm+s(Ω) setting.

Beyond the previous Wm,p(Ω) results, we turn therefore in this subsection to weakforms in Hilbert spaces Hm(Ω) = Wm,2(Ω). This modifies the approach in Section 2.2for studying differential operators of order 2m. We test the differential operator oforder 2m with elements in Hm(Ω), often in Hm

0 (Ω), and

A : V → V ′, V = Hm(Ω) ⊂ W = L2(Ω) = W ′ ⊂ V ′ = H−m(Ω), (2.133)

with Ω satisfying (2.5) and the scalar product (u, v)L2(Ω). For Sobolev spaces, seeSubsection 1.4.3, we use the scalar product and norm in (2.74). In this context it is


appropriate to reformulate Au in divergence12 form and the corresponding ellipticitycondition.

Definition 2.42. Divergent linear differential operator: For aαβ ∈W |β|,∞(Ω) andaαβ ∈ L∞(Ω) the strong and weak form A = As and A are, see Definition 2.25,

As : H2m(Ω) → L2(Ω), Asu :=∑

|α|,|β|≤m

(−1)|β|∂β(aαβ∂αu) (2.134)

and A : Hm(Ω) → H−m(Ω), defined by

a(u, v) := 〈Au, v〉H−m(Ω)×Hm(Ω) :=∫

Ω

∑|α|,|β|≤m

aαβ∂αu∂βvdx. (2.135)

The symbol, S, principal part, Ap, and the different types of ellipticity for A, aredefined as obvious modifications of (2.77)–(2.79), possibly in D ⊂ Ω, as

λ(x)|ϑ|2mn ≤

∑|α|=|β|=m

aα,β(x)ϑαϑβ ≤ Λ(x)|ϑ|2mn ∀ϑ ∈ Rn, a.e. x ∈ D. (2.136)

We repeat the three sets of m boundary conditions imposed in the trace sense, seeRemark 2.1 and Theorem 1.37, ∀x ∈ ∂Ω, or different conditions in complementarysets:

Dirichlet conditions: ∂iu/∂νi = 0 ∀ 0 ≤ j ≤ m− 1, (2.137)

Neumann conditions: ∂iu/∂νi = 0 ∀ m ≤ j ≤ 2m− 1, (2.138)

General conditions: Bju :=∑

|α|≤mj

bj,α∂αu, order mj < 2m. (2.139)

Natural conditions: for m = 1 : Ba u :=j=1,...,n∑i=0...,n

νjaij∂i u. (2.140)

The latter are generalized to m > 1, in Lions and Magenes [478], p. 120, and Oden andReddy [518], p. 290 ff., see below. For smooth ∂Ω, the Dirichlet boundary conditions(2.137) are equivalent to, analogously for Wm,p

0 (Ω),

u ∈ V := Hm0 (Ω) = {u ∈ Hm(Ω) : ∂αu(x) = 0 ∀|α| ≤ m− 1, x ∈ ∂Ω} for Ω ∈ Cm−1.

In (2.41) and (2.63) we had introduced the concepts of V-coercive and V-ellipticbilinear forms. We recall that, e.g. for V = Hm

0 (Ω) the bounded bilinear form a(·, ·) :V × V → R is called V-coercive, and V-elliptic, respectively, if constants α ∈ R+, Cc ∈R exist such that, e.g. for V = Hm

0 (Ω),W = L2(Ω)

a(u, u) > α‖u‖2V and ≥ α‖u‖2V − Cc‖u‖2W ∀u V. (2.141)

The second inequality is a so-called Garding inequality.

12 The (−1)|β| in (2.134) usually yields an opposite sign compared to (2.76) and (2.112). This

(−1)|β| produces the standard sign for a(·, ·) in (2.135).


For the following coercivity results for the principal part Ap, we have to imposeeither the Dirichlet conditions on a subset ∂Ω1 of ∂Ω of positive measure, hencemeasn−1(∂Ω1) :=

∫∂Ω1

1ds > 0, such that ∂Ω = ∂Ω1 ∪ ∂Ω2, otherwise for naturalboundary conditions the coercivity results holds for Ap + c, c > 0. We impose condi-tions on the aαβ in (2.135) and for Ω as in (2.5) for guaranteeing the Hm

0 (Ω)-ellipticityof the a(·, ·) in (2.135). We summarize Hackbusch’s [387] Theorems 7.2.2, 7.2.3, 7.2.11,7.2.13, Lemma 7.2.12 and extend his Theorem 7.2.7. Some of the following results havealready been formulated under different assumptions:

Theorem 2.43. Garding: V-coercivity and ellipticity, Fredholm alternative forDirichlet and natural boundary conditions: Let A in (2.134) have a strongly ellipticprincipal part, see (2.79), and satisfy aαβ ∈ L∞(Ω). Then

1. For m ≥ 1, and aαβ ∈ L∞(Ω), the a(·, ·) in (2.135) are bounded.2. For m = 1, the principal part ap(·, ·) (or for a00(x) ≥ 0 the ap(u, v) +

∫Ωa00uvdx)

and a(·, ·) are H10 (Ω)-coercive and H1

0 (Ω)-elliptic, respectively3. For m > 1 let, additionally, the aαβ be continuous in Ω for |α| = |β| = m. Then

ap(·, ·) and a(·, ·) are Hm0 (Ω)-coercive and Hm

0 (Ω)-elliptic, respectively4. For boundedly invertible A : H1

0 (Ω) → H−10 (Ω) or H1

0 (Ω)-coercive a(u, v) weobtain

∀f ∈ H−10 (Ω) : ∃1u0 ∈ H1

0 (Ω) : Au0 = f

⇐⇒ a(u0, v) = 〈f, v〉H−10 (Ω)→H1

0 (Ω)∀v ∈ H10 (Ω). (2.142)

5. If in a(·, ·) = a′(., .) + a′′(., .) the a′(., .) is Hm0 (Ω)-coercive or -elliptic and

a′′(u, v) =|α|+|β|<2m∑|α|,|β|≤m

∫Ω

aαβ∂αu∂βvdx,

contains only derivatives of orders ≤ 2m− 1, then a(·, ·) is Hm0 (Ω)-elliptic.

6. These results remain correct, if V = Hm0 (Ω) is replaced by Hm

0,p(Ω) ⊂ Hm(Ω)

V := Hm0,p(Ω) = {u : ∂αu(x) = 0 ∀|α| ≤ m− 1, x ∈ ∂Ω1 ⊂ ∂Ω} (2.143)

for, e.g. a connected ∂Ω1 ∈ Cm−1 with measn−1(∂Ω1) =∫

∂Ωds > 0. In ∂Ω2 :=

∂Ω \ ∂Ω1 natural or other boundary conditions and possibly a modified functional,f , cf. (2.38), might be imposed.

7. For natural boundary conditions, explicitly formulated in Section 2.2 only form = 1, the ap(·, ·) and V = Hm

0 (Ω) are replaced by the Hm(Ω)-coercive ap(u, v) +∫Ωc0uvdx with c0 ≥ ε > 0 and V by Hm(Ω). Then the previous results remain

correct.

Vice versa, if aαβ ∈ L∞(Ω) holds and aαβ ∈ C(Ω) for |α| = |β| = m, then theHm

0 (Ω)-ellipticity of a(u, v) implies a strongly elliptic principal part, see (2.79).


Proof. For m = 1, we indicate the nearly trivial proof. The strong ellipticity condition(2.136) yields, for ∂iu(x) = ϑi ∈ R, ϑ = (ϑ1, . . . , ϑn) ∈ Rn,

ε|∇u(x)|2n = ε|ϑ|2n ≤n∑

i,j=1

aij(x)ϑiϑj =n∑

i,j=1

aij(x)∂iu(x)∂ju(x) ∀ x ∈ Ω. (2.144)

Integration over Ω yields a(u, u) ≥∫Ωε|∇u(x)|2dx ≥ ε‖u(x)‖2H1(Ω) by the equivalence

of the semi- or energy-norm and ‖ · ‖H1(Ω) in H10 (Ω).

For m > 1, Fourier transforms, cf. Hackbusch’s [387] proof of Theorem 7.2.7, arecombined with the technique of frozen coefficients. In Theorem 2.104 we will give theproof for systems for the case of constant (frozen) aαβ for |α| = |β| = m, and aαβ ≡ 0for 0 ≤ |α|+ |β| ≤ 2m− 1. We do not give all the details, cf. Wloka [667], p. 282 ff. orHormander [403], but present only the main idea. By the continuity of the aαβ in Ωwe can choose, for an ε > 0, a finite number of points xi ∈ Ω, i = 1, . . . , N, and theirneighborhoods Uxi

such that ∀|α|+ |β| = 2m

Ω ⊂ ∪Ni=1Uxi

with ac,iαβ := aαβ(xi) :

∥∥∥ac,iαβ − aαβ

∥∥∥C(Uxi

)< ε ∀i = 1, . . . , N. (2.145)

We choose a first generation of the i such that, after an appropriate renumbering,Uxi

∩ ∂Ω �= ∅, i = 1, . . . , N1 ≤ N. Then the standard technique allows us to prove, forthe strongly elliptic principal part, see (2.79), and for u ∈ Hm

0 (Ω), the coercivity ofap on each of the Uxi

, i = 1, . . . , N1. Now choose the next inner generation of Uxi

intersecting the union of the first generation, so

Uxi∩(∪N1

i=1 Uxi

)�= ∅ ∀i = N1 + 1, . . . , N2.

The normal derivatives ‖∂ku/∂νk‖Hm−k−1/2(∂Uxi), i = 1, . . . , N1, k = 0, . . . ,m− 1 can

be estimated by Remark 2.1 and the Trace Theorem 1.38, (1.67), by ‖u‖Hm(Uxi), i =

1, . . . , N1. Each of these ∂ku/∂νk|Hm−k−1/2(∂Uxi), k = 0, . . . ,m− 1, can be employed

proving, for each Uxi, i = N1 + 1, . . . , N2 in the next generation, the coercivity of ap

for the above nontrivial boundary values ‖∂ku/∂νk‖Hm−k−1/2(∂Uxi), k = 0, . . . , m− 1,

etc.The other claims are proved in [387]. We only indicate the case of natural boundary

conditions. Then the strong elliptic principal part and∫Ωc0u

2dx induce norms equiv-alent to the seminorm in Hm(Ω) and to L2(Ω), their sum induces a norm equivalentto that in Hm(Ω), hence ap(u, u) +

∫Ωc0u

2dx is Wm,p(Ω)-coercive. �

Corollary 2.44. V-coercive and V-elliptic a(·, ·) induce a boundedly invertible A andimply the Fredholm alternative, respectively, cf. Theorem 2.21:

1. For, V = Wm,p(Ω) or V = Wm,p0 (Ω), and a V-coercive a(·, ·), cf. 3., 4 below,

Theorem 2.15 guarantees the unique existence of a solution u0 for

Au0 = f ∈ V ′, cf. (2.135),(2.107). (2.146)


If a(·, ·) is even symmetric, it solves, by Theorem 2.16, the minimal problem(2.61) as well.

2. For V-elliptic a(·, ·), e.g. induced by an elliptic operator A with V-coerciveprincipal part ap(·, ·), Theorem 2.21 implies the Fredholm alternative with indexA = 0. So, for an eigenvalue 0 of A, infinitely many solutions for Au0 = fexist under additional conditions for f , otherwise exactly one solution exists and‖A−1‖V←↩V′ <∞, hence

∀ f ∈ V ′ ∃1 solution u0 ∈ V s.t. Au0 = f ∈ V ′ and ‖u0‖V ≤ C‖f‖V′ . (2.147)

3. All Hm(Ω)-coercivity and ellipticity based existence, uniqueness and Fredholmresults, including the numerical stability of all our discretization methods, remaincorrect with the following modifications: As a consequence of the continuousembedding of the Wm,p(Ω) ↪→ Hm(Ω), 2 ≤ p ≤ ∞, cf. Theorem 1.26, ellipticoperators defined on Wm,p(Ω) have a Hm

0 (Ω)-coercive principal part. For theHm(Ω)-coercivity we add μ

∫Ωuvdx with sufficiently large μ.

4. These Wm,p(Ω), 2 ≤ p ≤ ∞, results are most important for the later linearizationof operators and their stability, cf. Section 2.7. Consequently, we will get forour discretization methods stability and convergence with respect to the discreteHm(Ω) norms for 2 ≤ p ≤ ∞. This applies to each of the different cases inTheorem 2.43.

In fact, these last claims are a consequence of the continuous embeddings. So letthe bilinear form a(u, v) be Hm(Ω)-coercive. Then the embeddings

Wm,p(Ω) ↪→ Hm(Ω) ↪→Wm,q(Ω), 2 ≤ p ≤ ∞, 1 ≤ q ≤ 2, imply (2.148)

‖u‖W m,q(Ω) ≤ C‖u‖Hm(Ω) ≤ C ′‖u‖W m,p(Ω) for C,C ′ ∈ R+.

So for 1 ≤ p ≤ 2, the Wm,p(Ω)-coercivity would be a consequence of the Hm(Ω)-coercivity. Unfortunately the bilinear forms, cf. (2.135), usually are not defined forthis case. For 2 ≤ p ≤ ∞ we still obtain Hm(Ω)-coercivity, and well-defined bilinearforms.

Again regularity results are valid for smooth or convex Ω with corners and for theinterior of Ω and under conditions relaxed compared to Subsections 2.4.2, 2.4.3. Forour Theorems 2.45–2.49 see [386], Theorems 9.1.16, 9.1.21, 9.1.22, 9.1.26. For a(·, ·)and its coefficients we impose the conditions:

Let a(·, ·)in (2.135) be Hm0 (Ω)− elliptic, and ∀α, β with |α|, |β| ≤ m (2.149)

∂γaα,β ∈ L∞(Ω) ∀ γ with |γ| ≤ max{0, t + |β| −m} for t ∈ N,

or for t �∈ N let aα,β ∈ Ct+|β|−m(Ω) for |β| > m− t, else aα,β ∈ L∞(Ω),

finally, let Ω ∈ Ct+m in (2.5) for some t ≥ 0.

Theorem 2.45. Regular solutions and estimates:

1. Let a(·, ·) in (2.135) and its coefficients satisfy (2.149). Let s ≥ 0 and for t ∈N let s + 1/2 �∈ {1, 2, . . . ,m} and 0 ≤ s ≤ t or for t �∈ N let 0 ≤ s < t. Every (not


necessarily unique) weak solution u0 ∈ Hm0 (Ω) of

a(u0, v) = 〈f, v〉V′×V ∀ v ∈ V = Hm0 (Ω), for f ∈ H−m+s(Ω), (2.150)

see Corollary 2.44, is in Hm+s(Ω) ∩Hm0 (Ω) and satisfies the estimate

‖u0‖Hm+s(Ω) ≤ Cs

(‖f‖H−m+s(Ω) + ‖u0‖Hm(Ω)

). (2.151)

2. If u0 ∈ Hm(Ω) solves the inhomogeneous Dirichlet problem

∂lu0/∂νl = ϕl ∈ Hm+s−l−1/2(∂Ω), l = 0, . . . ,m− 1, (2.152)

then again u0 ∈ Hm+s(Ω) and we have to add to (2.151) the last term

‖u0‖Hm+s(Ω) ≤ Cs

(‖f‖··· + ‖u0‖··· +

m−1∑l=0

‖ϕl‖Hm+s−l−1/2(∂Ω)

). (2.153)

3. Specifically, for m = 1 and f ∈ H−1+s(Ω), let u0 ∈ V = H1(Ω) solve the inho-mogeneous natural boundary value problem, cf. (2.150),

a(u0, v) = 〈f, v〉V′×V∀v ∈ V, f ∈ H−1+s(Ω), Bau0 = ϕ ∈ Hs−1/2(∂Ω). (2.154)

Then u0 ∈ H1+s(Ω) satisfies, cf. (2.151), (2.153)

‖u0‖H1+s(Ω) ≤ Cs

(‖f‖H−1+s(Ω) + ‖u0‖H1(Ω) + ‖ϕ‖Hs−1/2(∂Ω)

).

Theorem 2.46. Regular solutions and estimates under weaker conditions, Necas [509,510]: Let Ω in C0,1, and a(·, ·) in (2.135) and its coefficients satisfy (2.149) for 0 <s < t ≤ 1/2 and f ∈ H−m+s(Ω). Then every (not necessarily unique) weak solutionu0 ∈ Hm

0 (Ω) of (2.150) is in Hm+s(Ω) ∩Hm0 (Ω) and satisfies the estimate (2.151).

Theorem 2.47. Regular solutions for convex Ω: Let Ω in (2.5) be convex, henceΩ ∈ C0,1 and a(·, ·) in (2.135) and its coefficients satisfy (2.149) for the case |α|+|β| < 2m, 1 = t ∈ N, and

aα,β ∈ C0,1(Ω) for |α| = |β| = m. (2.155)

Then the above result (2.151) is valid for s = 1.

The transition from the weak to the strong form, so from Au0 = f ∈ H−m(Ω) toAsu0 = f0 ∈ L2(Ω), is only possible for smooth enough coefficients and right-handside. So for m = 1 and the boundary condition Bau0 = ϕb this requires instead off ∈ H−1(Ω) the special form, cf. (2.109),

〈f, v〉V′×V :=∫

Ω

f0vdx +∫

∂Ω

ϕbvds ∀ v ∈ V := H1(Ω).

For our later studies of so-called nonconforming methods we have to be able torestrict the normal derivatives or more general boundary operators of functions to theboundary of or to interior curves in Ω. Then Remark 2.1 and Theorem 1.39 consider


the continuous tracing γ∂αu in the sense

Ω ∈ Ct, and |α|+ 1/2 < s ≤ t ∈ N, or s < t �∈ N =⇒ (2.156)

γ∂α ∈ L(Hs(Ω),Hs−|α|−1/2(∂Ω)).

Theorem 2.48. Traces of u ∈ Hs(Ω) to Hs−|α|−1/2(∂Ω): Let Ω0 ∈ C0,1, u ∈H3/2+ε(Ω), and |α| ≤ 1. Then γ∂αu ∈ Hε(∂Ω).

Related results are found in, e.g. Taylor [618], Chapter 4, Proposition 4.5; Jonssonand Wallin [425]; Jerison and Kenig [419]; Schwab [575]; and Grisvard [373].

For the analysis of nonconforming methods we restrict the discussion to secondorder problems, so to m = 1. Then we usually assume either H2(Ω) or H3/2+ε(Ω), cf.Remark 2.1.

In contrast to Theorem 2.45 the following result relates data and interior regularity,cf. [618], Chapter 5, Theorem 11.1.

Theorem 2.49. Interior regularity: Let Ω0 ⊂⊂ Ω1 ⊂⊂ Ω in (2.5), let a(·, ·) in (2.135)be Hm

0 (Ω)-elliptic. Let the coefficients satisfy (2.149) with Ω replaced by Ω1, s ≥ 0, andt ≥ s ∈ N or t > s. For f |Ω1 ∈ H−m+s(Ω1) let u0 be a weak solution of (2.150). Thenu0|Ω0 ∈ Hm+s(Ω0) satisfies a modified (2.151),

‖u0‖Hm+s(Ω0) ≤ C(s,Ω0,Ω1,Ω)(‖f‖H−m+s(Ω1) + ‖u0‖Hm(Ω)

). (2.157)

The following result, answering the question of existence and uniqueness for a largeclass of boundary value problems, is a simple consequence of the equivalence of normsin Sobolev spaces. We only indicate its proof and assume

A special Dirichlet system Bju :=∑

|α|≤j−1

bj,α(x)∂αu hence (2.158)

m1 = 0 < m2 = 1 < · · · < mm = m− 1, and let bj,α ∈ L∞(Ω).

Theorem 2.50. Modified boundary conditions:

1. Let a(·, ·) in (2.135) be Hm0 (Ω)-elliptic, cf. Theorem 2.43. Assume (2.158) such

that the principal part of the boundary operator, the Bp,j, defined by the highestderivatives ∂α, |α| = j − 1, multiplied by the bj,α, in the boundary operator yieldsthe rearranged13 form (2.159). The normal and the complementary tangentialderivatives ∂j−1u/∂νj−1 and ∂βu/∂ tβ , cf. (2.26), (2.33), are split as

Bp,ju(x) :=∑

|α|=j−1

bj,α(x)∂αu = bj(x)∂j−1u

∂νj−1+∑

|β|=j−1

cj,β(x)∂βu

∂ tβ

0 �= bj ∈ C(∂Ω), j = 1, · · · ,m, bj unique. (2.159)

13 The rearranged form is obtained by a transformation of the original coordinates into a systemof ν and orthogonal elements t1, . . . , tn−1 in the tangential plane of ∂Ω. The ∂αu : |α| = j − 1 are

transformed into ∂j−1u/∂νj−1 and the complementary ∂βu/∂ tβ , and ∂ tβ = ∂ tβ11 · · · ∂ t

βn−1n−1 ∂ νβn

with β1 · · · + βn−1 > 0.


Then for a given g = (Bju = gj)mj=1 unique ϕj exist, such that, in the trace sense,

g = (Bju = gj)mj=1 ⇐⇒ (∂ju)/∂νj = ϕj , j = 0, · · · ,m− 1.

2. The a(·, ·) in (2.135) is simultaneously Hm0 (Ω)- and Hm

B (Ω)-elliptic with

HmB (Ω) := {u ∈ Hm(Ω) : Bj(u)(x) = 0 ∀x ∈ ∂Ω, j = 1, · · · ,m}.

3. A combination with the index, regularity and embedding results in Theorems 2.28and 2.45, 2.47 and 1.26, respectively, shows: index A = 0 for a Hm

0 (Ω)-elliptica(·, ·) in (2.135) and for all the linear operators, A, induced by a(·, ·), studied inthis section.

Proof. By Theorem 2.43 it suffices to prove the claim for the principal part of thedifferential and boundary operators. Obviously the Dirichlet conditions in the tracesense, see Remark 2.1 and Theorem 1.37, (∂ju/∂νj) = 0, j = 0, . . . ,m− 1, determine∂αu = 0, |α| ≤ m− 1, and hence the conditions Bj(u) = 0. This remains correct fornonhomogeneous boundary conditions as well.

Vice versa we prove by induction that Bj(u) = 0 or = g imply the homogeneousor inhomogeneous Dirichlet conditions (∂j(u)/∂νj) = 0 or = ϕj , j = 0, · · · ,m− 1 : infact, B1(u) = 0 or = g1 is equivalent to u = 0 or = (g1/b1). This information allows usto compute all the tangential partial derivatives ∂αu/∂ tα, |α| = 1 along ∂Ω, vanishingfor homogeneous Dirichlet conditions. We combine these known ∂αu/∂ tα, |α| = 1 withB2(u) = 0 or = g2 determining the ∂u/∂ν, again vanishing along ∂Ω for homogeneousDirichlet conditions. Next, we combine the known u, ∂u/∂ν, ∂αu/∂ tα, |α| = 1, fordetermining in a first step all the ∂βu/∂ tβ with |β| = 2. Then B3(u) = 0 or = g3 yields∂2u/∂ν2 along ∂Ω, etc. Finally we use Lemma 2.26 to transform the inhomogeneousto homogeneous Dirichlet conditions. Then by Theorem 2.43, the principal partap(·, ·) is Hm

0 (Ω)-coercive. The next claims are an immediate consequence of thisresult. �

We finish by indicating the relation between As and A for the order 2m and generalboundary conditions. Similarly to Section 2.2 we have extended A = As in (2.134) toA in (2.135). For finding this relation we would have to determine a complementarysystem Sj , j = 1, . . . ,m, of boundary operators for the Bj , j = 1, . . . ,m. For theseBj , Sj , j = 1, . . . ,m, a second pair of Cj , Tj , j = 1, . . . ,m, can be determined such thata generalized Green’s formula is valid, see e.g. Lions and Magenes [478] and Oden andReddy [518], pp. 162 ff.

If we restrict the discussion to Dirichlet boundary conditions we can avoid allthese technicalities, since these boundary operators are multiplied with the van-ishing ∂αu(x) = 0 ∀|α| ≤ m− 1, x ∈ ∂Ω anyway. Multiplying Asu in (2.134) withv ∈ Hm

0 (Ω), integrating over Ω and iteratively applying the Green’s formula we get,generalizing the special case m = 1 in (2.9), (2.16)–(2.18), the weak form A. Note that

2.5. Nonlinear elliptic equations 77

we only require boundary conditions for the v, but not for u in∫Ω

vAsudx =∫

Ω

v∑

|α|,|β|≤m

(−1)|β|∂β(aαβ∂αu)dx ∀ u ∈ H2m(Ω), v ∈ L2(Ω),

= (Asu, v)L2(Ω) with aαβ ∈W |β|,∞(Ω) (2.160)

= a(u, v) := 〈Au, v〉V′×V =∫

Ω

∑|α|,|β|≤m

aαβ∂αu∂βvdx

∀ u ∈ V = Hm(Ω), v ∈ Vb = Hm0 (Ω), aαβ ∈ L2(Ω)

with |〈Au, v〉V′×V | ≤ max|α|,|β|≤m

‖aαβ‖L∞(Ω)‖u‖Hm(Ω)‖v‖Hm(Ω).

By once more using the Green’s formula we obtain for v ∈ Hm0 (Ω) ∩H2m(Ω)

(Asu, v)L2(Ω) = a(u, v) = 〈Au, v〉V′×V = 〈u,Adv〉V×V′ =(u,Ad

sv)L2(Ω)

(2.161)

∀u, v ∈ H2m(Ω) ∩ Hm0 (Ω), with Ad

sv =∑

|β|,|α|≤m

((−1)|α|∂α(aαβ∂βv),

for aα,β ∈Wmax{|α|,|β|},∞(Ω). Summarizing these results we get, see (2.160),

As, Ads : Vs := H2m(Ω) ∩Hm

0 (Ω) →W := L2(Ω) for aα,β ∈Wmax{|α|,|β|},∞(Ω)

with ‖As‖W←Vs≤ sup

‖u‖H2m(Ω)=1

⎧⎨⎩ ∑|α|,|β|≤m

∫Ω

|∂β(aαβ(∂αu))|dx

⎫⎬⎭ (2.162)

≤ C∑

|γ|≤|β|,|α|≤m

‖∂γaαβ‖L∞(Ω) ≤ C∑

|α|,|β|≤m

‖aαβ‖Wmax{|α|,|β|},∞(Ω)

and similarly∥∥Ad

s

∥∥W←Vs

≤∑

|α|,|β|≤m

‖aαβ‖W |α|,∞(Ω),

A,Ad : V := Hm0 (Ω) → V ′ := H−m(Ω) for aα,β ∈ L∞(Ω) and

‖Ad‖V′←V , ‖A‖V′←V ≤ C∑

|α|,|β|≤m

‖aαβ‖L∞(Ω). (2.163)

2.5 Nonlinear elliptic equations

2.5.1 Introduction

There is a vast literature on this extremely active topic, e.g. we mention that a searchfor regular solutions of nonlinear elliptic problems yields ≈ 1000 hits.14

14 We list some of them: Alt and Luckhaus [23], Amann and Quittner [25], Bensoussan and Frehse[81], Boccuto and Mitidieri [99], Bochniak and Sandig [128], Chemin and Xu [169], Dachkovski [232],De Donno [271], Deng [283], Garello [337], Giaquinta, Necas, John and Stara [343], Girardi, Mastroeni


Already in Section 2.4 we have indicated the need for different concepts. We wantto demonstrate this for the the case of nonlinear elliptic operators:

−Δu = f(u) ∈ Ω, u|∂Ω = 0 with A = −Δ. (2.164)

This is a special case of the so-called semilinear elliptic operators. For f(u) = eu weobtain the famous Bratu equation, a standard example for bifurcation problems, e.g.Allgower and Jepson [22].

Either the f and Ω are smooth enough and A = −Δ is boundedly invertible, seeSection 2.4. Then the preceding linear theory for A can be combined via u = A−1f(u)with fixed-point theorems to get results for specific semilinear equations. This ispossible in Holder spaces A : C2,γ(Ω) → Cγ(Ω). No growth conditions for the f(u)for u→∞ are necessary. This is well presented by Zeidler [676], e.g. Sections 6.5,7.16, 8.12. We may relax the above smoothness conditions for A and impose growthconditions for the f(u). In some cases this still allows us to apply the strong formA : H2(Ω) → L2(Ω).

Or the f and Ω are not smooth. Then we have to generalize the weak theory inthe last sections to (2.164). The very powerful tool of weak solutions u ∈ H1(Ω)for the corresponding weak linear operator A requires Au− f(u) ∈ H−1(Ω). This isusually no longer correct for a nonlinear f . So we have to generalize A : H1(Ω) →H−1(Ω) to A : W 1,p(Ω) →W−1,q(Ω), 1/p + 1/q = 1, and have to impose appropriategrowth conditions on the nonlinear function f(u) for u→∞. This is necessarysince the W 1,p(Ω) is not a Banach algebra, so, e.g. u ∈W 1,p(Ω) does not implyu2 ∈W 1,p(Ω). A generalization of Sobolev spaces, e.g. towards Hr(Ω),W r,p(Ω), r ∈ R,or A : Wm,p(Ω) →W−m,q(Ω), or to Besov or Besov–Triebel–Lizorkin spaces allows adeeper analysis, see e.g. Runst and Sickel [561].

A major topic in this direction are general semilinear, quasilinear and fully nonlinearelliptic equations, more generally the so-called monotone and coercive and noncoerciveoperators. A thorough presentation in the Sobolev and Holder space setting, appropri-ate for our context, is given by Zeidler [676–678] and Taylor [618–620]. Second orderequations are studied in the classical Gilbarg and Trudinger [346], and in Showalter[589] and Chen and Wu [170]. Here we essentially study homogeneous Dirichlet bound-ary conditions. In [589] extensions to more general quasilinear equations and specialsystems and to non-Dirichlet boundary conditions are discussed. Showalter [589]considers Neumann type conditions, and boundary and interior inequality constraintsfor the solutions. Here we combine, for special cases, the classical, strong or weakformulation for linear operators, e.g. A : C2,γ(Ω) → Cγ(Ω), with Au− f(u) = 0 for

and Matzeu [347], Grunau and Sweers [377], Johnsen [424], Lakkis [468], Lassoued [470], Le Dung[308], Laurence and Stredulinsky [468], Luckhaus [481], Muller [506], Naniewicz and Panagiotopoulos[507], Necas, John and Stara [511], Nedev [512], Nguyen [633], Pacard [522], Shahgholian [582], Guoand Li [379], Uchida [639], Zhang [682] Yang [672]; compare Kozlov, Maz’ya and Rossmann [452].Additional papers and books consider solution techniques, e.g. Altman [24]. A rather importantfeature of this theory of nonlinear evolution equations in Banach spaces is the following: Nonlinearproblems can be solved via linearized evolution equations with many additional tricks. Dong [299],lists solvability and regularity of solutions for many applications. In his Chapter 9, fully nonlinearparabolic equations are studied.


interesting existence results. Estimates for the solutions, in particular with respect tohigher norms, hence regularity, are highly complicated and often seem to be missing intextbooks. Sometimes a complete discussion in Hilbert spaces H1(Ω) is still possibledue to the specific structure of the nonlinearity. A famous example is the Navier–Stokesoperator in Section 2.8.

We present in this section a short introduction to semilinear, quasilinear andfully nonlinear elliptic equations, strongly inspired by Zeidler [676–678]. Follow-ing the essential definitions in Subsection 2.5.2 we discuss special semilinear equa-tions in Subsection 2.5.3, generalizing the above (2.164). Subsection 2.5.4 stud-ies quasilinear equations of second order. We continue with the so-called Nemy-ckii operators in Subsection 2.5.5. They generalize the above f(u) and allow thediscussion of semilinear and quasilinear elliptic operators in Subsection 2.5.6 oforder 2m. We finish with fully nonlinear operators in Subsection 2.5.7. They aremainly studied here for specific cases, see e.g. Example 2.78 and Gilbarg andTrudinger [346].

Quasilinear problems have been studied very intensively. So their existence anduniqueness results, applied to semilinear problems, might be more general than thoseformulated for semilinear problems, see e.g. Theorem 2.61.

Recently nonlinear elliptic problems have been considered with more than one solu-tion for specific nonlinearities, e.g. with continuation methods, cf. Allgower, Cruceanuand Tavener [18,19].

For the many gaps in our knowledge of solutions of nonlinear elliptic problems, theresults of this book open up new numerical techniques. Continuation and discretiza-tion methods with the “stability implies invertibility of linear operators” in Section3.6 allow at least indications for solutions and their properties under appropriateconditions for these problems.

2.5.2 Definitions for nonlinear elliptic operators

For orders 2 and 2m, we recall the notations, variables, and differential operators in(2.73) as, e.g.

ϑ0 = 1, ϑα,Θ0 ∈ R, ϑ,Θ = Θ1 ∈ Rn,Θk ∈ Rnk

and ∂0u = ∇0u = u, ∂αu,∇k = (∂α)|α|=k. (2.165)

In ∇2 the uxy and uyx are different. So α = (1, 1) appears twice, similarly in ∇2m,thus defining the n2m and N2m below. We extend the previous linear to fully nonlinearequations and introduce their ellipticity. For nonlinear operators we consider realvalued functions of the form

Gw : D(Gw) → R, w = (x,Θ0, . . . ,Θ2m) ∈ D(Gw) ⊂ Ω× RN2m (2.166)

where Ω ⊂ Rn satisfies (2.5), RN2n = R× Rn × · · · × Rn2n

, and

N2m := (n2m+1 − 1)/(n− 1) for n > 1, and N2m := 2m + 1 for n = 1.


For the important case 2m = 2 this reads as

Gw : D(Gw) → R, w = (x, ..,Θ2) ∈ D(Gw) ⊂ Uo := Ω× R× Rn × Rn×n. (2.167)

Sometimes we replace x,Θ0,Θ1,Θ2 by x, z, p, r, and usually we need only the n(n +1)/2-dimensional subspace of symmetric real valued matrices in Rn×n.

In all the following nonlinear problems only functions, u, are considered with

u : Ω → R and w(x) := wu(x) := (x, u(x),∇u(x), . . . ,∇2mu(x)) ∈ D(Gw). (2.168)

Preparing the following definitions, we explicitly formulate different (strong) formsG of quasi- and semilinear operators of order 2. The generalization to order 2m isobvious.15

G(·) : C2(Ω) → C(Ω), for u,∇u,G(u) we simplify the notation, writing

G(u) :=n∑

i,j=1

aij(·, u,∇u)∂i∂j u + a(·, u,∇u) (2.169)

is the quasilinear operator, usually with (aij)ni,j=1 symmetric,

G(u) := −n∑

j=1

∂j(bj(·, u,∇u)) + b(·, u,∇u) (2.170)

is the quasilinear divergent operator,

G(u) :=n∑

i,j=1

aij(·, u)∂i∂j u + b(·, u,∇u), is the semilinear operator, (2.171)

G(u) :=n∑

i,j=0

aij(·)∂i∂j u− f(u), is a special semilinear operator, (2.172)

G(u) := Gw(·, u,∇u,∇2u), is fully nonlinear, if it is none of the above. (2.173)

For G(·) : C2(Ω) → C(Ω) we need, for the quasilinear case, real valued aij , a, b ∈ C orbj ∈ C1. An extension G(·) to G(·) : W 2,p(Ω) → Lq(Ω) requires appropriate growthconditions for the aij(·, u,∇u), bj(·, u,∇u), a(·, u,∇u), b(·, u,∇u), Gw. We will comeback to this point below.

Definition 2.51. Nonlinear differential operators and ellipticity:

1. A nonlinear PDE of order 2m (analogously for order m) for a function u as in(2.168) and Gw as in (2.166) has the form and the solution u0

G(u) := Gw(wu) = Gw(·, u,∇u, . . . ,∇2mu) and G(u0)(x) = 0 ∀ x ∈ Ω (2.174)

or a.e. for x ∈ Ω. It is quasilinear if Gw is an affine function of the variablesΘ2m, see (2.169)–(2.170). It is semilinear if Θ2m and its coefficients in Gw, theprincipal part, define a linear operator in Θ2m, hence, these coefficients only

15 Note the − sign in (2.170) in contrast to the + sign in [346], p. 87.


depend upon x, u, . . . ,∇2m−1u, see (2.171)–(2.172). In all other cases it is fullynonlinear.

2. Let x ∈ D ⊂ Ω and w(x) = wu(x) ∈ D(Gw). Then we require G ∈ C1(D(Gw))(or e.g. ∈W 1,∞(D(Gw))) and a principal part (or the symbol) of the linearizedoperator, compare (2.73), (2.79), with the corresponding property. So it satisfies,with the Euclidean norm, |ϑ| = |ϑ|n, and (2.79)

λ(w)|ϑ|2mn ≤ (−1)m

∑|α|=2m

aα(w)ϑα ≤ Λ(w)|ϑ|2mn ∀ ϑ ∈ Rn,

(2.175)

∀x ∈ Ω or, a.e.∀x ∈ Ω, with w = wu(x) ∈ D(Gw), aα(w) := (Gwϑα) (w(x));

hence, 0 ≤ λ(w) for elliptic, 0 < ε ≤ λ(w) for strongly elliptic and 0 < ε ≤λ(w),Λ(w)/λ(w) bounded for a specific w = w(x) ∈ D(Gw), or w ∈ D, or ∀ w ∈D(Gw) for uniformly elliptic operators, G. The Gw

ϑα denotes the partial derivativeof the real valued function Gw with respect to ϑα, |α| = 2m. For Gw

ϑα(w) ∈L∞(D(G)) this condition is only required a.e. in Ω.

3. For 2m = 2 , Gw ∈ C2(D(Gw)) we get the symmetric matrix

−GwΘ2(w) := − (Gw

ϑα(w))|α|=2 := − (Gwϑiϑj (w))n

i,j=1 , ϑα = ϑiϑj ,∀x ∈ D. (2.176)

Then λ(w) and Λ(w) can be chosen as the minimal and the maximal (real)eigenvalues of Gw

Θ2(w(x)).

2.5.3 Special semilinear and quasilinear operators

The topic of this subsection is special equations of the form (2.169)–(2.172) and theirgeneralization to quasilinear elliptic equations of order 2m. We study three types ofresults. The common structure of the first two types is their definition in Holderspaces. This allows applying the Schauder fixed point theorem and the Schauderprinciple. In the special semilinear case, the nonlinearity only occurs in the lowestorder term as f(u) as in (2.172). For the quasilinear case, we restrict the presentationto 2m = 2 with special coefficients aij , a, bj , b in (2.169), (2.170), and (2.193), (2.194).The third type presents semilinear equations, where again the nonlinearity onlyoccurs in the lowest order term as f(u) as in (2.172), but only defined on nonlinearspaces.

We repeat: the existence and uniqueness results for quasilinear problems might bemore general than those for semilinear problems, see e.g. Theorem 2.61.

Special semilinear operators in Sobolev and Holder spaces

We study the orders 2m and 2 and impose C∞(Ω) and Ck,γ(Ω) conditions, respectively,with the norms ‖u‖Ck,γ(Ω), see (2.87). So the operators, considered here, use the linearoperator A as in Section 2.4, homogeneous boundary and smoothness conditions as,


see (2.88), (2.93), (2.117),

G(u) := Au− cf(u) = 0 in Ω, Bu = (Bju)mj=1 = 0 on ∂Ω with k ∈ N0 (2.177)

G : C2m+k,γ(Ω) → Ck,γ(Ω), Bj : C2m+k,γ(∂Ω) → C2m+k−mj ,γ(∂Ω),mj < 2m,

Au :=∑

|α|≤2m

aα∂αu, Bju =

∑|α|≤mj

bjα∂αu, j = 1, . . . ,m, and

B satisfies the complementary condition, either Dirichlet or as (2.183), with

C2m+k,γB,0 (Ω) := {u ∈ C2m+k,γ(Ω) : Bu|∂Ω = 0}, k ∈ N0, (2.178)

aα ∈ C∞(Ω), bjα ∈ C∞(∂Ω), ∀|α| ≤ 2m, j = 1, . . . ,m, ∂Ω ∈ C∞, (2.179)

A : C2m+k,γB,0 (Ω) → Ck,γ(Ω) is strongly (so uniformly) elliptic and of index 0.

We use the notation C2m+k,γB,0 (Ω) indicating the more general boundary conditions

allowed in this context. The conditions (2.177)–(2.179) and the following (2.180)–(2.181) are formulated for k ∈ N0. In both cases we start with k = 0. The generalk ∈ N0 will be needed for higher regularity results, see Theorem 2.56 below. Examplesfor (2.178), (2.179) are (2.80) and Theorem 2.50.

Similarly as above, it is possible to reduce the conditions for the coefficients anddomain for the special case of differential equations of order 2, cf. Theorem 2.37 ff.and (2.83), (2.112). We assume for

2m = 2,m1 = 1 let A : C2,γB,0(Ω) → Cγ(Ω) with index A = 0, (2.180)

Au :=n∑

i,j=0

aij(x)∂i∂ju, Bu = b0u +n∑

i=1

ci∂iu,

aij ∈ Ck,γ(Ω), b0, cj ∈ C1+k,γ(Ω), 0 < γ < 1 ∀i, j = 0, · · · , n, (2.181)

∂Ω ∈ C2+k,γ , k ∈ N0, for c = (c1, · · · , cn) ≡ 0 let b0(x) �= 0, otherwise

|(c(x), ν(x))n| > κ > 0 ∀ x ∈ ∂Ω, A is strongly (so uniformly) elliptic.

Leung [473] discusses different examples of interacting population reaction diffusionsystems. These are PDEs of this type coupled with ODEs.

For solvability results for the semilinear equations Au− f(u) = 0 in (2.177) we haveto extend some results for the linear equations Au− g = 0. We present the main ideafor the proofs of the following theorems, but only indicate the necessary modificationsof the proofs for the R2 results in [676], Proposition 6.4, to the Rn case: We start witha boundedly invertible A : C2m,γ

B,0 (Ω) → Cγ(Ω), see (2.178), and Proposition 2.54, withk = 0. Its inverse can be extended to a K : C(Ω) → C1,β

B,0(Ω), hence K|Cγ(Ω) = (A)−1.This yields the two equivalent formulations

G(u)(x) = Au− cf(u) = 0 ⇔ u = Kcf(u), (2.182)


for c, f , see Theorem 2.55. Then Schauder type results for the linear K : C(Ω) →C1,β

B,0(Ω) are combined with fixed-point theorems. This yields the existence anduniqueness of solutions for (2.182).

We combine the results in Subsections 2.4.2 and 2.4.3 for linear equations of orders2m and 2 in Rn, respectively, to obtain the continuous extension K : C(Ω) → C1,β

B,0(Ω),similarly to Zeidler [676]. We start summarizing conditions from Subsections 2.4.2 and2.4.3 which guarantee a boundedly invertible linear A : C2m,γ

B,0 (Ω) → Cγ(Ω). Theorem2.28 shows that for order 2m we cannot directly guarantee a boundedly invertibleA. For order 2 the situation is modified due to the maximum principle reflectedin Theorem 2.37. For orders 2m and 2 and k = 0 we obtain Theorems 2.52 and2.53 by combining Theorems 2.21, 2.28, 2.33, 2.39, 2.40, 2.43, Corollary 2.44 andTheorem 2.50.

In Theorem 2.50 we found a special Dirichlet system satisfying the complementarycondition with index A = 0. We had transformed it into the form (j = 1, . . . ,m),

Bju = bj∂j−1u

∂νj−1+∑

|β|=j−1

cj,β∂βu

∂ tβ+

∑|α|≤j−2

bj,α∂αu, with 0 �= bj(x)∀x ∈ ∂Ω. (2.183)

Theorem 2.52. Results for the boundary conditions in (2.177)–(2.179):

1. For an A : C2m,γB,0 (Ω) → Cγ(Ω) of order 2m assume (2.177)–(2.179) and let index

(A) = 0. If λ = 0 is not an eigenvalue of A, it is boundedly invertible. Otherwise(A + μ) with sufficiently small or large enough μ ∈ R is boundedly invertible. Thefollowing estimates are valid for Au0 = g ∈ Cγ(Ω) : If A is boundedly invertible,compare Theorem 2.37, the ‖u0‖Cγ(Ω) is dropped in

‖u0‖C2m,γ(Ω) ≤ C(‖u0‖Cγ(Ω) + ‖g‖Cγ(Ω)

). (2.184)

2. The preceding results are correct if the complementary or Dirichlet conditions arereplaced by these modified conditions Bj in (2.183).

For the particularly important case of second order equations, 2m = 2, the boundaryconditions in (2.183) can be generalized, by Theorem 2.37, as one of the

Bu = B1u = u, or B2u = ∂u(x)/∂ν, or B3u = bu +n∑

i=1

ci∂iu on ∂Ω (2.185)

with C2,γB,0(Ω) := {u ∈ C2,γ(Ω) : Biu|∂Ω = 0}, for one of the i = 1, 2, 3. (2.186)

Theorem 2.53. Results for 2m = 2 and Bi in (2.185), (2.186): For A : C2,γB,0(Ω) →

Cγ(Ω) of order 2 assume (2.180)–(2.181) with k = 0, (2.185), (2.186). Then index(A) = 0 and the results in Theorem 2.52 are valid for 2m = 2. If additionally, λ = 0is not an eigenvalue of (A,B), e.g. the aij satisfy aij ≡ 0, for ij = 0, i + j > 0, and

a00(x) ≥ 0 ∀ x ∈ Ω for B1u and for B3u with b(x) > 0 ∀ x ∈ ∂Ω, or (2.187)

a00(x) > 0 ∀ x ∈ Ω for B2u ∀ x ∈ ∂Ω,

then A : C2,γB,0(Ω) → Cγ(Ω) is boundedly invertible and (2.184) holds for 2m = 2.


We combine these results, in particular Theorems 2.32, 2.37, 2.52 and 2.53, withthe results in Zeidler [676], pp. 233 ff., see his Proposition 6.4.

Proposition 2.54. Let A : C2m,γB,0 (Ω) → Cγ(Ω) in (2.177), (2.186), with B in

(2.178), be boundedly invertible, and yield K = (A)−1 : Cγ(Ω) → C2m,γB,0 (Ω). This is

correct under the conditions in Theorems 2.52 and 2.53, hence for orders 2m and 2,respectively Then K can be uniquely extended to a continuous linear operator

K : C(Ω) → C1,βB,0(Ω) for a fixed β, 0 < β < 1. (2.188)

Proof. There are only minor differences compared to the proof in Zeidler [676], pp.238, for 2m = 2. We combine Theorems 2.30 and 2.33, in particular (2.94) and (2.98)to generalize Zeidler’s condition (A) as

‖Kg‖W 2m,p(Ω) ≤ C‖g‖Lp(Ω) ∀g ∈ Lp(Ω), 1 < p. (2.189)

By Theorem 1.26 the embeddings

C(Ω) ↪→ Lp(Ω) and W 2m,p(Ω) ↪→ C1,β(Ω)

are continuous for p > n/(2m− 1− β). So (2.189) is modified into

‖Kg‖C1,β(Ω) ≤ C‖g‖C(Ω)∀g ∈ Cγ(Ω).

By the Weierstrass approximation theorem, Cγ(Ω) is dense in C(Ω). So K can beextended as claimed. �

Now we are ready for applying this result to special semilinear equations, see Zeidler[676], pp. 233 ff., his Proposition 6.7, and Corollaries 6.8, 6.9:

Theorem 2.55. Existence and uniqueness of solutions for semilinear equations:

1. Let A : C2m,γB,0 (Ω) → Cγ(Ω) in (2.177), with B in (2.178), satisfy the conditions

of Proposition 2.54 and assume:

for fixed R > 0 let f : [−R,R] → R, f ∈ Cβ , 0 < β < 1. (2.190)

Then for every sufficiently small |c|, c ∈ R, the semilinear boundary value problem

(Au− cf(u))(x) = 0∀x ∈ Ω, Bju|∂Ω = 0, j = 1, . . . ,m, (2.191)

has a unique solution u0 ∈ C2m,γ(Ω) ∩ CmB,0(Ω) with 0 < γ < 1, see (2.186).

2. For c ≥ 0 and f(u) ≥ 0 on [0, R], the solution is nonnegative, u0 ≥ 0 on Ω.3. For a Lipschitz-continuous f there exists exactly one solution with |u0(x)| ≤ R

on Ω. It can be determined by the convergent iteration process

uk ∈ C2m,γB,0 (Ω) := C2m,γ(Ω) ∩ Cm

B,0(Ω), u0 := 0, Auk+1 = cf(uk), k ∈ N0.

Zeidler’s proof [676], pp. 240,241, remains unchanged, if only his Proposition 6.4 isreplaced by our Proposition 2.54. Modifications for f are discussed by von Wahl [658],and Leung [473], pp. 82 ff.

Higher regularity for this and later theorems is obtainable via different kinds of so-called bootstrap arguments or inherited regularity, cf. [676]. They are chosen according


to the results available for the specific situation, usually based upon inductive argu-ments. Our approach here uses the regularity results for the corresponding linearelliptic equations, presented in Section 2.4, see Theorems 2.28, 2.30–2.33, 2.38, 2.39.We modify a technique of Hackbusch [387], Remark 9.1.2. We will refer to this resultlater on, hence, we formulate it as

Theorem 2.56. Regularity, inherited to compact and semilinear perturbations:

1. Choose k ∈ N0 in (2.177)–(2.179) for A of order 2m or (2.180)–(2.181) for order2. Then Theorems 2.52, 2.53 and the estimates (2.184) have the stronger formA : C2m+t,γ

B,0 (Ω) → Ct,γ(Ω), 0 ≤ t ≤ k,

Au0 = g satisfies ‖u0‖C2m+t,γ(Ω) ≤ C(‖u0‖C2m,γ(Ω) + ‖g‖Ct,γ(Ω)

). (2.192)

Note that for boundedly invertible A the term ‖u0‖C2m,γ(Ω) is deleted.2. Let δA be a linear or nonlinear differential operator of order ≤ 2m− 1, such that

δA : Cr+2m,γ(Ω) ∩ C2m,γB,0 (Ω) → Cr+1,γ(Ω) is bounded for all r = 0, 1, . . . , k − 1.

This implies A + δA : C2m+t,γB,0 (Ω) → Ct,γ(Ω) and (2.192) for the solution u0 of

(A + δA)u0 = g.3. Under the conditions of Theorem 2.55 let the f in (2.190) satisfy

f (r) : u ∈ Cr+2m,γ(Ω) → f (r)(u) := f(u(·)) ∈ Cr+1,γ(Ω), r = 0, 1, . . . , k − 1,

e.g. these f (r) are in Cr+1,γ(R). Then the solutions u0 of (2.191) are u0 ∈C2m+t,γ

B,0 (Ω). The norms of u0 can be estimated as in (2.192).

Proof. By induction: For k = t = 0 the result and estimates (2.192) are a consequenceof Theorems 2.45 and 2.53. Now we inductively assume (2.192) for the solution(A + δA)u0 = g ∈ Ct−1,γ(Ω) and for 0 ≤ t− 1 ≤ k − 1 for k ∈ N. Consequently, u0 ∈Ct−1+2m,γ(Ω) solves Au0 = g := g − δAu0 as well. By our inductive assumption forA + δA and finally the condition for A we estimate

‖u0‖Ct−1+2m,γ(Ω) ≤ C ′t−1

(‖u0‖C2m,γ(Ω) + ‖g‖Ct−1,γ(Ω)

), hence,

‖g‖Ct,γ(Ω) ≤ ‖δA‖Ct,γ(Ω)←Ct−1+2m,γ‖u0‖Ct−1+2m,γ + ‖g‖Ct,γ(Ω)

≤ C ′′t−1

(‖u0‖C2m,γ(Ω) + ‖g‖Ct,γ(Ω)

)and, by (2.192) for A,

‖u0‖Ct+2m,γ(Ω) ≤ C ′t

(‖u0‖C2m,γ(Ω) + ‖g‖Ct,γ(Ω)

), hence, the claim.

The final remark about the solutions u0 of (2.192) is obvious. �

Special quasilinear operators of order 2 in Holder spaces

The previous results were, modulo the differences in Theorems 2.52 and 2.53, valid fororders 2m and 2. The following theorem is restricted to quasilinear uniformly ellipticequations of order 2 in R2, a special form of (2.169). So we assume

n = 2 : G(u) :=2∑

i,j=1

aij(x, u,∇u)∂i∂j u, (aij)2i,j=1 symmetric. (2.193)


Similarly to (2.180), (2.181) we assume for the domain, the defining coefficients andthe inhomogeneous boundary function, g,

for 0 < γ < 1, let Ω in (2.5), ∂Ω ∈ C2,γ be a single curve with (2.194)

positive curvature ∀ x ∈ ∂Ω, and in (2.193), let aij ∈ Cγ(Ω× R3),

∀i, j = 1, 2, G be strongly (hence uniformly) elliptic, and g ∈ C2,γ(Ω).

Theorem 2.57. Existence and uniqueness of solutions for (2.193), Zeidler [676],pp. 246 ff., Gilbarg and Trudinger [346], pp. 312: Let the nonlinear G in (2.193)satisfy (2.194). Then the boundary value problem

G(u)(x) = 0 ∀ x ∈ Ω, u(x) = g(x) ∀ x ∈ ∂Ω (2.195)

has a solution u0 ∈ C2,γ(Ω) ∩ C0(Ω) with supx∈Ω |∇u(x)| ≤ K.

For regularity results we refer to the discussion at the end of this subsection and of2.5.4 and 2.5.6. Many other related results are listed, e.g. in Zeidler [676], Gilbarg andTrudinger [346], and Runst and Sickel [561]; or the linearization techniques in Section2.7 can be combined with the Fredholm alternative and the inherited regularity inTheorem 2.56 extending these results.

Special semilinear operators in nonlinear spaces

For this last type of results we return to semilinear equations. For aαβ ∈W |β|,∞(Ω)we assume the linear operator A in strong divergence form, cf. (2.134)

A = As : H2m(Ω) → L2(Ω), Au =∑

|α|,|β|≤m

(−1)|β|∂β(aαβ∂αu). (2.196)

We want to solve the Dirichlet boundary value problem

G(u)(x) := (Au− f(u))(x) = g(x) ∀ x ∈ Ω (2.197)

∂αu(x) = 0 ∀|α| ≤ m− 1, ∀ x ∈ ∂Ω. (2.198)

As usual we multiply (2.197) with v ∈ Hm0 (Ω), integrate over Ω and iteratively use

partial integration, see (2.9). We find the following generalized form, related to

Au0 − f(u0) = g, u0 ∈ V = Hm0 (Ω),

(Asu− f(u)− g, v)L2(Ω) = a(u, v)− a1(u, v)− (g, v)∀u ∈ H2m(Ω), v ∈ V

a(u, v) = 〈Au, v〉V′×V =∫

Ω

∑|α|,|β|≤m

aαβ∂αu∂βvdx with aαβ ∈ L∞(Ω), (2.199)

a1(u, v) =∫

Ω

f(u(x))v(x)dx, (g, v) =∫

Ω

g(x)v(x)dx. (2.200)

This shows that we certainly have to impose conditions upon f, g and choose appro-priate u, v such that a1(u, v), (g, v) are well defined. So we assume for G and a(·, ·) in


(2.197) and (2.199),

a(·, ·) is V − coercive, aαβ ∈ L∞(Ω) ∀ |α|, |β| ≤ m, Ω satisfies (2.5),

f : R → R continuous with a ∈ R fixed, (f(u) + a)u ≤ 0 ∀u ∈ R, (2.201)

V = Hm0 (Ω),D(G) := {u ∈ V : h := (f(u) + a)u ∈ L1(Ω)} �= V and (2.202)

W = Hm0 (Ω) ∩Hk(Ω), k > n/2, k′ := max{k,m} and ‖v‖W = ‖v‖Hk′ (Ω).

We still require (2.198), but instead of (2.197), we solve the generalized problem

determine u0 ∈ D(G) s.t. a(u0, v)− a1(u0, v)− (g, v) = 0 ∀v ∈ W. (2.203)

The a(u0, v), a1(u0, v), (g, v) are well defined for u0 ∈ D(G), v ∈ W but the domainD(G) is not a linear space. Then Zeidler’s proof of his Proposition 27.21, [678], p. 605,for Au = −Δu can be easily generalized to our situation:

Theorem 2.58. Existence and uniqueness for solutions of (2.197), (2.198): Underthe conditions (2.201), (2.202), e.g. for f(u) = −eu, this boundary value problem hasfor all g ∈ L2(Ω) a generalized solution in the sense of (2.203).

Remark 2.59. The condition (f(u) + a)u ≤ 0 ∀u ∈ R in (2.201) is satisfied if f :R → R is continuous and monotonically decreasing, e.g. for f(u) = −eu, a = f(0). ForAu = −Δu this is a standard example in (numerical) bifurcation, the so-called Bratuequation, cf. e.g. Allgower and Jepson [22] and Govaerts [357], p. 21 ff.:

−Δe + eu = 0 in Ω and u = 0 on ∂Ω. (2.204)

We do not go into details, but only indicate some properties: This equation has mainlybe discussed on the basis of very rough difference methods. So it is not quite clearwhether the bifurcations scenarios in the many papers go back to spurious solutions,which do not correspond to solutions of the exact equation, or reflect the exact (2.204)scenarios. In fact Peitgen, Saupe and Schmitt [525] have shown that difference methodsapplied to (2.204) always posess spurious solutions, moving to ∞ with the step sizeh→ 0. For n = 1, 2 one finds the situation indicated in Govaerts [357], p. 21 ff., witha turning and a bifurcation point. For n = 3 the solutions oscillate around a specificvalue of λ infinitely often with decreasing amplitudes. For n ≥ 4 there are no longerany solutions. For analytical results, cf. Joseph and Lundgren [428].

Here the situation is very similar to Theorem 2.56. There we studied A :C2m+t,γ

0 (Ω) → Ct,γ(Ω); here we consider the weak form A : Hm+s(Ω) ∩Hm0 (Ω) →

H−m+s(Ω). We nevertheless formulate, for the reader’s convenience, the updated resultonce more. We omit the proof, since it is nearly identical in both cases, except thatthe C2m+t,γ

0 (Ω) have to be replaced by Hm+s(Ω), etc.


Theorem 2.60. Regular solutions for linear and semilinear equations:

1. Choose t ∈ N0 such that the conditions (2.149) and in Theorem 2.45 are satisfied.Then A : Hm+s(Ω) ∩Hm

0 (Ω) → H−m+s(Ω) for 0 ≤ s ≤ t (or even for 0 ≤ s <t �∈ N) and we obtain for the solution

u0 of Au0 = g that ‖u0‖Hm+s(Ω) ≤ C(‖u0‖Hm(Ω) + ‖g‖H−m+s(Ω)

). (2.205)

Note that for boundedly invertible A the term ‖u0‖Hm(Ω) is deleted.2. Let δA be a linear or nonlinear differential operator of order ≤ 2m− 1 such that

δA : Hr+m(Ω) ∩Hm0 (Ω) → Hr+1−m(Ω) is bounded for all r = 0, 1, . . . , s− 1.

This implies the same A + δA : Hm+s(Ω) ∩Hm0 (Ω) → Hs−m(Ω) and (2.205) for

the solution u0 of (A + δA)u0 = g. In particular, the condition for δA is satisfied,if the linear δA is induced by the lower order terms |α|+ |β| < 2m of the bilinearform (2.199) and the aαβ are smooth enough, aαβ ∈ Cmax{0,s−2m+|β|}(Ω).

3. Under the conditions of Theorem 2.58 let the f in (2.201) and its derivativessatisfy

f (r) : u ∈ Hr+m(Ω) → f (r)(u(·)) ∈ L∞(Ω), r = 0, 1, . . . , s− 1,

e.g. these f (r) are bounded. Then the solutions u0 of (2.191) are u0 ∈ C2m+t,γ0 (Ω).

The norms of u0 can be estimated as in (2.205).

2.5.4 Quasilinear elliptic equations of order 2

Throughout this subsection γ always denotes a real 0 < γ < 1. Furthermore, we willuse Wm,q(Ω) instead of Wm,p(Ω), since we need the p in the triple x, z, p below.

For quasilinear equations of second order, two competing approaches have beenworked out in the monographs consulted here, essentially Ladyzenskaja and Uralceva[464], Gilbarg and Trudinger [346], Zeidler [678], Taylor [620], Showalter [589], andChen and Wu [170]. We present in this subsection results based upon the ellipticityof these equations in the sense of Definition 2.51. Then maximum principles, fixedpoint theorems, Holder estimates for the gradient, and boundary gradient estimatesare combined yielding global, interior and boundary estimates and correspondingexistence results. These comparison results yield fewer uniqueness results than thesecond approach in Subsection 2.5.6. This uses Nemyckii and monotone operators. Itreplaces the ellipticity of the equations, strongly related to the coercivity in the senseof Definition 2.65, by a generalized coercivity in the sense of (2.286) and monotonicityin (2.289). Relation (2.289) implies the standard ellipticity condition only for q = 2,hence in Hm(Ω), cf. Proposition 2.121. The theory is in many respects independentof the order 2 or 2m. So we will formulate it in Subsection 2.5.6 below, only for theorder 2m. It allows us existence results for a large class of operators and uniquenessunder natural additional conditions. This difference influences our numerical meth-ods. The approach via ellipticity allows the proofs for stability and convergence vialinearization. For the monotonic situation this remains possible for all cases as well,but modifications for stabiliy have to be imposed, see Corollary 2.44 and Section 2.7below.


We repeat for order 2 the different quasilinear equations, cf. (2.170), (2.169),

Gu =n∑

i,j=1

aij(x, u,∇u)∂i∂ju + a00(x, u,∇u) = 0, u ∈ C2,γ(Ω), general (2.206)

Gu =n∑

i=0

(−1)i>0∂i(Ai(x, u,∇u)) = 0 on Ω, u ∈W 2,q(Ω),divergent form, (2.207)

〈Gu, v〉V′×V =∫

Ω

n∑i=0

Ai(x, u,∇u)∂ivdx = 0 ∀ v ∈ V = W 1,q0 (Ω), weak (2.208)

form, with u ∈ V = W 1,q0 (Ω), sometimes u ∈W 1,q(Ω), u|∂Ω = ϕ|∂Ω.

The following results are examples of existence and uniqueness results, serving asan illustration of the scope of the theory in Gilbarg and Trudinger [346]. They areobtained by appropriate combinations of tools formulated in the introductory remarksof this subsection. For special cases, [346], Theorems 10.1 and 10.7 immediatelyimply uniqueness. This is combined below with uniqueness results from Taylor [620].Simplifying the notation we use, in this subsection, the standard (z, p) ∈ Rn+1 insteadof (Θ0,Θ) ∈ Rn+1 with N1 = n + 1, cf. (2.73). Denote the minimal and maximal realeigenvalues of {aij (x, z, p)}n

i,j=1 , cf. (2.206), an assumed positive matrix, as λ(x, z, p)and Λ(x, z, p) and ϑ ∈ Rn. We called (2.206) elliptic, etc. in (x, z, p), iff

0 < λ(x, z, p)|ϑ|2 <

n∑i,j=1

aij(x, z, p)ϑiϑj < Λ(x, z, p)|ϑ|2∀0 �= ϑ ∈ Rn, (2.209)

with aij(x, z, p), Ai(x, z, p) ∈ R, (x, z, p) ∈ Uo ⊂ Ω× R× Rn, p = (p1, · · · , pn).Obviously (2.207) implies (2.206) for Ai(·, ·, ·) ∈ C1 with aij(x, z, p) =

−∂(Ai(x, z, p))/(∂pj). Conversely, it is usually not possible to transform (2.206) into(2.207).

If in (2.209) the (x, z, p) is replaced by (x, u(x),∇u(x)) then this (uniform) ellipticityholds (locally) with respect to u. For the following growth conditions, we introducethe operators:

For u ∈ C1(Ω× R× Rn) let δu :=

(∂z + |p|−2

n∑i=1

pi∂xi

)u, δu :=

(n∑

i=1

pi∂pi

)u.

(2.210)

Uniformly elliptic equations

We distinguish uniformly and nonuniformly elliptic equations in general and indivergence form. For the different forms we require growth and regularity conditions,assuming for (2.211)–(2.219), |p| → ∞, uniformly for x ∈ Ω, and bounded z, with


(x, z, p) ∈ Uo, cf. (2.209), (2.210).

For (2.206): G unif. ell. on Ω, a00, aij ∈ C1(Ω× R× Rn), ∂Ω ∈ C2,γ , (2.211)

ϕ ∈ C2,γ(∂Ω) and assume the estimates:

aij , δaij = O(λ), δaij = o(λ), a00, δa00 = O(λ|p|2), δa00 = o(λ|p|2).For (2.207): G unif. ell. on Ω, Ai ∈ C1,γ(Ω× R× Rn), A0 ∈ Cγ(Ω . . .), (2.212)

∂Ω ∈ C2,γ , ϕ ∈ C2,γ(∂Ω), and assume the estimates: ∃τ > −1 :

|p|τ = O(λ),n∑

i=1

∂piAi = O(|p|τ ), |p|∂zAi,

n∑i=1

∂xiAi, A0 = O(|p|τ+2)

Specific results are possible for the following special case of the general form (2.206),noting that i, j ≥ 1:

Gu = Gu =n∑

i,j=1

aij(x, u)∂i∂ju + a00(x, u,∇u) = 0 on Ω, aij ∈ C1(Ω,R). (2.213)

Then a transformation into divergence form (2.207) is possible by choosing

Ai(x, z, p) := −n∑

j=1

aij(x, z)pj , for i = 1, . . . , n,

A0(x, z, p) = −a00(x, z, p) +n∑

i,j=1

∂zaij(x, z)pipj +n∑

i,j=1

∂xiaij(x, z)pj .

For (2.213): G unif. ell. on Ω, aij ∈ C1(Ω× R), a00 ∈ Cγ(Ω× R× Rn) (2.214)

∂Ω ∈ C2,γ , ϕ ∈ C2,γ(∂Ω), and the above p, x, z, assume a00 = O(|p|2),

∃μ1, μ2 > 0 :a00(x, z, p)sign z

μ1|p|+ μ2|p|2 ≤ E(x, z, p) :=

n∑i,j=1

aij(x, z, p)pipj . (2.215)

Note that for (2.213) these aij(x, z, p), i, j = 1, . . . , n, are independent of p.

Nonuniformly elliptic equations

We no longer require G to be uniformly elliptic on Ω, and Λ/λ is no longer bounded.This situation is important, e.g. for minimal surface operators. We assume the originalaij(x, z, p) can be split, according to

aij(x, z, p) = a∗ij(x, z, p) + (picj(x, z, p) + pjci(x, z, p))/2, i, j = 1, . . . , n, (2.216)

with aij , a∗ij , ci ∈ C1(Ω× R× (Rn \ {0}), and a positive symmetric matrix

(a∗ij)ni,j=1

.

Furthermore, let λ∗ indicate the lower bound in (2.209) defined by the matrix(a∗ij)ni,j=1


instead of the original (aij)ni,j=1 and use E in (2.215). We impose the structure and

regularity conditions, modifying the above (2.211), (2.212)

For (2.206): The first two lines in (2.211) remain, except delete “unif.ell.on”,

change the last line into |p|aij , a00, δa00 = O(E), δa∗ij = O(√λ∗E/|p|),

δa∗ij = o(√λ∗E/|p|), δa00 = o(E), μ1, μ2 > 0 exist, s.t. (2.217)

a00(x, z, p)sign z

μ1|p|+ μ2|p|2 ≤ E(x, z, p) =

n∑i,j=1

aij(x, z, p)pipj ∀(x, z, p �= 0) ∈ Uo.

If we replace uniformly elliptic by elliptic, we impose

For (2.206): The first two lines in (2.211) remain, but require G elliptic on Ω,

change the last line into Ω is uniformly convex (2.218)

a00 = o(|p|Λ), δa00 ≤ |O(a200/|p|2T ∗) , T ∗ := trace

[a∗ij].

Finally we consider the divergence form (2.207). In the last two lines of (2.219) we usethe general form (2.206) and introduce a family of equations, depending upon σ with0 ≤ σ ≤ 1. We assume:

For (2.207): The first two lines in (2.212) remain, but require G elliptic on Ω,

change the last line into Ω is uniformly convex (2.219)

∃τ ∈ R s.t. |p|τ = O(λ), |p|∂zAi,n∑

i=1

∂xiAi, A0 = o(|p|τ+1), for

Gσu :=n∑

i,j=1

aij(x, u,∇u)∂i∂ju + σa00(x, u,∇u) = 0 in Ω, u = σϕ on ∂Ω,

let the set of solutions u0 = u0,σ ∈ C2,γ(∂Ω) be uniformly bounded in Ω. (2.220)

Existence for these quasilinear elliptic equations

We summarize the existence results of the main cases in Gilbarg and Trudinger’s [346]Theorems 15.10–15.15. We omit his results for problems with boundary curvatureconditions.

Theorem 2.61. Existence for quasilinear operators: For (2.206), (2.207) or (2.213)on a bounded Ω in (2.5), let one of the corresponding conditions be satisfied: For uni-formly elliptic equations impose one of (2.211), (2.212) or (2.214), for nonuniformlyelliptic equations impose one of (2.217), (2.218), (2.219). Then there exists a solutionu0 ∈ C2,γ(Ω) of the Dirichlet problem Gu0 = 0 in Ω, and u0|∂Ω = ϕ.

For regularity results we refer to the end of this section and to subsection 2.5.6.


Uniqueness results

For the following results, we have to impose additional conditions:

For (2.206): G unif. ell. with respect to u, consider solutionsu ∈ V = C(Ω) ∩ C2(Ω),

aij independent of z, a00 nonincreasing in z for each (x, p) ∈ Ω× Rn,

and a00, aij ∈ C1 with respect to the p variable in Uo ⊂ Ω× R× Rn. (2.221)

A combination of (2.211) with (2.221) shows that δaij = o(λ), δa00 = o(λ|p|2) in(2.211) would be implied by

∑nk=1 ∂xk

aij = O(λ),∑n

i=1 ∂xia00 = O(λ|p|2).

For (2.207): G elliptic in Ω, consider solutions u ∈ V = C1(Ω), (2.222)

A0, Ai ∈ C1 with respect to the z, p variables in Ω× R× Rn,

A0 is nonincreasing in z for each fixed (x, p) ∈ Ω× Rn

finally, one of the following conditions (1), (2), (3) is satisfied:

(1) Ai, i = 1, . . . , n, is independent of z; or(2) A0 is independent of p; or(3) the following (n + 1)× (n + 1) matrix satisfies componentwise(

[∂pjAi(x, z, p) −∂pj

A0(x, z, p)]ni,j=1

∂zAi(x, z, p) −∂zA0(x, z, p)

)≥ 0 in Ω× R× Rn.

By considering a special case of (2.206), Taylor [620] obtains uniqueness under, insome sense, less restrictive conditions, see (2.223)–(2.5.4). We define the

Taylor case : Gu =n∑

i,j=1

aij(∇u)∂i∂ju = 0 on Ω, u ∈ V = C∞(Ω), (2.223)

cf. [620], Chapter 14, (10.1). It is combined with the following different conditions(2.224), (2.225), (2.226) for (2.223), or a generalization with (2.5.4). So we assume

For (2.223): ellipticity 0 < λ(p)|ϑ|2 <

n∑i,j=1

aij(p)ϑiϑj < Λ(p)|ϑ|2∀ϑ �= 0,

aij ∈ C∞(Rn), ∂Ω ∈ C∞, ϕ ∈ C∞(∂Ω), and let (2.224)

the solutions uσ forn∑

i,j=1

aij(∇u)∂i∂ju = 0 on Ω, u|∂Ω = σϕ

satisfy |∂xiuσ(x)| ≤ K ∀1 ≤ i ≤ n, x ∈ ∂Ω,∀0 ≤ σ ≤ 1.


For (2.223): For strictly convex Ω, retain lines 1,2 in (2.224) (2.225)

and cancel lines 3,4.

For (2.223): For uniformly elliptic G, hence 0 < Λ(p)/λ(p) ≤ K ∀p (2.226)

retain lines 1,2 in (2.224) and cancel lines 3,4.

Allowing an additional zero order term a00(u), cf. [620], Chapter 14, (10.49), weconsider the

General Taylor case: Gu =n∑

i,j=1

aij(∇u)∂i∂ju + a00(u) = 0 on Ω, u ∈ V = C∞(Ω),

unif. ellipt.: 0 < Λ(p)/λ(p) ≤ K ∀p ∈ Rn2, let |a00(z)| ≤ A1 + A2|z|, A1, A2 > 0,

a′00(z) ≥ 0∀z ∈ R, aij ∈ C∞(Rn), a00 ∈ C∞(R), ∂Ω ∈ C∞, ϕ ∈ C∞(∂Ω). (2.227)

Here we summarize Gilbarg and Trudinger’s [346] uniqueness results from Theorem10.7 and the remarks following Theorem 15.10, see (2.221), (2.222), and Taylor [620],cf. Chapter 14, Theorems 10.2, 10.6, 10.7, Proposition 10.9, see (2.224)–(2.5.4).

Theorem 2.62. Uniqueness: For the quasilinear operators (2.206) or (2.207) or(2.208) or (2.223) on a bounded domain Ω ⊂ Rn, let one of the corresponding con-ditions (2.221), (2.222) or (2.224)–(2.5.4) be satisfied. Then the solution u0 ∈ V ofthe Dirichlet problem Gu0 = 0 in Ω, and u0|∂Ω = ϕ exists, possibly except for (2.222),and is unique.

Regularity results under Ladyzenskaja–Uralceva conditions

Similarly to Subsection 2.6.2 below, see Theorems 2.106, 2.107, regularity resultsdistinguish estimates for the interior or for the entire Ω. To give a flavor for thetype of available results, for second order equations, we summarize Theorems 1.1,2.1, 3.1., 5.1., 6.4., 6.5 in Chapter 4, from the great old ladies Ladyzenskaja andUralceva [464]. We formulate the necessary modifications of the later growth andcoerciveness conditions (2.284), (2.285), (2.286) in Subsection 2.5.6, here for the specialcase m = 1 instead of a general m. For avoiding misunderstandings with p ∈ Rn weagain formulate the results for W 1,q(Ω) with 1 < q <∞, cf. the beginning of thissubsection. We consider the strong divergence form

Gu0 =n∑

i=0

(−1)i>0∂i(Ai(x, u0,∇u0)) = 0 on Ω, and its weak form (2.228)

〈Gu0, v〉V′×V =∫

Ω

n∑i=0

Ai(x, u0,∇u0)∂ivdx = 0 ∀ v ∈ V = W 1,q0 (Ω), (2.229)

for u0 ∈ V = W 2,q(Ω) and u0 ∈ V = W 1,q(Ω), respectively The Ai : (x, z, p) → R aresufficiently differentiable in x, z, p ∈ Ω× R1+n. Boundary conditions u|∂Ω = ϕ|∂Ω areimposed below for some of the results. We impose the following conditions forx ∈ Ω, z ∈ R, and p = (p1, . . . , pn), p′ = (p′1, . . . , p

′n) ∈ Rn : Let μ and ν be positive


nondecreasing and nonincreasing functions.

1. Growth cond.n∑

i=1

|Ai(x, z, p)|(1 + |p|) + |A0(x, z, p)| ≤ μ(|z|)(1 + |p|)q, (2.230)

Coerciveness:n∑

i=1

Ai(x, z, p)pi ≥ ν(|z|)|p|q − μ(|z|), see (2.269), (2.231)

Ellipticity: ν(|z|) ≤ (1 + |p′|)2−q|p|−2n∑

i,j=1

∂Ai(x, z, p′)∂p′j

pipj ≤ μ(|z|), (2.232)

2. Growth cond .

[n∑

i=1

(|Ai|+

∣∣∣∣∂Ai

∂z

∣∣∣∣+ ∣∣∣∣∂A0

∂p′i

∣∣∣∣) (1 + |p|) ,

+|A0|+∣∣∣∣∂A0

∂z

∣∣∣∣] (x, z, p) ≤ μ(|z|)(1 + |p|)q, (2.233)

3. Growth cond .

[n∑

i=1

(|Ai|+

∣∣∣∣∂Ai

∂z

∣∣∣∣) (1 + |p|)

+n∑

i,j=1

∣∣∣∣∣∂Ai

∂p′j

∣∣∣∣∣+ |A0|

⎤⎦ (x, z, p) ≤ μ(|z|)(1 + |p|)q, (2.234)

4. Growth cond .

⎡⎣⎛⎝ n∑i,j=1

∣∣∣∣∂Ai

∂xj

∣∣∣∣ +n∑

i=1

∣∣∣∣∂A0

∂p′i

∣∣∣∣⎞⎠ (1 + |p|) ,

∣∣∣∣∂A0

∂z

∣∣∣∣+ n∑i=1

∣∣∣∣∂A0

∂xi

∣∣∣∣]

(x, z, p) ≤ μ(|z|)(1 + |p|)q, (2.235)

Boundedness:n

maxx∈Ω,i,j=1

{∣∣∣∣∣∂Ai

∂p′j

∣∣∣∣∣ ,∣∣∣∣∂Ai

∂z

∣∣∣∣ , ∣∣∣∣∂Ai

∂xj

∣∣∣∣ , |A0|}

(x, z, p) ≤ μ1 <∞, (2.236)

Condition A for ∂Ω : ∃ 0 < a1, a2, a2 < 1 : ∀Bρ(x), x ∈ ∂Ω, ρ ≤ a1,

∀ components Ω ⊂ Bρ(x) ∩ Ω : measure Ω ≤ (1− a2) measure Bρ(x),(2.237)

Integr. cond.∫

Ω

(1 + |∇u|)q−2

⎛⎝ n∑i,j=1

∣∣∣∣ ∂2u

∂xi∂xj

∣∣∣∣2⎞⎠ dx,

∫Ω

(1 + |∇u|)q+2dx <∞.

(2.238)

These (2.231)–(2.235) are not independent. For example, (2.232), (2.233) imply partsof (2.231), in particular, the standard coercivity, see Ladyzenskaja and Uralceva [464],p. 255. This fact is generalized in Lemma 2.77 below. The above (2.232), and (2.289)below, are in some sense competing. Obviously, the above (2.232) indicates the


ellipticity for the linearized differential equation. By different combinations of theseconditions different results are achieved, see Theorems 1.1, 2.1, 5.2, 6.4, 6.5 in Chapter4 [464]. The existence of solutions u0 ∈W 1,q(Ω) is discussed in Chapter 4, Section8 [464], under highly technical additional conditions. There are some interestingdifferences between the conditions (2.230)–(2.232), imposed for m = 1, and for m ≥ 1,see (2.284)–(2.289) below.

Theorem 2.63. Regular, bounded solutions and uniqueness on small balls: Weassume a generalized bounded solution u0 ∈W 1,q(Ω) in the sense of (2.229) withM0 := ‖u0‖L∞(Ω) <∞, 0 ≤ α ≤ 1.

1. Bounded generalized solutions: Under the conditions (2.230) and (2.231), weget u0 ∈ C0,α(Ω) with α depending upon M0, q, μ(M0), ν(M0). For an arbitraryΩ1 ⊂⊂ Ω the ‖u0‖C0,α(Ω1) can be estimated from above by a constant dependingupon the previous constants and dist(Ω1, ∂Ω).If additionally ∂Ω satisfies (2.237) and if u|∂Ω = ϕ|∂Ω ∈ C0,β(∂Ω) for α ≤ β,then ‖u0‖C0,α(Ω) over the entire Ω can be estimated from above by a con-stant depending upon the previous constants and the a1, a2, in (2.237), β and‖ϕ‖C0,β(∂Ω) (Theorem 1.1).

2. Uniqueness in the small: Assume (2.232), (2.233) and let u1, u2 ∈W 1,q(Ω) betwo bounded solutions of (2.229), coinciding on the surface of a ball Bρ ⊂ Ω1 ⊂⊂Ω with radius ρ < ρ1, and max{‖u1‖L∞(∂Ω), ‖u2‖L∞(∂Ω)} =: M1 <∞. Then theycoincide in Bρ as well. This ρ1 is determined by the M1, q, μ(M1), ν(M1) anddist(Ω1, ∂Ω) (Theorem 2.1).

3. A bound for ‖∇u0‖L∞(Ω1) over Ω1 ⊂⊂ Ω and u0 ∈ H2(Ω1): Assume dif-ferentiable Ai(x, z, p), i = 0, . . . , n, and (2.232), (2.234), (2.235), and for x ∈Ω, |u| ≤M , and arbitrary q the (2.238). Then for an arbitrary Ω1 ⊂⊂ Ω we getu0 ∈ H2(Ω1). It satisfies (2.228) almost everywhere in Ω and ‖∇u0‖L∞(Ω1) can beestimated by constants depending only upon M0, q, μ(M0), ν(M0), ∂Ω, ‖ϕ‖C2,0(Ω),and dist(Ω1, ∂Ω) (Theorem 5.2).

4. Bounds for ‖u0‖Cl,α(Ω), l ≥ 2. Let M2 ≥ ‖∇u0‖L∞(Ω) and M := {(x, z, p) ∈Ω× R× Rn : |z| ≤M0, |p| ≤M2}. If, in addition, Ai(x, z, p) ∈ Cl−1,α(M), i =1, . . . , n, A0(x, z, p) ∈ Cl−2,α(M), and ∂Ω ∈ Cl,α, ϕ ∈ Cl,α(Ω), then u0 ∈Cl,α(Ω) and the ‖u0‖Cl,α(Ω) are bounded by constants depending uponM0,M2, q, μ(M0), ν(M0), ‖ϕ‖Cl,α(Ω), ∂Ω and ‖Ai‖Cl−1,α(M), ‖A0‖Cl−2,α(M)

(Theorem 6.5).

Finally, we summarize regularity results from Chen and Wu [170] for quasilinearequations (2.206). Essentially based upon the Leray–Schauder theorem, Chen andWu get solutions u0 ∈ C2,γ(Ω) under modified conditions. We present their Theorem5.1 in Chapter 5, for divergent quasilinear second order equations, but only refer toTheorem 4.1 in Chapter 7 for nondivergent quasilinear second order equations. Weuse the remark on p. 117 in Chapter 7 for fully nonlinear second order equationsformulating the higher regularity results u0 ∈ C2+k,γ(Ω), k > 0, in the last part ofthe following Theorem 2.64, compare [170], p. 78. Their Theorem 7.4. in Chapter 7for fully nonlinear second order equations will be discussed as Theorem 2.82 below.


Additional results are available as special cases of Theorem 2.91 below. Chen andWu study divergent equations of the form (2.207) and require ellipticity and otherconditions in the form

Ai(x, z, p) ∈ R,∀ (x, z, p) ∈ Uo ⊂ Ω× R× Rn, 0 �= p′ = (p′1, . . . , p′n) ∈ Rn,

∃Λ > λ > 0 : 0 < λ|p′|2 <

n∑i,j=1

∂Ai

∂pj(x, z, p)p′ip

′j < Λ|p′|2, (2.239)

|Ai(x, z, 0)| ≤ g(x) ∀i = 1, · · · , n, g ∈ Lq(Ω), q > n, and finally[(1 + |p|2)

∣∣∣∣∂Ai

∂pj(·)∣∣∣∣+ (1 + |p|)|

(∣∣∣∣∂Ai

∂z(·)∣∣∣∣+ |Ai(·)|

)+∣∣∣∣∂Ai

∂xj(·)∣∣∣∣ (2.240)

+ |A0(·)|]

(x, z, p) ≤ μ(|z|)(1 + |p|2) ∀i, j = 1, · · · , n,∀(x, z, p) ∈ Uo,

−A0(x, z, p) sign z ≤ Λ[|p|+ f(x)] for an f ∈ Lq∗(Ω), q∗ := nq/(n + q).

The relation for the ellipticity conditions in (2.232) and (2.239) is discussed in Lemma2.77 below for order 2m.

Theorem 2.64. Existence and regularity of solutions:

1. Let the conditions (2.239)–(2.240) be satisfied and assume ∂Ω ∈ C2,γ , Ai ∈C1,γ(Ω× R× Rn), A0 ∈ C0,γ(Ω× R× Rn), ϕ ∈ C2,γ(Ω) with 0 < γ < 1. Thenthere exists a solution u0 ∈ C2,γ(Ω) for the Dirichlet problem (2.206).

2. If additionally Ai ∈ Ck+1,α(Uo), A0 ∈ Ck,α(Uo), k ≥ 1, and ∂Ω ∈ C2+k,γ ,ϕ ∈ C2+k,γ(Ω), then the solution u0 ∈ C2,γ(Ω) ∩ Ck+2,α(Ω), forAi ∈ Ck+1,α(Uo), A0 ∈ Ck,α(Uo), even u0 ∈ Ck+2,γ(Ω).

Further regularity results are formulated in Subsection 2.5.7 and are studied byLadyzenskaja and Uralceva [463, 464, 466], Koshelev and Chelkak [449–451], Morrey[500], Necas [510], and Giaquinta and Hildebrandt [341,342].

These quasilinear equations are elliptic equations in the sense of Definition 2.51,since the linearized equation satisfies one of the ellipticity conditions (2.209), (2.214),(2.217), (2.224), (2.232), (2.239). In Lemma 2.77 below, we will discuss the relationof growth and coercivity conditions versus ellipticity for quasilinear equations oforder 2m.

2.5.5 General nonlinear and Nemyckii operators

In the last Subsections 2.5.3, 2.5.4 we studied nonlinear operators. We had to eval-uate, e.g. f(u) or aij(x, u,∇u) usually for given continuous functions. Therefore theevaluation was obvious. The situation changes if u ∈Wm,p(Ω), 1 ≤ p ≤ ∞. This isnecessary for studying monotone operators. They are, in some sense, evaluations ofquasilinear elliptic operators and the main tool for analytical results for their solutions.So we define different monotone and Nemyckii operators and summarize their essentialproperties, see Skrypnik [590,591], Zeidler [676–678], and Runst and Sickel [561].


For the next definition we need a function, see Zeidler [678], pp. 500 ff.

a : R+ → R+ used in (2.244), (2.253), continuous and strictly increasing with

a(0) = 0, limt→∞

a(t) =∞, e.g. a(t) = c|t|p−1, 0 < c, 1 < p. (2.241)

Definition 2.65. Let X be a Banach space, A : X → X ′ an operator. The followingconditions are imposed ∀u, v ∈ X . Then we call A16

monotone ⇔ 〈Au−Av, u− v〉 ≥ 0, (2.242)

strictly monotone ⇔ 〈Au−Av, u− v〉 > 0 for u �= v, (2.243)

uniformly monotone ⇔ 〈Au−Av, u− v〉 (2.244)

≥ a(‖u− v‖X )‖u− v‖X , e.g., ≥ c‖u− v‖pX ,

strongly monotone ⇔ 〈Au−Av, u− v〉 ≥ c‖u− v‖2X , c ∈ R+, (2.245)

(nonlinear) coercive ⇔ lim‖u‖X→∞

〈Au, u〉‖u‖X

= ∞, (2.246)

weakly coercive ⇔ lim‖u‖X→∞

‖Au‖X ′ =∞ for Hilbert spaces, (2.247)

where we identify X = X ′,

Lipschitz-continuous ⇔ ∃L > 0 : ‖Au−Av‖X ′ ≤ L‖u− v‖X , (2.248)

continuous ⇔ ∀ε > 0∃δ > 0 : ‖Au−Av‖X ′ < ε ∀‖u− v‖X < δ, (2.249)

hemicontinuous ⇔ t→ 〈A(u + tv), w〉 is continuous on [0, 1], (2.250)

strongly continuous ⇔ un ⇀ u ⇒ Aun → Au for n→∞, (2.251)

bounded ⇔ A maps bounded sets into bounded sets, (2.252)

stable ⇔ ‖Au−Av‖X ′ ≥ a(‖u− v‖X ). (2.253)

In (nonlinear) coercivity we often omit the “nonlinear”.

Remark 2.66. Obviously strong monotonicity implies uniform implies strict andimplies monotonicity, so (2.245)⇒ (2.244)⇒ (2.243)⇒ (2.242). Lipschitz-continuityimplies continuity and hemicontinuity, so (2.248) ⇒ (2.249) ⇒ (2.250). Obviously, auniformly monotone A is coercive and stable.

The preceding properties of A allow important consequences, see Zeidler [677],pp. 455 ff: Let X be a reflexive Banach space. We replace the bilinear forms a(·, ·)and the induced linear differential operators A : X → X ′ by nonlinear operatorsA : X → X ′, and the corresponding nonlinear forms. This is motivated by the results in

16 Note that for (2.251), un ⇀ u ∈ X is defined by 〈un − u, v〉 → 0∀v ∈ X ′.


Subsection 2.5.3. For generalizing Proposition 2.10 we start, cf. [677], pp. 314 ff., with a

nonlinear form a(·, ·) : X × X → R, s.t. |a(u, v)| ≤ C(u)‖v‖X ∀u, v ∈ X . (2.254)

Proposition 2.67. Under the condition (2.254) there exists a usually nonlinear butunique operator, such that, for all u ∈ X ,

〈Au, v〉X ′×X := a(u, v) ∀ v ∈ X , denoted as A : X → X ′, Au ∈ X ′. (2.255)

Then for any f ∈ X ′ the following three problems are equivalent:

Determine u0 ∈ X s.t. Au0 = f ∈ X ′ ⇔ a(u0, v) = 〈f, v〉X ′×X∀v ∈ X (2.256)

⇔ 〈Au0, v〉X ′×X = 〈f, v〉X ′×X∀v ∈ X .

Obviously (2.254) and (2.255) and the following considerations can be generalized toa(·, ·) : D(a) ⊂ X × X → R, and A : D(A) ⊂ X → R(A) ⊂ X ′, respectively.

Theorem 2.68. Properties of nonlinear operators, Zeidler [678], based upon Zaran-tonello, Browder, and Minty [150, 491, 674]: Let X be a real Banach space andA : X → X ′ a (nonlinear) operator. Then

1. If A is linear and monotone, then A is continuous, see [678], Proposition 26.4.2. If A is monotone, coercive, and hemicontinuous on a reflexive Banach space, then

A is surjective and, for each f ∈ X ′, the set of solutions of Au = f is bounded,convex and closed, cf. [678], Theorem 26A(a) [150,491].

3. If A is additionally strictly monotone, the above solution is unique. Then A−1 :X ′ → X exists and, cf. [678], Theorem 26A(c), (d) [150,491],

a uniformly monotone A implies a continuous A−1;a strongly monotone A implies a Lipschitz-continuous A−1.

4. Let X be a real Hilbert space and A be strongly monotone and Lipschitz-continuous with constant L. Then for each f the Au = f ∈ X ′ has a unique solu-tion, depending continuously upon f , more precisely, see [674], [678], Theorem25B,

‖u1 − u2‖X ≤ L−1‖f1 − f2‖X ′ for Au1 = f1, Au2 = f2, f1 ∈ X ′.

Definition 2.69. For given functions f and appropriate u ∈ D(F ), define F (u) as

f : D(f) ⊂ Ω× Rk → R, u : Ω → Rk : F (u)(x) := f(x, u1(x), · · · , uk(x)), (2.257)

with (x, u1(x), · · · , uk(x)) ∈ D(f). This F is called a Nemyckii operator.

Now we formulate two conditions guaranteeing that F (u)(x) makes sense. TheCaratheodory and the growth condition enforce measurability and appropriate growth


of the composite function F ; more precisely

Caratheodory condition: For f in (2.257) let (2.258)

Ω ⊂ Rn be nonempty, measurable and x �→ f(x, u) measurable on Ω ∀u ∈ Rk

u �→ f(x, u) is continuous on Rk almost everywhere (a.e.) for x ∈ Ω.

Growth condition: ∀(x, u) ∈ Ω× Rk let |f(x, u)| ≤ a(x) + b

k∑i=1

|ui|pi/q, (2.259)

with fixed reals 0 < b, 1 ≤ pi, q <∞ and fixed a ∈ Lq(Ω), 0 ≤ a(·)a.e.

Theorem 2.70. Estimates for nonlinear operators, Zeidler [678], Proposition 26.6:If (2.258), (2.259) are satisfied, then

F : Πki=1L

pi(Ω) → Lq(Ω)

is continuous and bounded (this is not equivalent for nonlinear operators!) such that

‖F (u)‖Lq(Ω) ≤ C

(‖a‖Lq(Ω) +

k∑i=1

‖ui‖pi/qLpi (Ω)

)∀u ∈ Πk

i=1Lpi(Ω). (2.260)

Particularly important is the case k = 1. Then the Nemyckii operator has the form

F (u)(x) := f(x, u(x)) and f : D(f) ⊂ Ω× R → R,

mainly F : X = Lp(Ω) → X ′ = Lq(Ω), 1/p + 1/q = 1. (2.261)

We summarize appropriate conditions for f . The Caratheodory condition is onlymodified by replacing Rk by R. We specialize the growth condition as

Growth condition: For fixed reals 0 < b, 1 < p, q <∞,

1/p + 1/q = 1 ⇔ p− 1 = p/q, a ∈ Lq(Ω), 0 ≤ a(·) a.e., (2.262)

let |f(x, u)| ≤ a(x) + b|u|p−1 ∀ (x, u) ∈ Ω× R.

We assume u, v ∈ R, and ∀x ∈ Ω, and denote the function f in (2.261) to be

monotone ⇔ f(x, u) ≤ f(x, v) for u ≤ v, (2.263)

strictly monot. ⇔ f(x, u) < f(x, v) for u < v, (2.264)

coercive ⇔ ∃d > 0, g ∈ L1(Ω) : f(x, u)u ≥ d|u|p + g(x), (2.265)

positive ⇔ f(x, u)u ≥ 0, (2.266)

asympt. positive ⇔ ∃R > 0 : f(x, u)u ≥ 0 for |u| ≥ R,meas Ω <∞. (2.267)


Theorem 2.71. Properties of Nemyckii operators, Zeidler [678], Proposition 26.7:For the function f in (2.261) we assume (2.258) with Rk replaced by R, (2.262) andlet F : X = Lp(Ω) → X ′ be the Nemyckii operator defined by f, cf. (2.261).

1. Then F is continuous and bounded with

‖F (u)‖Lq(Ω) ≤ C(‖a‖Lq(Ω) + ‖u‖p−1

Lp(Ω)

)and (2.268)

〈F (u), u〉X ′×X =∫

Ω

f(x, u(x))u(x)dx∀ u ∈ X .

2. A (strictly) monotone f , cf. (2.263), (2.264), implies a (strictly) monotone F.3. A coercive f , see (2.265), implies a coercive F and

〈F (u), u〉X ′×X ≥ d‖u‖pLp(Ω) +

∫Ω

g(x)dx∀ u ∈ X .

4. A positive f , see (2.266), implies a positive F, hence

〈F (u), u〉X ′×X ≥ 0 ∀ u ∈ X .

5. An asymptotic positive f , see (2.267), implies an estimate for F in the form

〈F (u), u〉X ′×X ≥ − c ∀ u ∈ X for a constant c > 0.

These properties of general nonlinear and Nemyckii operators are essential tools inthe study of divergent quasilinear elliptic equations.

2.5.6 Divergent quasilinear elliptic equations of order 2m

We start with the strong form of a quasilinear elliptic operator, G. The presentationis motivated by the properties of Nemyckii operators in the last Subsection 2.5.5. Atthe end of this subsection, we will show that, for the most important cases, underslightly stronger conditions, these quasilinear equations are elliptic equations in thesense of Definition 2.51. This allows us to prove convergence of our discrete solutionsto the exact solutions by our general techniques with the correct rate of convergencefor all the classes of space discretization methods, considered here, and the problems inWm,p(Ω), with 2 ≤ p ≤ ∞. A general convergence theory for monotone and quasilinearoperators with its “pros and cons” is elaboreted in Section 4.5 for 1 < p <∞.

We only consider Dirichlet boundary conditions on bounded domains Ω, compareTheorem 2.50, and recall the standard notation in (2.73), (2.165), (2.166):

ϑ ∈ Rn, Θ0, ϑi, ϑα ∈ R, Θk = (tα ∈ R)|α|=k ∈ Rnk , often = (ϑα)|α|=k, (2.269)

Aα defined as Aα : Ω× RNm → R, w = (x,Θ0, . . . ,Θm) =: (x,Θ≤m) ∈ Ω× RNm ,

with RNm = R× Rn × · · · × Rnm , norms, e.g. |ϑ|, |Θm|, and

Nm := (nm+1 − 1)/(n− 1) for n > 1, Nm := m + 1 for n = 1, partials

∂αu :=∂|α|

∂xα11 . . . ∂xn

αnu,∇0u := ∂αu := u for |α| = 0,∇ku := (∂αu)|α|=k.


We discuss the general form of quasilinear elliptic equations with Dirichlet boundaryconditions, see Skrypnik [590,591] and Zeidler [678]:

Gu− f = Gsu− f =∑

|α|≤m

(−1)|α|∂α(Aα(x, u, · · · ,∇mu)− fα) = 0 on Ω, (2.270)

=∑

|α|,|β|≤m

Aα,β(x, u, · · · ,∇mu)∂α∂βu− (−1)|α|∂αfα = 0 on Ω, with, e.g.

Aα,β(x, u, · · · ,∇mu) = (−1)|α| ∂mAα(x, u, · · · ,∇mu)

∂ϑβ∀|α| = |β| = m

Ω a bounded nonempty measurable domain in Rn and (2.271)

Dirichlet boundary conditions ∂iu(x)/∂νi = 0 ∀ 0 ≤ i ≤ m− 1, on ∂Ω

⇔ ∂αu|∂Ω = 0 ∀|α| ≤ m− 1 in the trace sense for smooth enough ∂Ω.(2.272)

A possible condition u ∈W 2m,p(Ω) is still too restrictive. Below, we will list condi-tions allowing us to multiply (2.270) with v ∈ V, and to integrate over Ω, where

V := Wm,p0 (Ω) := {v ∈Wm,p(Ω) : ∂αv = 0 ∀|α| ≤ m− 1 on ∂Ω}, 1 < p <∞.

We iteratively use the fundamental Green’s formula (2.9). For smooth enough w1, w2

and |α| ≤ m− 1 for w1 we get

w1 = ∂αv, w2 = ∂α(Aα(x, u, · · · ,∇mu)) :∫

Ω

∂(w1w2)/∂xidx =∫

∂Ω

(w1w2)νids.

For v, w ∈ H1(Ω) we obtain by partial integration and with the outer normal ν =(ν1, . . . , νn)T on a Lipschitz-continuous manifold ∂Ω, the fundamental Green’s formula,cf. Ciarlet [174], p. 14,∫

Ω

w∂v/∂xidx +∫

Ω

v∂w/∂xidx =∫

Ω


∂Ω

vwνids, (2.273)

∂Ω ∈ CL,∀w, v ∈ H1(Ω).

This yields the following weak form for G defined for u, v ∈ V = Wm,p0 (Ω).

as(u, v) =∫

Ω

vGsudx =∫

Ω

∑|α|≤m

v(−1)|α|∂α(Aα(x, u, · · · ,∇mu))dx, u ∈W 2m,p

=∫

Ω

∑|α|≤m

Aα(x, u, · · · ,∇mu)∂αvdx =: a(u, v) =: 〈Gu, v〉V′×V ∀ u, v ∈ V

(2.274)

〈Gu0, v〉V′×V = 〈f, v〉V′×V =∫

Ω

∑|α|≤m

fα∂αvdx ∀ u0 ∈ V ∀ v ∈ V.

(2.275)

The Aα(x, u, . . . ,∇mu) are Nemyckii operators, studied in the last subsection.


These weak forms of quasilinear equations are often obtained as Euler equations forvariational problems. For a smooth function F (x, u, . . . ,∇mu), a functional

J(u) :=∫

Ω

F (x, u, . . . ,∇mu)dx (2.276)

has to be (locally) minimized. The first variation of this functional applied to v yields aform as in (2.274), for more details cf. Subsection 2.6.4. This is one of the reasons thatfor the class of quasilinear differential equations the most complete and final resultsare available.

Example 2.72.

Important model problems of second order:

1. For this example it is important that [678], p. 592 ff., H10 (Ω) is compactly

embedded into L4(Ω) for dimension n = 1, 2, 3:

Gu = −Δu + α

n∑i=1

sin(u)∂iu = f ∈ L2(Ω) on Ω, (2.277)

n = 1, 2, 3, and the Dirichlet condition u(x) = 0 on ∂Ω. (2.278)

We define linear, bilinear and nonlinear forms related to Gu = f as

b(v) :=∫

Ω

fvdx, a1(u, v) :=∫

Ω

n∑i=1

∂iu∂ivdx ∀u, v, w ∈ H10 (Ω) = V,

c(u,w, v) :=∫

Ω

n∑i=1

(sinu)∂iwvdx, a2(u, v) := c(u, u, v), and (2.279)

determine u0 ∈ H10 (Ω) s.t. a1(u0, v) + αa2(u0, v) = b(v) ∀v ∈ H1

0 (Ω).

2. We assume 2 ≤ p <∞, 1/p + 1/q = 1, R � t ≥ 0, and Dirichlet boundary con-ditions, see [678], p. 567 ff., for

Gu = Gsu = −n∑

i=1

∂i(|∂iu|p−2∂iu) + tu = f ∈ Lq(Ω) on Ω. (2.280)

With the above b ∈ L2(Ω) we obtain the weak form of G∫Ω

vGsudx =∫

Ω

v

(−

n∑i=1

∂i(|∂iu|p−2∂iu) + tu

)dx

= a(u, v) := 〈Gu, v〉W−1,q(Ω)×W 1,p(Ω) (2.281)

:=∫

Ω

(n∑

i=1

|∂iu|p−2∂iu∂iv + tuv

)dx and determine

u0 ∈W 1,p0 (Ω) s.t. a(u0, v) = 〈f, v〉V′×V ∀v ∈W 1,p

0 (Ω). (2.282)


3. Minimal surface equation, see Gilbarg and Trudinger [346], p. 339: Let theminimal surface be represented by the function u : Ω → R. Then, with prescribedboundary conditions u = ϕ on ∂Ω, it has the form

Gu = Gu = (1 + |∇u|2)Δu +n∑

i,j=1

∂iu∂ju∂i∂ju = 0 on Ω. (2.283)

This is an example where the Schauder techniques with boundary gradientestimates are applied, see [346]. �

Now we impose conditions which guarantee that G(u) in (2.270), (2.277), (2.280),(2.283), and its weak form (2.274), (2.283) are well defined. The Caratheo-dory andgrowth conditions enforce measurability and appropriate growth of the compositefunction Aα, see [678], p. 571 ff. and (2.269) for the notation. More generally thanthe coercivity for bilinear forms we introduce a nonlinear coercivity, monotonicity andgrowth condition for these quasilinear operators, cf. (2.230)–(2.235), (2.242)–(2.247),(2.258), (2.259). This is again essential for the existence of solutions.

Caratheodory condition: Ω is bounded and measurable in Rn,

Aα : (x,Θ≤m) ∈ Ω× RNm → R,∀ |α| ≤ m,

Θ≤m �→ Aα(x,Θ≤m) is continuous in RNm a.e. for x ∈ Ω, (2.284)

x �→ Aα(x,Θ≤m) is measurable in Ω ∀ Θ≤m ∈ RNm , e.g. for continuous Aα.

For the following conditions, we assume 1 < p <∞, 1/p + 1/q = 1, the special formΘ≤m = (ϑγ)|γ|≤m, Θ≤m

1 = (ϑγ1)|γ|≤m ∈ RNm and ∃ c, d, C ∈ R+, such that

∃ a ∈ Lq(Ω), h ∈ L1(Ω), s.t. ∀ x ∈ Ω,Θ≤m,Θ≤m1 ∈ RNm , ϑ0, ϑα, ϑα

1 ∈ R.

Growth condition: |Aα(x,Θ≤m)| ≤ C(a(x) +∑

|γ|≤m

|ϑγ |p−1) ∀|α| ≤ m. (2.285)

(Nonlinear) coercivity:∑

|α|≤m

Aα(x,Θ≤m)ϑα ≥ c∑

|γ|=m

|ϑγ |p − h(x). (2.286)

Monotonicity:∑

|α|≤m

(Aα(x,Θ≤m)−Aα(x,Θ≤m1 )) (ϑα − ϑα

1 ) ≥ 0. (2.287)

Strict monotonicity:∑

|γ|≤m

|ϑγ − ϑγ1 | > 0 implies (2.288)

∑|α|≤m


1 ) > 0.

Uniform monotonicity: (2.289)∑|α|≤m


1 ) ≥ C∑

|γ|=m

|ϑγ − ϑγ1 |

p.


Other types of coercivity estimates towards (2.293) are discussed, e.g. by Showalter[589], p. 38 ff. By [678], Proposition 26.12, (2.284)–(2.287) imply the conditions of hisTheorem 26A.

Theorem 2.73. Generalized solutions for (2.270), (2.275), [678], pp. 572 ff.: Let(2.284)–(2.285) be satisfied and choose V = Wm,p

0 . Let

Fα : V = Wm,p0 (Ω) → Lq(Ω), (Fαu)(x) := Aα(x, u(x), · · · ,∇mu(x)).

1. Then for all |α| ≤ m,∀x ∈ Ω, these Nemyckii operators Fα are well defined,continuous and bounded, see [678], Proposition 26.12, Proof.

2. With the above Aα(x, u(x), . . . ,∇mu(x)) define the a(u, v) as in (2.274). Thenthere exists exactly one bounded operator G, cf. Proposition 2.67, such that, see[678], Proposition 26.12,

G : V → V ′ : a(u, v) = 〈Gu, v〉V′×V ∀u, v ∈ V = Wm,p0 (Ω) (2.290)

and ‖Fαu‖Lq(Ω), ‖Gu‖W−m,q(Ω) ≤ C

(‖a‖Lq(Ω) +

(‖u‖W m,p

0 (Ω)

)p/q).

Similarly to (2.275) and with 〈f, v〉V′×V , the original problem (2.270) correspondsto the generalized problem: Determine for a(u, v) in (2.290)

u0 ∈ V = Wm,p0 (Ω) s.t. a(u0, v) = 〈f, v〉V′×V∀v ∈ V ⇐⇒ Gu0 = f. (2.291)

3. Let, additionally, all the Aα be coercive and monotone, see (2.286) and (2.287).Then G is monotone, coercive, continuous and bounded, see [678], Proposition26.12. So by Remark 2.66, parts 2.−4. of Theorem 2.68 are valid for for (2.291).

4. In particular, for any f ∈W−m,q(Ω) = V ′ the (2.291) has a solution u0 ∈Wm,p

0 (Ω) and the set of solutions is bounded, convex and closed. If all the Aα

are additionally strictly monotone, see (2.288), Theorem 2.68, then G is strictlymonotone as well and the above solution is unique. So the inverse G−1 : V ′ → Vexists and is strictly monotone and bounded, see [678], Theorem 26A.

5. If all the Aα are, and hence G is uniformly monotone, see (2.289), or if Gis strongly monotone, see (2.245), this G−1 is continuous or even Lipschitz-continuous.

Remark 2.74. For our above model problems (2.277), (2.280) and (2.283) withhomogeneous Dirichlet boundary conditions we find:

1. There exists an α0 > 0, such that for all α ∈ R, |α| ≤ α0, (2.279), correspondingto (2.277), has exactly one solution u0 ∈ H1

0 (Ω), see Proposition 27.11 [678].2. (2.281), corresponding to (2.280), has exactly one solution u0 ∈W 1,p

0 (Ω). (2.281)is equivalent to Gu0 = b and G is uniformly monotone, coercive, continuous andbounded, see Proposition 26.10 [678].

3. More complicated conditions for (2.283) are discussed in [346], Chapters 12–15.

We modify the preceding results, see Zeidler [678], Problem 27.6, by requiring, fora quasilinear operator G, some conditions only for the principal part:


Theorem 2.75. Modified coercivity and monotonicity only for the principal part:For (2.270), let Ω be bounded, and choose V = Wm,p

0 (Ω). Assume the Caratheodory,the growth and the coercivity conditions, (2.284)–(2.286). Additionally impose the fol-lowing modified coercivity (2.293) and the special case (2.294) of the strict monotonic-ity (2.288), both only for the principal part: With, cf. (2.269),

Θ≤m = (Θ0, . . . ,Θm−1,Θm) =: (d,Θm),Θm = (ϑα)|α|=m, (2.292)

let a.e. ∀x ∈ Ω ∀Θm ∀ bounded sets H ⊂ RNm−nm :

lim|Θm|→∞

supd∈H

∑|α|=m

Aα(x, d,Θm)ϑα

|Θm|+ |Θm|p−1= ∞, (2.293)

∑|α|=m

(Aα(x, d,Θm)−Aα (x, d,Θm1 )) (ϑα − ϑα

1 ) > 0,∀Θm �= Θm1 ∈ Rnm . (2.294)

Then ∀f ∈ Lq(Ω) the Gu = f has a generalized solution u0 in the sense of

u0 ∈Wm,p0 (Ω) : a(u0, v) = b(v) ∀v ∈Wm,p

0 (Ω). (2.295)

Uniqueness results are available via linearization in Section 2.7.

The following result, again for special semilinear operators, is based upon theCaratheodory and growth conditions for the involved Nemyckii operator. We combinethe linear operator A in strong divergence form, see (2.196),

Au =∑

|α|,|β|≤m

(−1)|β|∂β(aαβ∂αu), u ∈ H2m(Ω) (2.296)

with Dirichlet boundary conditions. We replace in (2.197) the f(u) by f(x, u) anddetermine u0 such that

G(u0)(x) := (A + μ)u0(x) + f(x, u0(x)) = 0 a.e. ∀x ∈ Ω, u ∈ Hm0 ∩H2m(Ω). (2.297)

As usual we find the following generalized form, for (2.297)

with aμ(u, v) =∫

Ω

∑|α|,|β|≤m

aαβ∂αu∂βv + μuvdx determine (2.298)

u0 ∈ V := Hm0 (Ω) : aμ(u0, v)−

∫Ω

f(x, u0(x))v(x)dx = 0 ∀v ∈ V. (2.299)

For A, aμ(·, ·), f in (2.297), (2.298), and the domain Ω we impose the conditions

Ω bounded, aαβ : Ω → R measurable and bounded ∀|α|, |β| ≤ m, (2.300)

a(·, ·) is V-elliptic, ⇐⇒ aμ(·, ·) is V-coercive, for sufficiently large μ,

f : Ω× R → R satisfies the Caratheodory condition (2.258) for n = 1,

(e.g. f continuous) and the growth condition (2.262) for p = 2, so fixed

b ∈ R+, 0 ≤ a ∈ L2(Ω) exist, s.t. |f(x, u)| ≤ a(x) + b|u| ∀(x, u) ∈ Ω× R.


Theorem 2.76. Existence and uniqueness for semilinear problems, [678], Proposi-tion 28.9: Assume the conditions (2.300) for a V-coercive aμ(·, ·), and for the boundaryvalue problem (2.297) or its generalized form (2.299).

1. Existence and uniqueness. Let the function u→ f(x, u) be monotone increasingon R for all x ∈ Ω. Then (2.299) has exactly one solution.

2. Existence. Equation (2.299) has a solution, if the function (x, u) → f(x, u)satisfies

∃ R > 0 s.t. ∀u with |u| ≥ R : f(x, u)u ≥ 0 ∀ (x, u) ∈ Ω× R.

We return to the question of whether quasilinear equations are elliptic equationsin the sense of Definition 2.51, hence, the linearized equation is an elliptic equation.This has several important implications for our later discussion of the convergenceof discretization methods and for the Fredholm alternative for quasilinear equations,see Section 2.7. For analytical results and our applications in discretizations, differenttypes of conditions are imposed. We want to show that the above monotonicity impliesellipticity and we relate two important concepts.

We prove that an ellipticity condition for the linearization of the principal partessentially implies the nonlinear coercivity in the sense of (2.293) for the originalquasilinear equation, cf. Theorem 2.75. Generalizing a remark by Ladyzenskaja andUralceva [464], p. 255, we formulate the following lemma, cf. (2.285), (2.289), (2.292)–(2.294).

Lemma 2.77. Quasilinear elliptic and nonlinear coercive equations: Let the equa-tion (2.270), (2.274) satisfy the following simplified growth and nonstandard ellip-ticity condition:17 There exist positive continuous functions γ,Γ : RNm−1 → R+ andμ : Rnm → R+ such that with

|Aα(x, d,Θm = 0)| ≤ Γ(d) ∀|α| ≤ m, and μ(Θm) := (1 + |Θm|)p−2, p ≥ 2, (2.301)

γ(|d|)μ(Θm)∑

|γ|=m

|ϑγ |2 ≤∑

|α|=|β|=m

∂Aα

∂ϑβ(x, d,Θm)ϑβϑα ≤ Γ(|d|)μ(Θm)

∑|γ|=m

|ϑγ |2.

1. If 0 ≤ γ ≤ γ(|d|) and Γ(|d|)/γ(|d|) is bounded, we obtain uniform ellipticity. Thenthe principal part of the linearization evaluated in u0, hence in (x, d,Θm) =(x, ∂αu0, |α| ≤ m), is uniformly elliptic, cf. Definition 2.51.

2. For p ≥ 2, the principal part is nonlinear coercive for m ≥ 1, cf. (2.293) and, form = 1 in the sense of (2.231) only for bounded |Θm|. More precisely,∑

|α|=m

Aα(x, d,Θm)ϑα ≥ C (γ(|d|)|Θm|p − Λ(d)|Θm|) . (2.302)

Proof. The uniform ellipticity of the principal part of the linearization is an immediateconsequence of (2.301) combined with the continuity of the γ,Γ, μ evaluated in abounded subdomain defined by (x, d,Θm) = (x, ∂αu0, |α| ≤ m), x ∈ Ω.

17 We have called (2.301) a nonstandard ellipticity condition, since we have chosen the nonstandardform of γ(|d|)μ(Θm) instead of the usual ε.


For the coercivity, we modify the proof in [464], p. 255, and start with

(1 + x)α ≥ 1 + xα, ∀x ≥ 0, 1 ≤ α ∈ R. (2.303)

We rescale, for 1 < x, the (1 + x)α = xα(1 + 1/x)α and 1 + xα = xα(1 + (1/x)α). Soit suffices to apply the Taylor formula and find, with 0 < θ < 1, for

∀0 ≤ x ≤ 1, 1 ≤ α ∈ R : (1 + x)α − 1 = α(1 + θx)α−1x ≥ x ≥ xα.

The last inequality α(1 + θx)α−1x ≥ xα certainly does not hold ∀0 ≤ x, α ≤ 1. In factfor a fixed x > 0 we get

limα→0+

α(1 + θx)α−1x = 0 < limα→0+

xα = 1.

Combining (2.303) with directional derivatives, integration and (2.301) we find withμ(Θm) = (1 + |Θm|)p−2 and the mean value theorem 1.43 for operators, with Θm =(ϑβ |β| = m),

Aα(x, d,Θm) = Aα(x, d, 0) +∑

|β|=m

ϑβ

∫ 1

0

∂Aα

∂ϑβ(x, d, τΘm)dτ, hence,

∑|α|=m

Aα(x, d,Θm)ϑα

=∑

|α|=m

(Aα(x, d, 0) +

∑|β|=m

∫ 1

0

∂Aα

∂ϑβ(x, d, τΘm)dτϑβ

)ϑα

≥ −CΓ(|d|)|Θm|+ γ(|d|)∫ 1

0

((1 + τ |Θm|)p−2

)dτ∑

|γ|=m

|ϑγ |2

≥ −CΓ(|d|)|Θm|+ γ(|d|) 1|Θm|(p− 1)

(1 + τ |Θm|)p−1|10∑

|γ|=m

|ϑγ |2

≥ −CΓ(|d|)|Θm|+ γ(|d|) 1|Θm|(p− 1)

((1 + |Θm|)p−1 − 1p−1)∑

|γ|=m

|ϑγ |2

(2.303)

≥ −CΓ(|d|)|Θm|+ γ(|d|) |Θm|p−1

|Θm|(p− 1)|Θm|2, for p ≥ 2, p ∈ R,

≥ −CΓ(|d|)|Θm|+ γ(|d|)|Θm|p.

γ,Γ are continuous with respect to d, so they are bounded for all bounded sets H ⊂RNm−nm , see (2.73), implying (2.293) since with p ≥ 2

lim|Θm|→∞

supd∈H

Cγ(|d|)|Θm|p − Γ(|d|)|Θm|

|Θm|+ |Θm|p−1=∞ ∀ bounded H ⊂ RNm−nm .

This proves the claim. �


General regularity results of an as “simple” form as in Section 2.4 do not seemto exist or did not find their way into the general monographs summarized here. InSubsection 2.5.4 we presented some examples for order 2, and in Subsection 2.6.6 andTheorem 2.116 we will present results for general systems of order 2m.

For a lot of other results we refer to the literature. Morrey [500] and Giaquinta andHildebrandt [341, 342] study quasilinear equations induced by variational problems.Giaquinta and Hildebrandt [341, 342] essentially consider second order problems.Then the functional in (2.276) is often assumed in the form J(u) :=

∫ b

aF (u,∇u)dt

where u = u(t), a ≤ t ≤ b, indicates a curve on a surface. Under appropriate condi-tions a minimizing curve is then u0 ∈ C2([a, b]), cf. [341, 342], Chapter 1, Section3, Proposition 3 and Chapter 8, Section 4, Proposition 2. Relations of minima ofconvex functionals or duality and conjugate functionals and quasilinear equationsof orders 2 and 2m are discussed as well by Zeidler [675], Sections 42.7 and 51.7.Koshelev and Chelkak [449, 450] study regularity for systems, and Ladyzenskajaand Uralceva [463, 464, 466], for second order problems of elliptic and parabolictype. Ladyzenskaja [462] discusses boundary value problems of mathematical physics.Morrey in his many papers uses multiple integrals and the calculus of variations,and studies differentiability of solutions for nonlinear elliptic equations and systems,e.g. [500–504].

2.5.7 Fully nonlinear elliptic equations of orders 2, m and 2m

We turn to the fully nonlinear case and start with the most important order 2, seeGilbarg and Trudinger [346] and Chen and Wu [170]. We require Ω ⊂ Rn satisfying(2.5) and a real valued function, Gw, cf. (2.165), (2.166), of the form

Gw : D(Gw) → R, w = (x, z, p, r) ∈ D(Gw) ⊂ Uo = Ω× R× Rn × Rn2; (2.304)

we restrict the elements in Rn×n to the n(n + 1)/2-dimensional subspace of symmetricreal valued n× n matrices. Choose functions u : Ω→ R, such that

D(G) := {u ∈ H2(Ω) : wu(x) := (x, u(x),∇u(x),∇2u(x)) ∈ D(Gw)∀x ∈ Ω}. (2.305)

Remark. Fully nonlinear PDE of order 2 and 2m

1. A fully nonlinear PDE of order 2 is none of the above special cases (2.169)–(2.172). It has, for a Dirichlet boundary value problem, the strong classical form,hence

G : D(G) ⊂ H2(Ω) → L2(Ω), with u, u0 ∈ D(G) s.t. (2.306)

L2(Ω) � G(u0) := Gw(·, u0(·),∇u0(·),∇2u0(·)) = 0 on Ω, u0 = ϕ on ∂Ω.

In particular, the spaces of ansatz functions, u, u0 ∈ D(G) ⊂ H2(Ω), and of testfunctions, v ∈ L2(Ω), are different.

2. The same situation holds for the equations of order 2m and m at the end of thissubsection and the systems in Subsection 2.6.8.


In Definition 2.51 we have introduced (uniformly) elliptic G. For second orderequations, m = 1, the −1 in (2.175), (2.176) is often omitted. Choose w = (x, z, p, r) ∈D(Gw) as in (2.304), a u1 with its w1(x) := (x, u1(x),∇u1(x),∇2u1(x)) ∈ D(Gw), andw ∈ Uw1 , a neighborhood of w1. We assume for 0 �= ϑ ∈ Rn

0 < λ|ϑ|2 ≤n∑

i,j=1

∂Gw

∂rij(w)ϑiϑj ≤ Λ|ϑ|2∀w = (x, u, p, r) ∈ Uw1 ⊂ D(Gw), (2.307)

for its principal part. For a real symmetric matrix ∂Gw/∂rij(w) with its real eigen-values the minimal and maximal eigenvalues satisfy λ ≤ λ(w) ≤ Λ(w) ≤ Λ.

Example 2.78. The two of the most important examples:

1. Monge–Ampere equationThis is, for u : Ω → R and n ≥ 2

0 = G(u) := det∇2u− f(x, u,∇u) = uxxuyy−u2xy−f(x, u,∇u) for n = 2. (2.308)

We claim that this G is uniformly elliptic in u1 if ∇2u1 is strictly positive definiteand symmetric in Ω: standard textbooks show that, for a strictly convex u1 ∈C2(Ω) on a convex Ω, this is correct. Furthermore, a matrix and its inverseare simultaneously strictly positive definite and have a positive determinant.We define the cofactor of the element ri,j = ∂i∂ju = ∂2u/∂xi∂xj of the matrix∇2u = r = [rij = ∂i∂ju]ni,j=1, symmetric for u ∈ C2(Ω): cancel row i and columnj in ∇2u to get Rij and define, independently of ri,j = ∂i∂ju, the cofactor of∂i∂ju as (−1)i+j detRij . Then

det∇2u =n∑

j=1

∂i∂ju cofactor ∂i∂ju =n∑

j=1

∂i∂ju∂ det∇2u

∂rij(u)

⇒ (det∇2)′(u1)u =n∑

j=1

∂i∂ju∂ det∇2u

∂rij(u1). (2.309)

The inverse of a nonsingular matrix is obtained by the matrix of its cofactors as:

with B := [ cofactor ∂i∂ju1]ni,j=1 we get (∇2u1)−1 = BT /det∇2u1.

This (2.309) implies with (2.175), (2.307),

ϑT (det∇2)′(u1)ϑ =n∑

i,j=1

ϑiϑj ∂ det∇2u

∂rij(u1) = det∇2u1 ϑT (∇2u)−1(u1) ϑ,

hence our claim by (2.307).So the G(u) in (2.308) is uniformly elliptic in u1 for a symmetric strictly positivematrix r = ∇2u1 in Ω, hence for a function u1 ∈ C2(Ω), strictly convex ∀x ∈ Ωon a convex Ω. With det(∇2u) > 0, uniformly convex solutions are only possiblefor positive f , [345], pp. 441 ff. For partially nonelliptic generalizations, see [345],p. 467ff. and Courant and Hilbert [214], p. 324. According to Haltiner andKasahara the leading term in the balance equation in dynamical meteorology


has the form (2.308) [389,437]. In Westcott [664] it is related to geometric optics.Among the many applications in geometry we mention the

2. Equation for a surface with prescribed Gauss curvature K = K(x) at xThis is closely related to the Monge–Ampere Equation. Equation (2.308) has tobe modified as

G(u) := det∇2u−K(1 + |∇u|2)(n+2)/2 = 0, K = K(x) > 0. (2.310)

Again G(u) is elliptic only for uniformly convex functions u ∈ C2(Ω) in a convexΩ. Our FEMs for this class of problems are interesting for aspects of globalgeometry. The existence of surfaces with prescribed Gauss curvature is knownfor appropriate K. Specific properties would become more flexibly available viaour FEMs than by the available finite difference methods, see e.g. Crandall andLions [227]. Our FEM allows us FEs in C1 ∩W on quasiuniform grids. It yieldshigher orders of convergence, cf. Theorem 5.18, than the difference methods oforder 2, e.g. in Chapter 8 or in [227]. Both methods are only defined on equidistantgrids. �

For existence and uniqueness results we have to impose growth conditions. Wedistinguish the cases n = 2 and n ≥ 2 and evaluate the derivatives Gw

x , Gwz , G

wp , for Gw

r ,compare (2.307). We assume nonnegative λ = λ(·) and Λ = Λ(·), μ = μ(·) in (2.307),(2.311)–(2.312) depending upon |z|, nonincreasing and nondecreasing in z ∈ [0,∞),respectively, and require for w = (x, z, p, r), x ∈ ∂Ω,

for n = 2 : |Gwz (w)| ,

∣∣Gwp (w)

∣∣ ≤ λμ

|Gwx (w)| ≤ μλ(1 + |p|+ |r|) and (2.311)

for n ≥ 2 : |Gwz (w)| ,

∣∣Gwp (w)

∣∣ , |Gwrx(w)| ,

∣∣Gwpx(w)

∣∣ , |Gwzx(w)| ≤ λμ

|Gwx (w)| , |Gw

xx(w)| ≤ μλ(1 + |p|+ |r|). (2.312)

Theorem 2.79. Existence and uniqueness for fully nonlinear problems in R2, [346],Theorem 17.12: Assume n = 2, ∂Ω ∈ C3, ϕ ∈ C3(Ω). Let G satisfy the conditions(2.307), (2.311), and Gw

z ≤ 0 in Uo, see (2.304). Then the classical Dirichlet problem(2.306) has a unique solution u0 ∈ C2,α(Ω) for all 0 < α < 1.

Ω is called satisfying an exterior sphere condition for every ξ ∈ ∂Ω, if there exist aball B = BR(y) ⊂ Rn \ Ω such that B ∩ Ω = {ξ}.

Theorem 2.80. Regular solutions for fully nonlinear problems in Rn, n ≥ 2, [346],Theorem 17.17: Assume n ≥ 2, ϕ ∈ C(∂Ω). Let Ω satisfy an exterior sphere conditionfor every ξ ∈ ∂Ω, Gw ∈ C2(Uo) be concave (or convex) with respect to (z, p, r),nonincreasing with respect to z and let G satisfy the conditions (2.307), (2.312). Thenthe classical Dirichlet problem (2.306) has a unique solution u0 ∈ C2(Ω) ∩ C(Ω).

More detailed results for the above special cases in Example 2.78 are presented in[346].


Skrypnik [591], Theorem 3.2.3, considers the Monge–Ampere equation (2.308) onthe two-dimensional ball Ω = BR(0) with center 0 and imposes the boundary condition(u0 − g)|∂BR(0) = 0. He assumes the existence of functions

φ : BR(0) → R+, f0 : R2 → R+, s.t., with m := maxx∈BR(0)

g(x)

f(x, u, p) ≤ φ(‖x‖2)f0(‖p‖2) for x, p, s ∈ R2, R � u ≤ m and let (2.313)∫BR(0)

φ(‖x‖2)dx ≤∫

R2inf

‖s−p‖2<MH

1/f0(‖s‖2)dp,

where MH is the lower winding number of the curve defined by (u0 − g)|∂BR(0) = 0,see, e.g. Bakelman [55]. Finally, let G := {(x, u, p) : x ∈ BR(0), u ∈ R, p ∈ R2} and

H4+(Ω) := {u ∈ H4(Ω) : (∂20u∂02u− (∂11u)2)(x) > 0, ∂20u(x) > 0∀x ∈ BR(0)}.

Theorem 2.81. Regular solutions for the Monge–Ampere equation, Skrypnik [591],Theorem 3.2.3: Let (2.313) be satisfied and f ∈ C2,γ(G), g ∈ C4,γ(∂Ω). Then (2.308)has at least one solution in H4

+(Ω).

We continue with some existence and regularity results, elaborated by Chen andWu [170], Chapter 7, Theorem 7.4 and Remark on p. 117, for general fully nonlinearequations (2.306) of order 2. They impose six conditions (F1)− (F6) for (2.306):

Let G(x, z, p, r) ∈ R, (x, z, p, r) ∈ D(G) ⊂ Uo = Ω× R× Rn × Rn2, and (2.314)

assume a constant μ and nonnegative, nondecreasing μi : [0,∞) → R+, i = 1, 2, 3, s.t.

(F1) : ∃0 < λ = λ(x, z, p),Λ = Λ(x, z, p) s.t., with the identity matrix I

λ I ≤(

∂G

∂rij

)n

i,j=1

≤ Λ I, and Λ/λ ≤ μ1(|z|) on Uo;

(F2) : |G(x, z, p, 0)| ≤ λμ2(|z|)(1 + |p|2);(F3) :

[(1 + |p|)−1|Gx|+ |Gz|+ (1 + |p|)|Gp|

](x, z, p, r)

≤ λμ3(|z|)(1 + |p|2 + |r|)∀(x, z, p, r) ∈ Uo;

(F4) : on Uo : G(x, z, p, r) is concave with respect to r;

(F5) : G(x, z, p, 0) sign z ≤ λμ(1 + |p|);(F6) : Gz(x, z, p, r) ≤ 0 ∀(x, z, p, r) ∈ Uo. (2.315)

Theorem 2.82. Regular solution for fully nonlinear problems:

1. Let the conditions (2.314)–(2.315) be satisfied, Λ be bounded on Ω× R× Rn.Then there exists 0 < α < 1 with the following property: For any 0 < γ < α < 1with ∂Ω ∈ C2,γ ϕ ∈ C2,γ(Ω), for the Dirichlet problem (2.306) there exists aunique solution u0 ∈ C2,γ(Ω). This α depends only upon n, μ1, μ2, μ3, ‖ϕ‖C2,γ(Ω)

and Ω.


2. If G ∈ Ck,α(Uo), k ≥ 1, and ∂Ω ∈ C2+k,γ , ϕ ∈ C2+k,γ(Ω), then the solution u0 ∈C2,γ(Ω) ∩ Ck+2,α(Ω). For G ∈ Ck,α(Uo), it is u0 ∈ Ck+2,γ(Ω).

We turn to some existence and regularity results for fully nonlinear equations oforder 2m. Skrypnik [591] seems to be the author of the only textbook studying theseproblems. He combines monotone operators and degree theory to get his results forquasilinear and fully nonlinear elliptic equations. He discusses some of these problemson narrow strips and in perforated domains. To give a flavor of his existence andregularity results we cite one of his theorems, a “conditional” result. For the originalproblem, he formulates a family of parameter dependent problems (t ∈ [0, 1]) with t =0 and t = 1 for the original problem and a problem with appropriate degree properties.He determines a solution u0 for the nonlinear Dirichlet problem. Let a function G begiven such that, with N2m as in (2.285), cf. Remark concerning (2.306)

Gw ∈ C(D(Gw)), D(Gw) := [0, 1]×D (Gw0 ) = [0, 1]× Γ ⊂ [0, 1]× Ω× RN2m ,

with appropriate N2m, let Gw(0, x,−ΘN2m) = −Gw(0, x,ΘN2m), and

∀t ∈ [0, 1] : Gw(t, . . .) ∈ Cl+1 (D (Gw0 )) , l ≥ [n/2] + 1, where (2.316)

u, u1 ∈ D(G) if, e.g. w1(x) := (x, ∂αu1(x), |α| ≤ 2m) ∈ D (Gw0 ) ∀x ∈ Ω,

G(u) := G(t = 0, u) := Gw(t = 0, ·, u(·), . . . ,∇2mu(·))G : D(G) ⊂ U := Hm

0 (Ω) ∩H2m(Ω) → V := L2(Ω), determine u0 from (2.317)

G(u0(·)) = Gw(t = 0, ·, u0, . . . ,∇2mu0)(·) = 0, u0 ∈ X0 := Hm0 (Ω) ∩H2m+l(Ω).

He assumes the existence of constants 0 < k, 0 < γ < 1, 0 < Λ <∞ for (2.316), andmodifies the ellipticity condition, compare (2.232), such that

∃k ∈ R+, 0 < γ < 1 : ∀t ∈ [0, 1], v ∈ X0 : G(t, v) = 0 implies ‖v‖C2m,γ(Ω) ≤ k, and

k−1|ϑ|2m ≤∑

|α|=2m

[∂α(Gw(t, ·, v, · · · ,∇2mv)

)(x)]ϑα ≤ Λ|ϑ|2m ∀x ∈ Ω. (2.318)

Theorem 2.83. Regular solutions, Skrypnik [591], Theorem 3.2.2: Let G satisfy(2.316), (2.318) and let ∂Ω ∈ C∞ for a bounded domain in Rn. Then (2.317) has, fort = 0, at least one solution u0 ∈ U .

We finish with a nonlinear PDE of order m, see Definition 2.85, in the form

G(u) := G(x, u,∇u, . . . ,∇mu) and G(u)(x) = g(x) ∀ x ∈ Ω. (2.319)

Theorem 2.84. Local solvability and regularity result – Taylor, [620], Chapter 14,Theorem 3.1: Let g ∈ C∞(Ω), and let u1 ∈ C∞(Ω) satisfy (2.319) at x = x0,

G(u1)(x) = g(x) for x = x0

and let G be m-elliptic at u1. Then for any � there exists a u ∈ C�(Ω) such that

G(u)(x) = g(x) in a neighborhood of x0.

2.6. Linear and nonlinear elliptic systems 113

2.6 Linear and nonlinear elliptic systems

2.6.1 Introduction

In this section we discuss systems of q elliptic differential equations essentially in theHilbert Sobolev space setting for two reasons: Practically all available results in thetextbooks consulted are formulated for the Hilbert case. It is straightforward as inSubsection 2.4.4 to extend the following results, based upon the Hm-coercivity, to theWm,p case for 2 ≤ p <∞. The 1 ≤ p < 2 usually have to be excluded. As in Corollary2.44, 3., 4., we modify them for 2 < p ≤ ∞: these Wm,p(Ω) ↪→ Hm(Ω) are densely andcontinuously embedded. Consequently, all Hm-coercivity based existence, uniquenessand Fredholm results, including the numerical stability and convergence of all ourdiscretization methods, remain correct for the Wm,p case for 2 < p ≤ ∞ with respectto (discrete) Hm norms. We will explicitly formulate this fact for the particularlyineresting Theorems 2.89 and 2.104.

Comparing equations with systems shows an astonishing difference: the definitionof ellipticity for linear and nonlinear equations coincides for nearly all aspects. Fororder 2m they are direct generalizations of those for second order. This is no longercorrect for systems. In Subsection 2.6.2, Definition 2.85, we introduce the conceptof systems of elliptic differential equations, a concept of ellipticity, pretty differentfrom those previously introduced. It is motivated by important examples, includingthe Navier–Stokes and von Karman equations. Starting with Subsection 2.6.3, andexcluding Navier–Stokes equations, we turn back to the definitions of ellipticity, similarto or directly generalizing those for the case m = q = 1. However, depending upon thechosen goal, e.g. existence and uniqueness or regularity, different types of ellipticitybecome relevant. Subsection 2.6.3 is devoted to linear second order systems, cf. (2.343),extended to quasilinear second order systems in Subsection 2.6.4, cf. (2.370), (2.371)for existence and (2.384) for regularity. The last three Subsections 2.6.5, 2.6.6 and 2.6.8generalize some of these results to order 2m of linear, cf. (2.394), (2.401), quasilinear,cf. (2.437), and fully nonlinear elliptic systems, cf. (2.446). Not much seems to beknown about unified conditions for the different aspects and orders.

Systems of elliptic differential equations are intensively studied in different mono-graphs, e.g. Hormander [402], Garabedian [336], Lions and Magenes [478–480], Chenand Wu [170], Taylor [618–620], and Koshelev and Chelkak [449–451]. For systemsthe ellipticity is not uniformly defined. In particular, existence and regularity arediscussed by Ladyzenskaja and Uralceva [463–466], Koshelev and Chelkak [450, 451],Skrypnik [590, 591], Morrey [500], Agmon, Douglis and Nirenberg [3], Taylor [620],Zeidler [675–678], Giaquinta and Hildebrandt [341], Necas [510], and Kozlov Maz’yaand Rossmann [452]. Here we include some papers as well: Wahl and Grunau [378,659],Luckhaus and Alt [23,481], Nedev [512], and Boccuto and Mitidieri [99]. These booksand papers show, e.g. [23,99,481,659], that usually generalizations of regularity resultsfrom the linear to the nonlinear case require new techniques and a careful analysis.Regularity for nonlinear elliptic systems with interesting applications to stochasticgames, ergodic control, semiconductors, plasticity, and the stationary Navier–Stokesequations are studied by Bensoussan and Frehse [81]. The regularity results for the


stationary Navier–Stokes equations would be interesting for Section 2.8. However,in [81] the main goals are specific “maximum-like solutions” for higher dimensionsn > 3, which might give insight into the nonstationary Navier–Stokes equations. Wesummarize some essential results of Koshelev and Chelkak [449–451] and Skrypnik[591], in Subsections 2.6.4 and 2.6.6, 2.6.8.

2.6.2 General systems of elliptic differential equations

In this subsection we give information for general systems of elliptic equations in thesense of Agmon, Douglis and Nirenberg [3]. The stationary Navier–Stokes equationis one of the most important of these systems of elliptic differential equations.It certainly motivated the new definition of ellipticity in this and the next subsections.Although we discuss the Navier–Stokes equation in a separate section, we memorizeit here. We want to motivate the general systems, discussed here in contrast to thefollowing subsections. As before we usually omit the index s for the strong form,possible for a regular enough situation. The context indicates the appropriate form.In this subsection essentially all operators are in strong form.

Already in Chapter 1 we introduced the Navier–Stokes equation. For the stationaryform, the solution, �u0, p0 satisfies

G(�u0, p0) :=

⎛⎝−νΔ�u0 +n∑

i=1

u0i ∂

i�u0 +∇ p0

− div �u0

⎞⎠ =(�f0

)in Ω, (2.320)

�u0 = 0 on ∂Ω,∫

Ω

p0dx = 0, Ω satisfies (2.5) and

�u = (u1, . . . , un)T, �v = (v1, . . . , vn)T, �f = (f1, . . . , fn)T : Ω ⊂ Rn → Rn,

div �u := ∂u1/∂x1 + · · ·+ ∂un/∂xn, p : Ω → R, �u, p, . . . sufficiently smooth,

where �u, p and �f denote the velocity, pressure and forcing term of an incompressiblemedium and �v a test function. The condition

∫Ωp0dx = 0 enforces a unique p0. Note

that (2.320) is a system of n + 1 differential equations for n + 1 unknown functionssubmitted to the boundary conditions and an integral constraint. For physical appli-cations only n ≤ 3 is interesting, for higher n, cf. [81].

The derivative G′ of G, evaluated at (�u, p), �u �= 0, and applied to an increment(�w, r), is the linearized Navier–Stokes operator, again an operator with n + 1components,

G′(�u, p)(�w, r) =

(−νΔ�w +

∑ni=1(wi∂

i�u + ui∂i �w) +∇r

− div �w

). (2.321)

The special case G′(�u ≡ 0, p0) and ν = 1 applied to an increment (�u, p) yields theStokes operator, S, and the corresponding Stokes equation,

G�u(0, p0)(�u, p) = S(�u, p) =

(−Δ�u +∇p− div �u

)=(�f0

)in Ω, (2.322)

�u = 0 on ∂Ω,∫Ωpdx = 0.


The nonlinear system (2.320) and its linearizations (2.321) and (2.322) do not allow adirect generalization of the previous framework (in contrast to the situation in the nextsubsections). Again, the following notation and even some results for linear systems ofq equations for q functions �u = (u1, . . . , uq) are often similar to the concepts of linearalgebra, cf. (2.73), (2.269), for ϑ ∈ Rn and ϑα, ∂αuj .

Definition 2.85. Elliptic systems – Agmon et al. [2, 3]:

1. A linear system of differential equations has the form,

L�u :=

⎛⎝ q∑j=1

Lijuj

⎞⎠q

i=1

= �f :=(f i)qi=1

, �u = (u1, . . . , uq)T defined in Ω ⊂ Rn,

Lij :=∑

|α|≤kij

aijα (x)∂α, i, j = 1, . . . , q, ∂αuj = uj for |α| = 0, (2.323)

usually with aijα ∈ C(Ω).18 We choose obviously not unique

m1, . . . ,mq, m′1, . . . ,m

′q s.t. kij ≤ mi + m′

j , i, j = 1, . . . , q, (2.324)

with the 2m in (2.328). The principal part of Lij or its symbol are defined by thecomponents of its “highest order” tems

Lijp :=

∑|α|=mi+m′

j

aijα ∂α or Lij

p (x, ϑ) :=∑

|α|=mi+m′j

aijα (x)ϑα, (2.325)

for i, j = 1, . . . , q. If kij < mi + m′j, these aij

α = 0, for |α| = mi + m′j , hence

Lijp = 0.

2. We define the symbol, Lp, and characteristic polynomial, detLp, as

Lp(x, ϑ) :=(Lij

p (x, ϑ))qi,j=1

∀x ∈ Ω, detLp(x, ϑ) = det(Lij

p (x, ϑ))qi,j=1

. (2.326)

3. A linear system (2.323) is called elliptic in x ∈ Ω, if Lp(x, ϑ) is invertible,

the q × q matrix Lp(x, ϑ)is invertible⇔ detLp(x, ϑ) �= 0 ∀ 0 �= ϑ ∈ Rn. (2.327)

Often, stronger than (2.327), ε > 0 exists, such that (only a.e. for aα ∈ L∞(Ω))

|detLp(x, ϑ)| ≥ ε|ϑ|2m ∀ × εΩ, 0 �= ϑ ∈ Rn with 2m :=q∑

i=1

(mi + m′i) . (2.328)

4. We call 2m : = maxi,j=1,...,q

(mi+m′j) the order of an elliptic system.

5. As in Definition 2.51 we call a nonlinear system elliptic if its linearization iselliptic. Obviously, we obtain the original ellipticity for 1 = q ≤ m = m1 = m′

1.

18 Several types of notation are used in the literature, each with pros and cons. For linear systems

(2.323) the notation L�u as a matrix of operators (Lij)qi,j=1 or coefficients (aij

α )qi,j=1 applied to a

vector �u = (u1, . . . , uq)T of functions is advantageous. This does not work too well for quasilinearsystems. In the Einstein convention the same index and exponent i indicates the sum over i. In(2.323) and (2.339) this could be used directly for the j and k, l, so

∑qj=1 and

∑nk,l=1, respectively,

would be omitted. This Einstein convention is not too common for numerical methods and can beapplied only in parts of this section. So we decided to use the componentwise or vectorial notation.


6. A modification is discussed by Taylor [618], pp. 245 ff. and pp. 379 ff.: Replacein (2.323) the |α| ≤ kij and in (2.325) the |α| = mi + m′

j by |α| ≤ m and|α| = m. We denote this modified linear system (2.323) as m-elliptic, if Lp(x, ϑ)is invertible ∀x ∈ Ω, ϑ ∈ Rn.

In the literature “strong ellipticity” is used in different ways. We use “strongellipticity” for (2.343) below, but not for (2.328).

Example 2.86. We list several examples of elliptic systems, in particular two systemsintroduced in Chapter 1. �

1. The Stokes operator L = S and the linearized Navier–Stokes operator: In (2.322)we choose

q := n + 1, �u = (u1, . . . , un, p)T, �f = (f1, . . . , fn, 0)T, (2.329)

Lii := −Δ, Liq := −Lqi = ∂i = ∂/∂xi for i = 1, . . . , n, otherwise Lij := 0.

Then kii = 2, kiq = 1 = kqi for i = 1, . . . , n and kij = 0 otherwise. Hence thefollowing mi,m

′i satisfy (2.324):

mi = m′i = 1 for i = 1, . . . , n = q − 1,mq = m′

q = 0.

Furthermore Lijp are independent of x and we get

Liip (ϑ) = −|ϑ|2, Liq

p (ϑ) = −Lqip (ϑ) = ϑi, for i = 1, . . . , n, else Lij

p (ϑ) = 0.

This yields finally, with the 2m in (2.328)

|detLp| =

∣∣∣∣∣∣∣∣∣det

⎛⎜⎜⎜⎝−|ϑ|2 0 ϑ1

0. . .

...−|ϑ|2 ϑn

−ϑ1 . . . −ϑn 0

⎞⎟⎟⎟⎠∣∣∣∣∣∣∣∣∣ = |ϑ|2m with 2m = 2n. (2.330)

Note that this 2m = 2n and the order 2 for the Stokes operator introduced inDefinition 2.85, 4. and Definition 2.102 are different.If we linearize the Navier–Stokes operator in an arbitrary (�u, p), the additionalterms are of lower order, and the corresponding Lij

p (ϑ) are ν× those of the Stokesoperator. Hence, (2.328) is satisfied with ε = 1 and ε = νn for the Stokes and thelinearized Navier–Stokes operator. So both are elliptic, with 2m = 2n.This ε = νn indicates one of the most fascinating properties of the Navier–Stokes equations: for ν → 0 the elliptic aspects are dominated by convectionphenomena, caused by

∑ni=1 ui∂

i�u terms in (2.320) and their linear counterpartsin (2.321). For the time-dependent form of the equations this corresponds to atransition from a parabolic to a hyperbolic system. We will come back to (2.320)in Section 2.8.

2. The von Karman equations, see Example 1.4: Berger and Five [82–84] haveelaborated the appropriate spaces for these equations and proved the existenceof solutions. Ciarlet [175] has identified these equations as the first term in an


asymptotic expansion, describing the mechanical bending of a plate. He analyzesthese equations and the corresponding FEMs.An external load λ compresses a plate located at Ω. The Airy stress functionw(x, y) at the point (x, y) and the deviation or deflection u(x, y) of the platefrom its flat state u(x, y) ≡ 0, u,w : Ω → R, are described by

G(u,w) :=

⎛⎝ Δ2u −[u,w] +λuxx

Δ2w + 12 [u, u]

⎞⎠ , G(u0, w0) =

⎛⎝ f

0

⎞⎠ (2.331)

in Ω. Here Δ2 = ΔΔ is the biharmonic operator in R2, in some papers the λuxx

is missing. The bracket operator [ ·, · ] is defined by

[u, v] = [v, u] = uxxvyy − 2uxyvxy + uyyvxx.

The derivative G′(u1, w1) applied to (v, z) has the form

G′(u1, w1)(v, z) :=

⎛⎝ Δ2v −([u1, z] + [v, w1]) +λvxx

Δ2z +[u1, v]

⎞⎠ .

Here we choose

q := 2, Lii := Δ2 for i = 1, 2, Lij := 0 for i �= j so

kii := 4 for i = 1, 2, kij := 0 for i �= j, mi = m′i = 2, and (2.332)

Liip (ϑ) := |ϑ|4 for i = 1, 2, Lij

p (ϑ) := 0 for i �= j, q = 2.

This yields finally

|detLp(ϑ)| = |detLp| =∣∣∣∣det

(|ϑ|4 00 |ϑ|4

)∣∣∣∣ = |ϑ|8. (2.333)

Hence, the von Karman equations are strongly (ε = 1) elliptic and 2m = 8. Wewill come back to this in Subsection 2.6.6.

3. The system of Lame: This describes the deflection function �u of an elastic plateby a linear system: with μ, λ > 0 we determine the solution �u0 from

�u0 : Ω ⊂ R3 → R3, μΔ�u0 + (λ + μ)∇ div�u0 = f in Ω, = ϕ on ∂Ω. (2.334)

Similarly as above one shows that |detLp| = μ2(2μ + λ)|ϑ|6, so 2m = 6. �

Agmon et al. [2, 3] strongly stimulated research on partial differential equationsand systems. It is absolutely impossible to appropriately summarize these results inthis chapter. Furthermore, some of these general systems require mixed discretizationmethods, e.g. the Navier–Stokes equations in Section 2.8, requiring special treatment.So, in the following subsections we formulate for special elliptic systems the resultsfor existence and regularity, similar to those for elliptic equations. We give a flavorfor general systems, by summarizing a few examples on local existence and regularityfrom Taylor [618] for linear systems in Subsections 2.6.3 and 2.6.5.


2.6.3 Linear elliptic systems of order 2

In this and the following subsections we will study special elliptic systems of order2 and 2m. The linear systems are obtained, very similarly to elliptic equations, ascompact perturbations of the usually Hm

0 -coercive principal parts of these operators.As in Corollary 2.44, 3., 4., this implies only the Hm

0 -coercivity for 2 ≤ p ≤ ∞. Thenexistence, uniqueness and Fredholm alternatives are easily obtainable by Section 2.3,see e.g. Theorem 2.21.

We elaborate the corresponding results for linear and divergent quasilinear ellipticsystems. Our presentation generalizes Chen and Wu [170] with respect to existence andregularity. Their regularity results are only valid for equations defined by the principalpart, but will be extended to the general case by the inherited regularity technique inTheorem 2.56. We refer to Subsections 2.6.4 and 2.6.6 for more general types of regular-ity results, even for quasilinear equations, see Taylor [618–620], Koshelev and Chelkak[449–451] and Skrypnik, [591]. For these systems we can replace the |α| ≤ kij in (2.323)by |α| ≤ m. So we do not need the mi,m

′j in (2.324) any more. This allows a simpler

matrix formulation for the equations. We start with the discussion of second orderlinear problems and extend it to quasilinear systems of second and then to higher order.

Before we introduce the following setting for homogeneous Dirichlet boundary con-ditions, we want to recall the above generalizations. For a nonhomogeneous Dirichletboundary function, extended to a function on Ω, Lemma 2.26, and the standardtrick, reduces the nonhomogeneous to homogeneous Dirichlet boundary conditions.Then similarly to the cases following (2.124), generalized second (modified Neumann),third or natural boundary value problems, compare (2.125), can be introduced andtreated for second order problems. Similarly to (2.38), different boundary conditionson different parts of ∂Ω are possible, but will not be formulated here.

In this subsection, considering systems of order 2, we have to apply the par-tial derivatives ∂l, l = 1, . . . , n, to q components uj , vi, i, j = 1, . . . , q. We mod-ify the original notation in (2.73), replacing reals ∂lu(x), ∂αu(x), ϑl, ϑα ∈ R, andn-vectors ∂u(x), ϑ = (ϑ1, . . . , ϑn) ∈ Rn by vectors ∂l�u(x), ∂α�u(x) �ϑl, �ϑα ∈ Rq, andthe corresponding n× q matrices ∂�u(x), �ϑ ∈ Rn×q, using just one vector sign to keepconsistency with, but indicate the difference to (2.73). Evaluated in x (ev. at x) weget

�u := (u1, · · · , uq), ∂l�u := (∂lu1, · · · , ∂luq), ∂α�u := (∂αu1, · · · , ∂αuq) ∈ Rq, ev. at x,

∇�u = (∇u1, · · · ,∇uq) = (∂u1, · · · , ∂uq) = ∂�u = (∂1�u, · · · , ∂n�u) ∈ Rn×q, ev. at x,

∇k�u := (∂α�u)|α|=kev. in x :∈ Rnk×q,∇≤k�u := (∂α�u)|α|≤k ∈ RNk×q, ev. at x,

with ∇k�u = ∂l�u = ∂α�u = �u for k = l = |α| = 0 and (2.335)

�ϑj =(ϑ1

j , . . . , ϑnj

)∈ Rn, j = 1, . . . , q, �ϑl =

(ϑl

1, . . . , ϑlq

)∈ Rq, l = 1, . . . , n, with

�ϑ := (�ϑ1, · · · , �ϑn) := (�ϑ1, · · · , �ϑq) ∈ Rn×q, and |�ϑ| = |�ϑ|nq ∈ R,

�ϑα :=(ϑα

1 , · · · , ϑαq

)∈ Rq, �Θk := (�ϑα)|α|=k ∈ Rnk×q, �Θ≤k := (�ϑα)|α|≤k ∈ RNk×q.


We recall and extend the usual Sobolev spaces and norms for Ω in (2.5).Sobolev (Hilbert) spaces and their inner products, norms, and seminorms aredefined as

Wm,p(Ω,Rq) := (Wm,p(Ω))q,Wm,p0 (Ω,Rq),Hm(Ω,Rq) := Wm,2(Ω,Rq), (2.336)

(�u,�v)m := (�u,�v)Hm(Ω,Rq) :=q∑

i=1

∑|α|≤m

(∂αui, ∂αvi)L2(Ω) and (2.337)

‖�u‖W m,p(Ω) := ‖�u‖W m,p(Ω,Rq) :=

⎛⎝ q∑i=1

∑|α|≤m

‖∂αui‖pLp(Ω)

⎞⎠1/p

in particular

Wm,p(Ω,Rq) ↪→ Hm(Ω,Rq) densely and continuously embedded for 2 ≤ p.

We return to the second order equation with a functional, f, in general form:

�f := (f0, · · · , fn) ∈ Lp′(Ω,Rq(n+1)), fk(x) =

(f1

k , · · · , fqk

)(x), ∂k�v(x) ∈ Rq, (2.338)

〈�f,�v〉V′×V :=∫

Ω

n∑k=0

(fk, ∂k�v)qdx, ∀�f ∈ V ′ := W−1,p′

(Ω,Rq), 1/p + 1/p′ = 1,

cf. (2.107), (2.339), (2.340), noting that the fk(x), . . . , are only defined a.e.

Remark 2.87. As a consequence of (2.337), (2.338), the following integrals aredefined for �u,�v ∈ W 1,p

0 (Ω,Rq), �f ∈W−1,p′

0 (Ω,Rq), p ≥ 2. Hence the following argu-ments remain valid for this case as well, possibly with a H1

0 (Ω,Rq)-coercivity.

The strong form L is defined for �u, �u0 ∈ H2(Ω,Rq) and with �f instead of f0:19

f0 =[f i0

]qi=1

= Ls�u0 = L�u0 :=

⎡⎣ q∑j=1

⎛⎝ n∑k,l=0

(−1)k>0∂k(aij

kl∂lu0

j

)⎞⎠⎤⎦q

i=1

(2.339)

=n∑

k,l=0

(−1)k>0∂k(Akl∂

l�u0) =

⎡⎣ q∑j=1

Lijs u0

j

⎤⎦q

i=1

, with Akl(x) =(aij

kl

)q

i,j=1,

for Akl ∈W 1−δk,0,∞(Ω,Rq×q) and |∂k(aij

kl

)(x)| ≤ Λ0.

For Dirichlet boundary conditions we take the standard scalar product of (2.339)with �v = (v1, . . . , vq) ∈ H1

0 (Ω,Rq) and integrate by parts. Then the weak solution�u0 = (u0

1, . . . , u0q) ∈ H1

0 (Ω,Rq) for (2.339) is determined, cf. (2.274), by equating the

19 Trying to keep the notation as for quasilinear equations, we have changed the order of differen-tiation in the strong form (2.339) compared to (2.134).


following linear and bilinear forms, 〈�f,�v〉, and a(�u0, �v):

�u0 ∈ V := H10 (Ω,Rq) s.t. 〈�f,�v〉 =

⟨[n∑

k=0

f ik∂

k

]q

i=1

, �v

⟩=∫

Ω

n∑k=0

(fk, ∂k�v)qdx (2.340)

= a(�u0, �v) := (L�u0, �v)V :=∫

Ω

n∑k,l=0

(Akl∂l�u0, ∂k�v)qdx∀ �v ∈ V, with

Akl ∈ L∞(Ω,Rq×q), |aijkl(x)| ≤ Λ0 (a.e.) ∀x ∈ Ω, f i

k ∈ L2(Ω) ∀i, j, k, l, or (2.341)

=⇒ �f ∈ H−1(Ω,Rq(n+1)), with ‖�f‖H−1(Ω,Rq(n+1)) ≤i=1,...,qmax

k=0,...,n{‖f i

k‖L2(Ω)}.

Note the change in notation for �f ∈ H−10 (Ω,Rq(n+1)) here to f i

k ∈ L2(Ω) above and inChen and Wu. The principal part and principal symbol Lp of L are defined as

principal part: (Lp�u,�v)V :=∫

Ω

n∑k,l=1

(Akl∂l�u, ∂k�v)qdx, and symbol: (2.342)

Lp(x, �ϑ) :=(Lij

kl(x, �ϑ))q

i,j=1=

⎛⎝ n∑k,l=1

aijkl(x)ϑl

jϑki

⎞⎠q

i,j=1

=n∑

k,l=1

(Akl(x)�ϑl, �ϑk)q.

Accordingly, for the Stokes operator the matrix in (2.330) would have to be modified byreplacing the last row and column by zeros. For our special linear system of divergentdifferential equations we further update the above ellipticity in Definition 2.85, see(2.328), towards the Legendre, sometimes called the strong Legendre condition, andthe Legendre–Hadamard condition.

Definition 2.88. Ls or L in (2.339) or (2.340) are called strongly elliptic anduniformly elliptic, respectively, if they satisfy the strong and uniform Legendrecondition. That is, there exist 0 < λ and 0 < λ < Λ <∞, respectively, independent ofx (sometimes only 0 < λ < λ(x) < Λ(x), Λ(x)/λ(x) < C <∞), such that a.e. ∀x ∈ Ω,cf. (2.342),

λ|�ϑ|2 = λ|�ϑ|2nq ≤q∑

i,j=1

Lijp (x, �ϑ) =

n∑k,l=1

(Akl(x)�ϑl, �ϑk)q ≤ Λ|�ϑ|2. (2.343)

So the Stokes operator violates (2.343). For our aijkl ∈ L∞(Ω), the inequality in

(2.343) indeed only makes sense a.e. ∀ x ∈ Ω. We will discuss the relation of thedifferent types of ellipticity in Remark 2.90.

For proving existence and uniqueness results for (2.340) we reinterpret the bilinearform a(�u,�v) as the principal part, see (2.342), and additional lower order terms

a(�u,�v) = ap(�u,�v) + b(�u,�v), with b(�u,�v) :=∫

Ω

n∑k,l=0,k×l=0

(Akl∂l�u, ∂k�v)qdx. (2.344)


We repeat the concepts of V-coercive and V-elliptic bounded bilinear forms, see (2.41)and (2.63). For V = Hm

0 (Ω,Rq),m ≥ 1, a(·, ·) is called V = Hm0 (Ω,Rq)-coercive, and

V-elliptic, respectively, if constants α ∈ R+, Cc ∈ R exist such that

a(�u, �u) ≥ α‖�u‖2V and ≥ α‖�u‖2V − Cc‖�u‖2W ∀�u ∈ V = Hm0 (Ω,Rq),W = L2(Ω). (2.345)

Applying Theorems 2.21 and 2.16 we find

Theorem 2.89. V-coercivity and ellipticity:

1. Let (2.340)–(2.341) be a uniformly elliptic system, hence let (2.343) with 0 <

λ ≤ Λ <∞ be satisfied. Then 〈�f,�v〉, ap(�u,�v), a(�u,�v) in (2.340), (2.344) arebounded with respect to V, and ap(�u,�v) and a(�u,�v) are V-coercive and V-elliptic,respectively.

2. All these results remain correct, if the above strong Legendre condition (2.343) isreplaced by the weaker Legendre–Hadamard condition (2.348).

3. For the principal part, Lp, and its ap(�u,�v), the unique solution �u0 of (2.346)solves the variational problem (2.61).

4. Finally, all H1-coercivity based existence, uniqueness and Fredholm resultsremain correct for the W 1,p case for 2 ≤ p ≤ ∞, however with respect to theH1 norm.

Under these conditions Corollary 2.44 applies. It implies for a V-coercive a(·, ·) theexistence of a unique generalized or weak solution �u0 and a boundedly invertible L,and for a V-elliptic a(·, ·) the Fredholm alternative with its consequences. Hence, fora boundedly invertible L, with the induced a(�u,�v),

∀�f ∈ V ′ = H−10 (Ω,Rq) ∃1�u

0 ∈ V : a(�u0, �v) = 〈�f,�v〉∀�v ∈ V. (2.346)

Proof. For completeness we include this nearly obvious proof: by (2.341) the〈�f,�v〉, ap(�u,�v), a(�u,�v) are bounded with respect to V. Furthermore (2.343) implies

ap(�u, �u) ≥ λ

∫Ω

|∂�u|2dx with |∂�u|2 =q∑

i=1

n∑k=1

|∂kui|2 ∀�u ∈ V. (2.347)

The ‖�u‖V and∫Ω|∂�u|2dx are equivalent norms in V, so ap(�u,�v) is V-coercive. Since

a(�u,�v)− ap(�u,�v) are lower order terms, a combination of Theorem 1.26 and Propo-sition 2.24 yields the V-ellipticity of a(�u,�v). Theorems 2.15 and 2.21 thus verify theabove claims.

The final remark concerning the strong Legendre–Hadamard condition is a specialcase of Theorem 2.104 below. The W 1,p-coercivity is a consequence of the initialdiscussion of Remark 2.87. �

Remark 2.90. We list some properties, relating the different concepts of ellipticsystems (q ≥ 1) to elliptic equations (q = 1) in (2.327)–(2.343).

1. We start by showing that (2.343) for 2m = 2 implies the Taylor modification of(2.327): In fact let (2.327) be violated for our special case. Then a nontrivial


solution y = (y1, · · · , yq) for the linear homogeneous system exists:

q∑j=1

Lijp (x, ϑ)yj = 0, y = (y1, · · · , yq) �= 0, hence,

q∑i,j=1

n∑k,l=1

αiaijkl(x)ϑl

jϑki yj = 0 ∀ αi ∈ R, a.e. x ∈ Ω.

By an appropriate rescaling of the αiϑljϑ

ki yj into new ϑl

j := ϑljyj , ϑ

ki := αiϑ

ki we

obtain the desired contradiction to (2.343).2. The linearized von Karman and Navier–Stokes operators satisfy and violate

(2.343), respectively. We will see in Section 2.8 that the Stokes operator (andall of its compact perturbations) is not coercive. So the results of this and thenext subsection are not applicable. Hence the Navier–Stokes operator and therelated saddle point problems require a separate treatment.

3. For q = 1 the coercivity in (2.347) for the principal part and the two precedingellipticity conditions (2.328) and (2.343) are essentially equivalent. This is nolonger correct for q > 1. Chen and Wu [170], see Chapter 8, Theorems 1.1, and1.2, show (compare the Taylor form of 2m-ellipticity in Definition 2.85, 6.):

under the condition (2.341) for the Akl =(aij

kl

)q

i,j=1the strong Legendre condi-

tion implies the coercivity in (2.345) for the principal part.4. Conversely, the coercivity of the principal part and (2.341) imply the so-called

strong Legendre–Hadamard condition: with Lp(x, ϑ) in (2.326)

∃λ > 0 : ∀x ∈ Ω, ϑ ∈ Rn, η ∈ Rq,

ηTLp(x, ϑ)η =q∑

i,j=1

n∑k,l=1

aijkl(x)ϑlϑkηjηi ≥ λ|ϑ|2|η|2. (2.348)

So instead of the above equivalence for q = 1 we find for q > 1 the followingimplication

For q > 1 assume (2.341) for Lp. Then: (2.343) =⇒ (2.347) =⇒ (2.348)(2.349)

or verbatim, for Akl = (aijkl ∈ L∞(Ω))q

i,j=1, the strong Legendre condition =⇒coercivity for ap(·, ·) =⇒ strong Legendre–Hadamard condition for Akl.

By the proof of Theorem 2.104 we get for aijkl ∈ C(Ω), k, l = 1, . . . , n, even the

equivalence (2.347) ⇐⇒ (2.348).

For our numerical applications again regularity results �u0 ∈ Hs(Ω,Rq), s ≥ 1, for thegeneral equation (2.340) are important. Similarly to the preceding sections, differentregularity results are known. In the remainder of this subsection we combine Chen andWu [170], and the proof of Theorems 2.56 and 2.60. For linear L, Chen and Wu provetheir results only for the solution �u0 of the equation (2.350) defined by the principal


part Lp and with the general �f in (2.340)

ap(�u0, �v) = 〈�f,�v〉∀�v ∈ H10 (Ω,Rq). (2.350)

Then the proof of Theorem 2.56 allows us the extension to the general problem (2.340).We summarize the results, obtained by this combination with Chen and Wu [170]:

Theorem 2.91 1. and 2. correspond to [170], Chapter 8, Theorems 2.3, 2.6, 2.7, andto Corollary 2.4. Theorem 2.91 1. yields �u0 ∈ Hs+2(Ω,Rq), s ≥ −1, Theorem 2.91 3.generalizes [170], Chapter 9, Theorems 2.7, 2.8. The resulting �u0 ∈ Cs,δ(Ω,Rq), s ≥ 1is needed in Subsection 2.6.4.

Theorem 2.91. Regular solution �u0 ∈ Hs+2(Ω,Rn×q) and ∂�u0 ∈ Cs,δ(Ω,Rn×q):

1. Let a(·, ·) be strongly elliptic, see (2.343), and �f of the form (2.340). For s ∈Z, s ≥ −1, let

∂Ω ∈ Cs+2, aijkl ∈W s+1,∞(Ω), f i

k ∈ Hs+1(Ω)∀i, j, k, l, (2.351)

equivalent to �f ∈ Hs+1(Ω,Rn). Then any weak solution �u0 ∈ H10 (Ω,Rq) of

(2.340) satisfies �u0 ∈ Hs+2(Ω,Rq) and

‖�u0‖Hs+2(Ω,Rq) ≤ C(‖�u0‖H1(Ω,Rq) + ‖�f‖Hs+1(Ω,Rq)

). (2.352)

2. We even get �u0 ∈ C∞0 (Ω,Rq) if (2.351) is replaced by

aijkl, f

ik ∈ C∞(Ω) ∀i, j, k, l, ∂Ω ∈ C∞. (2.353)

3. For a strongly elliptic a(·, ·) and the Dirichlet problem let

aijkl, f

ik ∈ Cs,δ(Ω), ∂Ω ∈ Cs+1,δ, 0 < δ < 1, 0 ≤ s, ∀i, j, k, l. (2.354)

Then for any weak solution �u0 ∈ H10 (Ω,Rq) of (2.340) the ∂�u0 ∈ Cs,δ(Ω,Rn×q).

Finally, the operator L, induced by a(., .), with L : H2+k(Ω,Rq) ∩H10 (Ω,Rq) →

Hk(Ω,Rq) in (2.340) is Fredholm.

Proof. We only prove Theorem 2.91, 1.; for 3. the proof is analogous. For s = 0 andap(·, ·) the (2.352) is Chen and Wu’s Theorem 2.6, however with ‖�u0‖H1(Ω,Rq) replacedby ‖�u0‖L2(Ω′,Rq). For s > 0 and �u0 ∈ Hs+2(Ω,Rq) it is their Theorem 2.7 with (2.352)missing, for s =∞ their Corollary 2.4. Applying the arguments in the modification ofthe proof of Theorem 2.56 now for Theorem 2.60 verifies these results for the generala(·, ·) in (2.340). In fact, (2.339) shows that the aij

kl are derived at most once, henceany �u ∈ Hs+2(Ω,Rq) yields L�u ∈ Hs(Ω,Rq) and thus (2.352). �

Since Hackbusch’s [387] proof for Remark 9.1.1 remains valid, we can improveTheorem 2.91 1. as

Theorem 2.92. Let (2.340) with �f ∈ V ′ have a unique solution �u0 ∈ V, let a(·, ·)induce L ∈ L(V,V ′), and let the conditions of Theorem 2.91 be satisfied. Then L−1 ∈L(Hs(Ω,Rq),Hs+2(Ω,Rq) ∩ V) for s ∈ Z, s ≥ −1, and the following estimates areequivalent for �f ∈ V

′−s := V

′ ∩Hs(Ω,Rq) :

‖L−1‖Hs+2(Ω,Rq)∩V←↩Hs(Ω,Rq) ≤ C ⇐⇒ ‖�u0‖Hs+2(Ω,Rn) ≤ C‖�f‖Hs(Ω,Rn). (2.355)


We finish with some Schauder type results. Again we generalize Chen and Wu’sresults for the principal part to the full operator. They distinguish regularity in theinterior and the closure of Ω. Theorem 2.93 1. combines the results [170], Chapter 9,Theorems 2.5, 2.6, Theorem 2.93 2. with [170], Chapter 10, Theorems 2.1, 2.2.

Theorem 2.93. Interior regularity for ∂�u0 ∈ C0,δ(Ω1,Rn×q) and ∈ Lp(Ω1,Rn×q):

1. Assume ap(·, ·) is strongly elliptic, see (2.343), and

∂Ω ∈ C1,δ, aijkl, f

ki ∈ C0,δ(Ω), 0 < δ < 1,

∣∣∣aijkl(x)

∣∣∣ ≤ Λ ∀x ∈ Ω, ∀i, j, k, l. (2.356)

Then for any weak solution �u0 ∈ H10 (Ω,Rq) of the original equation (2.340) its

derivative is ∂�u0 ∈ C0,δ(Ω,Rn×q). For any Ω1 ⊂⊂ Ω it is estimated as

‖∂�u0‖C0,δ(Ω1,Rn×q) ≤ C(‖∂�u0‖L2(Ω′,Rn×q) + ‖�f‖C0,δ(Ω′,Rn×q)

), (2.357)

where Ω′ := {x ∈ Ω : dist(x, ∂Ω) > dist(Ω1, ∂Ω)/2} and C depends on n, q, λ,Λ,δ,dist(Ω1, ∂Ω),

∥∥∥aijkl

∥∥∥C0,δ(Ω′)

, and diam Ω.

2. Assume ap(·, ·) is strongly elliptic and replace in (2.351) aijkl, f

ik ∈ C0(Ω) by

aijkl ∈ C0(Ω), =⇒

∣∣∣aijkl(x)

∣∣∣ ≤ Λ ∀x ∈ Ω, f ik ∈ Lp′

(Ω), p ≥ 2. Then for any weak

solution �u0 ∈ H10 (Ω,Rq) of (2.340) its derivative is ∂�u0 ∈ Lp(Ω,Rn×q). For any

Ω1 ⊂⊂ Ω it can be estimated as

‖∂�u0‖Lp(Ω1,Rn×q) ≤ C(‖�u0‖H1(Ω,Rq) + ‖�f‖Lp(Ω,Rn×q)

),

with C = C[n, q, p, λ,Λ,dist(Ω1, ∂Ω), modulus of continuity of the aij

kl

].

Similarly to Theorem 2.47 we obtain regularity results for convex domains, howeveronly for systems of order 2 and only in Rn with n = 2, 3. For a(·, ·) and its coefficientswe impose the conditions, cf. Rossmann [558], and Kozlov, Maz’ya and Rossmann[452], Sections 8.6 and 11.4:

Let a(·, ·) and its coefficient matrices Akl ∈ Rq×qin (2.344) satisfy (2.358)

Akl = A∗lk, Akl ∈W 1,∞(Ω), Akl ∈ C(Uv) ∀1 ≤ k, l ≤ n, and chosen

neighborhoods Uv ⊂ Rn ∀ vertices v ∈ ∂Ω, finally �f ∈W 1,∞(Ω,Cq),

for �f (j)(x) ∈ Cq, j ≤ n, letn∑

k,l=1

(Akl�f (l), �f (k))q ≥ c0

n∑j=1

|�f (j)|2q.

This last inequality obviously implies the coercivity of the principal part.

Theorem 2.94. Regular solutions for convex Ω: Let Ω ⊂ Rn, in (2.5) be convex,n = 2 or n = 3, and Ω ∈ C2, except in the vertices v ∈ ∂Ω, and a(·, ·) in (2.344) andits coefficients satisfy (2.358). Then the exact solution is �u0 ∈ H2(Ω,Rq).

Proof. This result is not an immediate consequence of results in Subsections 8.6 and11.4 of [452]. So we indicate the missing arguments, by locally modifying Ω into Ω, in


the neighborhoods, Uv, of the vertices, v. Similarly to [452], the system and coefficientsare modified such that Theorem 2.91 1. yields for s = 1 the regularity �u0 ∈ H2(Ω).For the local analysis, the condition Akl ∈ C(Ue) allows us the technique of frozencoefficients near the vertex, v. Then the local H2(Ω)-regularity is correct, if the frozensystem does not admit a singularity function rλU(ϕ) with 0 < Reλ ≤ 1 and −1/2 <Reλ ≤ 1/2 for two and three dimensional problems. In [452], Sections 8.6 and 11.4, it isshown that 1 < Reλ and 1/2 < Reλ if the angle between edges is < π. That is exactlythe situation in vertices of convex domains under the conditions of our theorem. �

Regularity estimates of a different type, not used in our numerical context, are dueto Morrey [500], Chen and Wu [170], pp. 137 ff., and Koshelev [449], pp. 108 ff. Forexample, for 1 ≤ p, 0 ≤ μ and Ω(x, ρ) := Ω ∩B(x, ρ) we define a norm as

‖�u‖Lp,μ(Ω) :=

[sup

x∈Ω,0<ρ<diam Ω

ρ−μ

∫Ω(x,ρ)

|�u(z)|pdz]1/p

. (2.359)

Then the space

Lp,μ(Ω) := {�u ∈ Lp(Ω) : ‖�u‖Lp,μ(Ω) <∞}

is a normed linear space, the Morrey space, with respect to the norm ‖�u‖Lp,μ(Ω).

Theorem 2.95. Regularity in Morrey spaces: Assume the aijkl in (2.342) satisfy

(2.343). If

aijkl(x) ∈ C0(Ω), f i

k ∈ L2,μ(Ω) ∀i, j, k, l, (2.360)

(hence (2.341) is valid!), then for any weak solution �u0 ∈ H10 (Ω,Rq) of (2.350) its

derivative is in ∂�u0 ∈ L2,μ(Ω,Rn×q). For Ω1 ⊂⊂ Ω it can be estimated as

‖∂�u0‖L2,μ(Ω1,Rn×q) ≤ C(‖∂�u0‖L2(Ω,Rn×q) + ‖�f‖L2,μ(Ω,Rn×q)

),

with C = C(n, q, λ, μ,Λ,dist(Ω1, ∂Ω), and the modulus of continuity of the aij

kl

).

2.6.4 Quasilinear elliptic systems of order 2 and variational methods

We generalize some of the results from the last Subsection 2.6.3 to quasilinear systemsand use the notation in (2.335). Our presentation again follows Chen and Wu [170],Chapter 11, and finally indicates some results from Leung [473] and Koshelev [449].We consider the following divergent quasilinear system in strong form:

Gs�u0(x) := G�u0(x) =

[n∑

k=0

(−1)k>0∂k(ai

k(x, �u0(x), ∂�u0(x)))]q

i=1

(2.361)

=n∑

k=0

(−1)k>0∂k(�ak(x, �u0(x), ∂�u0(x))) = 0,

with �u0, �u ∈ H2(Ω,Rq), �ak =[ai

k

]qi=1

,


and aik : Ω× Rq × Rn×q � w = (x, z, p) → ai

k(x, z, p) ∈ R. Often this divergent quasi-linear system characterizes a critical point of a variational problem as in (2.366). Thecorresponding strong form of the nondivergent quasilinear system, (2.361), originates,if (2.366) can be differentiated, see (2.369). In this sense, Taylor [620], pp. 198 ff.,presents his similar results on surfaces, and for smooth enough coefficients.

The standard multiplication of (2.361) with �v ∈ V := H10 (Ω,Rq), the scalar product

(·, ·)q in Rq, and integration by parts yields the corresponding weak system,

�u0 ∈ V : 〈G�u0, �v〉V′×V :=∫

Ω

q∑i=1

(n∑

k=0

aik(x, �u0(x), ∂�u0(x)), ∂kvi(x)

)dx (2.362)

=∫

Ω

n∑k=0

(�ak(x, �u0(x), ∂�u0(x)), ∂k�v(x))qdx =: a(�u0, �v) = 0 ∀�v ∈ V = H10 (Ω,Rq).

The most important examples for (2.361) or (2.362) are the so-called Euler systemsof a corresponding variational problem: We start with a function F , a functionalJ(�u), an admissible set A and formulate the following conditions essentially withrespect to F (note that the following Fp etc. indicates the partial with respect top etc.)

F : Ω× Rq × Rn×q � w = (x, �z, p) → F (x, �z, p) ∈ R (2.363)

J(�u) :=∫

Ω

F (x, �u(x), ∂�u(x))dx, J : H1(Ω,Rq) → R (2.364)

with boundary conditions in the form �g ∈ H1(Ω,Rq) s.t. J(�g) <∞e.g. choose A := {�u ∈ H1(Ω,Rq) : �u− �g ∈ V = H1

0 (Ω,Rq)} and (2.365)

determine �u0 ∈ A s.t. J(�u0) ≤ J(�v) ∀�v ∈ A. (2.366)

If F is convex in p, hence Fp(x, �z, p)(p− p) ≤ F (x, �z, p)− F (x, �z, p), then (2.366) andJ in (2.364) are called a regular variational problem and a regular integral functional.Obviously the admissible set A in (2.365) is a closed convex subset of H1(Ω,Rq). Theuniqueness of the solution, �u0, is an immediate consequence of the arguments on topof page 166 in [170].

Theorem 2.96. Variational problems, [170], Chapter 11, Theorem 2.1:

1. Let F, Fp be continuous in Ω× Rq × Rn×q, F convex in p, and F (x, �z, p) ≥ λ|p|2with λ > 0. Then the regular variational problem (2.366) has a solution �u0 ∈H1(Ω,Rq).

2. If F is strictly convex in p, hence Fp(x, �z, p)(p− p) < F (x, �z, p)− F (x, �z, p) forp �= p, then �u0 is unique.

So Chen and Wu [170] have shown the existence and uniqueness of a solution�u0 ∈ H1(Ω,Rq) for the regular variational problem (2.366). In their Chapter 11, 2.2


they elaborate the relation between this variational problem and quasilinearequations.

For the following considerations we need the derivatives of J(�u). We start withformal derivatives and present conditions in Theorem 2.97, justifying these derivatives.We introduce, similarly to (2.336), the notation for the components of �z ∈ Rq, p ∈Rn×q, without completely indicating their vector structure, as

�u(x) = (u1, . . . , uq)(x), �z = (z1, . . . , zq) ∈ Rq, and corresponding

p = (p1, . . . , pq) = (p1, . . . , pn), ∂�u(x) = (∂u1, . . . , ∂uq)(x) ∈ Rn×q, where

pi =(p1

i , . . . , pni

), ∂ui(x) =

(∂1ui, . . . , ∂

nui

)(x), x ∈ Rn, pk

i ∈ R,

pk =(pk1 , . . . , p

kq

), ∂k�u(x) =

(∂ku1, . . . , ∂

kuq

)(x) ∈ Rq, and for

w := (x, �z, p) ∈ Ω× Rq × Rn×q, i = 1, . . . , q, k = (0), 1, . . . , n,

F�z(w)�v =q∑

i=1

Fzi(w)vi, Fp(w)∂�v =

q∑i=1

n∑k=1

Fpki(w)∂kvi, (2.367)

F ′(w)�v = F�z(w)�v + Fp(w)∂�v, note that for w(x) = (x, �u(x), ∂�u(x)),

[∂Fp(w(x))]�v(x) = [∂Fp(x, �u(x), ∂�u(x))]�v(x) =q∑

i=1

n∑k=1

[∂kFpki(w(x))]vi(x)

=q∑

i,j=1

n∑k,l=1

[Fpk

i plj(w(x))∂k∂luj(x)

+Fpki zj

(w(x))∂kuj(x) + Fpki xj

(w(x))]vi(x).

Chen and Wu [170] derive two necessary conditions by the standard and a modifiedtrick in the classical calculus of variations:

Firstly, under appropriate growth condition, a solution �u0 ∈ A ∩ C1(Ω,Rq) of(2.366) satisfies the weak and strong form of the Euler system for J, see (2.368),(2.369) and compare Theorem 2.73. We formulate them componentwise and ina short form. With (2.367), (2.361) or (2.362) and �ak : Ω× Rq × Rn×q → Rq,we get

�ak(w) =(ai

k(w) := Fpki(w))q

i=1, for k > 0, and �a0(w) =

(ai0(w) := Fzi

(w))qi=1

.

With w0(x) = (x, �u0(x), ∂�u0(x)), we obtain the weak form

a(�u0, �v) : = (J ′(�u0), �v)H10 (Ω,Rq) =

∫Ω

(Fp(w0(x))∂�v + F�z(w0(x))�v(x)

)dx (2.368)

=∫

Ω

(F ′(w0)�v)(x)dx


=∫

Ω

n∑k=0

(�ak(w0(x)), ∂k�v(x))qdx = 0 ∀�v ∈ V = H10 (Ω,Rq) � �u0.

By standard partial integration (2.368) is transformed into∫Ω(· · · )�v(x)dx = 0 ∀�v ∈

H10 (Ω,Rq). This implies the system in its strong form as

(· · · ) = −∂Fp(w0(x)) + F�z(w0(x)) =n∑

k=0

(−1)k>0(∂k(�ak(w0(x))

=

⎡⎣Fzi(w0(x))−

q∑j=1

n∑k,l=1

(Fpk

i plj(w0(x))∂k∂luj

+Fpki zj

(w0(x))∂kuj + Fpki xj

(w0(x)))⎤⎦q

i=1

= 0. (2.369)

So the quasilinear equations (2.361), (2.369) and (2.362), (2.368) represent the strongand weak Euler system for J for the solution �u0. These Fpk

i(w0(x)), and their gradients

∂k(Fpki)(w0(x))), k > 0, . . . , are Nemyckii operators of a type similar to those studied

in Subsections 2.5.5 and 2.5.6. We will impose appropriate conditions to have themwell defined.

Secondly, for F ∈ C2, a solution �u0 of (2.366) necessarily satisfies, cf. [170], p. 169,the condition for local minima, the so-called Legendre–Hadamard condition for thecorresponding principal part, see (2.348).

q∑i,j=1

n∑k,l=1

Fpki pl

j(x, �u0(x), ∂�u0(x))ϑkϑlζiζj ≥ 0 ∀ ϑ ∈ Rn, ζ ∈ Rq. (2.370)

For our later linearization we assume, stronger than (2.370), cf. (2.349), that (2.368),(2.369) satisfies for �u0 the so-called strong Legendre condition, compare (2.342),(2.343), so F is strictly convex in w(x),

q∑i,j=1

n∑k,l=1

Fpki pl

j(w0(x))ϑk

i ϑlj ≥ λ|�ϑ|2nq ∀ �ϑ ∈ Rn×q. (2.371)

This implies, for Fpki pl

j(w0(x)) ∈ L∞(Ω,Rq×q) by Theorem 2.89, the coercivity of the

linearized operator. Similarly to quasilinear equations, Chen and Wu [170], pp. 163ff., guarantee solutions by imposing Caratheodory type and controllable and naturalgrowth conditions for F, measurable in x, cf. [170], pp. 171 ff.

F : Ω× Rq × Rn×q → R measurable in x,

let F ∈ C1with respect to(�z, p) a.e.x ∈ Ω. (2.372)


Controllable growth conditions: With constants λ,Λ, C ∈ R+ and functions20

gi ≥ 0, gi ∈ L1(Ω), f ik, fi ≥ 0, f i

k ∈ L2(Ω), fi ∈ Lr/(r−1)(Ω),

define V := (1 + |�z|2 + |p|2). We assume for F, Fxk, Fpk

i: (2.373)

λV − g1(x) ≤ F (x, �z, p) ≤ ΛV + g2(x), mind our V is V 2 in [170],∣∣Fxk(x, �z, p)

∣∣ ≤ C(|p|2(1−1/r) + |�z|r−1 + fk(x)), k = 1, . . . , n,∣∣∣Fpki(x, �z, p)

∣∣∣ ≤ C(|p|+ |�z|r/2 + f i

k(x)), i = 1, . . . , q. (2.374)

Natural growth conditions: For |�z| ≤M, and constants λ, μ,Λ = Λ(M) ∈ R+ let

λ|p|2 − μ ≤ F (x, �z, p) ≤ Λ(1 + |p|2) (2.375)

|Fxk(x, �z, p)| ≤ Λ(1 + |p|2),

∣∣∣Fpki(x, �z, p)

∣∣∣ ≤ Λ(1 + |p|). (2.376)

Theorem 2.97. J ′(�u)v is a weak Euler system, Chen and Wu, [170], Chapter 11,Theorems 2.3, 2.4.: Let (2.372)–(2.374) or (2.372), (2.375)–(2.376) be satisfied for�u ∈ H1(Ω,Rq) ∩ L∞(Ω,Rq). Then J(�u) is differentiable in �u and J ′(�u)v, a quasilinearoperator, has the form of the weak Euler system (2.368).

We call �u0 ∈ A ⊂ L∞(Ω,Rq), cf. (2.365), a generalized solution of (2.362) if forcontrollable growth conditions, (2.368) is satisfied for �u0 and for all �v ∈ A. For naturalgrowth conditions, we have to test (2.368) with all �v ∈ L∞(Ω,Rq). It is well knownthat F is strictly convex in p iff the strong Legendre condition (2.371) is satisfied.

Theorem 2.98. Generalized solution of (2.362), [170], Chapter 11, Summary: Underthe conditions of Theorems 2.96 and 2.97 and for F convex in p, there exists, and for Fstrictly convex there uniquely exists, a generalized solution �u0 of (2.362), characterizedas a weak solution of an elliptic quasilinear equation.

Now, we formulate regularity results, see Chen and Wu [170], Chapter 12, pp. 173ff., directly for the quasilinear systems in (2.361) or (2.362). Accordingly we introducemodifications of the above strong Legendre and growth conditions, obviously strongerthan (2.373), (2.374), and (2.375), (2.376), as

A :=(ai

k, e.g. aik = Fpk

i

) q n

i=1,k=1,�b :=

(ai0, e.g. ai

0 = Fzi

)qi=1

, A,�b ∈ C1, (2.377)

∃λ,Λ > 0 :q∑

i,j=1

n∑k,l=1

(ai

k

)pl

j

(w)ϑki ϑ

lj ≥ λ|�ϑ|2 ∀�ϑ ∈ Rn×q,with w = (x, �z, p),(2.378)

|A(w)|, |Ax(w)|, |�b(w)|, |�bx(w)| ≤ Λ(1 + |�z|2 + |p|2)1/2, (2.379)

|A�z(w)|, |Ap(w)|, |�b�z(w)|, |�bp(w)| ≤ Λ for the controllable growth conditions, and

20 r is the so-called Sobolev conjugate of 2, r = 2n/(n − 2) for n > 2, else r ∈ [2,∞).


|A(w)|, |Ax(w)|, |A�z(w)|, |�bp(w)| ≤ Λ(1 + |p|), |Ap(w)| ≤ Λ, (2.380)

|�b(w)|, |�bx(w)|, |�b�z(w)| ≤ Λ(1 + |p|2)∀|�z| ≤M, for the natural growth conditions.

The constants λ,Λ may depend upon M for natural growth conditions. We haveto slightly modify the concept of generalized solutions: For the two cases of con-trollable and natural growth conditions (2.379) and (2.380), we test a possiblegeneralized solution for (2.362), �u0 ∈ H1(Ω,Rq) and �u0 ∈ H1(Ω,Rq) ∩ C0(Ω,Rq), by∀�v ∈ V := V1 := H1

0 (Ω,Rq) and �v ∈ V := V1 ∩ L∞(Ω,Rq), respectively. Independentlyof boundary conditions we get:

Theorem 2.99. Regularity of solutions, [170], pp 173 ff., Theorems 1.1, 1.2:

1. Let the quasilinear equation (2.362) have smooth coefficients, A,�b ∈ C1, andsatisfy the strong (linearized) Legendre condition for �u0, see (2.377), (2.378),and the controllable or natural growth condition (2.379) or (2.380). Then ageneralized solution �u0 ∈ H1(Ω,Rq) is in H2(Ω,Rq).

2. The derivatives ∂��u0, � = 1, . . . , n satisfy some kind of linearized form of (2.362):

∫Ω

q∑i,j=1

n∑k,l=1

[(ai

k

)pl

j

w0(x)∂j(∂�u0

l

)+(ai

k

)zlw0(x)

(∂�u0

l

)+(ai

k

)x�w0(x)

−δi,�a0kw

0(x)]∂ivkdx = 0∀�v ∈ V with w0(x) := (x, �u0(x), ∂�u0(x)). (2.381)

For the two cases of controllable and natural growth conditions the chosen V aredense in H1

0 (Ω,Rq). Hence, the system (2.381) is, by (2.378), a linear elliptic systemfor each of the ∂��u0, � = 1, . . . , n. So we can achieve higher regularity by induction,cf. Theorems 2.38, 2.39, 2.45. We only discuss Theorem 2.91 1. Applications of 2. areonly indicated.

Theorem 2.100. Regular solution of (2.362) with ∂��u0 ∈ Hs+2(Ω,Rq):

1. Under the conditions of Theorem 2.99 the relation (2.381) is strongly elliptic. Weassume for a w1(x) := (x, �u1(x), ∂�u1(x)) with ∂�u1(x)) ∈ Hs+1(Ω,Rn×q) that, cf.(2.351),

∂Ω ∈ Cs+2,(ai

k

)pl

j

(w1(x)),(ai

k

)zj

(w1(x)) ∈W s+1,∞(Ω), (2.382)

�f :=

⎛⎝ n∑k,l=1

(ai

k

)x�

(w1(x))− δk,�a0i (w

1(x)) ∈ Hs+1(Ω)

⎞⎠q, n

i=1,k=1

.

2. Then any weak solution �u0 ∈ H1(Ω,Rq) and �u0 ∈ H1(Ω,Rq) ∩ C0(Ω,Rq) of(2.362) satisfies ∂��u0 ∈ Hs+2(Ω,Rn×q), � = 1, . . . , n, and∥∥∂��u0

∥∥Hs+2(Ω,Rn)

≤ C(‖∂��u0‖L2(Ω′,Rn) + ‖�f‖Hs+1(Ω,Rn)

). (2.383)


3. Theorem 2.91 2., 3. implies �u0 ∈ C∞0 (Ω,Rn×q) and ∂��u0 ∈ Cs,δ(Ω,Rn×q),

� = 1, . . . , n, if (2.382) is replaced by the C-conditions in (2.353) and (2.354),respectively

Specific systems allow �u0 ∈ C0,δ(Ω1,Rq) for Ω1 = Ω, or Ω1 ⊂⊂ Ω, [170], pp. 178,181 ff. is extended to the general form by our Theorem 2.91. Indirect methods, reverseestimates for ∂�u0 and singular sets, Ω \ Ω1, are studied.

Leung [473], Theorem 3.2-1, shows �u0 ∈ C2(Ω,Rq) ∩ C1(Ω,Rq) for the solution of aspecial case of our quasilinear systems, the semilinear systems of the form

Δ�u + f(x, �u) = 0 ∀x ∈ Ω, ∂�u/∂ν + g(x, �u) = b(x) ∀x ∈ ∂Ω.

Koshelev and Chelkak [449,450] present many different regularity results for ellipticand parabolic equations. For example, for special cases of the quasilinear equation(2.362), Holder continuity for the derivatives of solutions in weighted spaces, seeTheorem 3.2.1, is proved. A combination of Liouville’s theorem, generalized to ellipticproblems, with the method of elastic solutions describes appearing cracks. For theNavier–Stokes problem [450] estimates smoother norms for time dependent solutions.We give one example of [450] results. Again, we consider the above quasilinearboundary value problem (2.361), (2.365), see (2.16). In [449] he assumes a modifiedCaratheodory condition (2.284) for all terms, so all ak

i (x, �z, p), i = 1, . . . , q, k = 0, . . . , n,are measurable with respect to x for all finite (�z, p) ∈ Rn+nq and continuously dif-ferentiable with respect to (�z, p) a.e. for x ∈ Ω. As growth conditions he requires asharpened strong Legendre condition for the linearized operator, compare (2.378). Forall �ϑ ∈ Rn×q, and with w = (x, �z, p), let

λ(1 + |(�z, p)|2)(s−1)/2|�ϑ|2 (2.384)

≤q∑

i,j=1

n∑k,l=0

(∂ai

k/∂pjl

)(w)ϑk

i ϑlj ≤ Λ(1 + |(�z, p)|2)(s−1)/2|�ϑ|2,

for 1 < s ≤ 2, λ,Λ = const . > 0 and∣∣∣(∂aik/∂p

jl

)(w)∣∣∣ ≤ Λ(1 + |(�z, p)|2)(s−1)/2 ∀ i, j, k, l, and for some (2.385)

1 < p′ ∀�u ∈W 1,p′(Ω′) and Ω′ ⊂ Ω let ai

k(x, �u(x), ∂�u(x)) ∈ Lp′/(s−1)(Ω′).

Choose g ∈ H1α(Ω) and for 0 < γ < 1 let α := 2− n− 2γ and K satisfy

1− α(α + n− 2)2(m− 1)

> 0,K2

[1− α(n− 2)

m− 1

] [1− α(α + n− 2)

2(m− 1)

]−1

< 1, (2.386)

H1α(Ω) :=

{�u ∈ H1(Ω) : (2.387)

‖�u‖H1α(Ω) := sup

x0∈Ω

∫Ω

⎛⎝|�u(x)|2 +∑|α|=1

|∂α�u(x)|2|x− x0|α⎞⎠ dx <∞

}.


Theorem 2.101. Solution of (2.362) in H1α(Ω): Under the conditions (2.384)–

(2.386), the nonhomogeneous Dirichlet problem (2.362), (2.365) has a unique solution�u0 ∈ H1

α(Ω) and �u0 ∈ C(2−m−α)/2(Ω). Stronger results are possible for weighted normsin Sobolev spaces, extending the H1

α(Ω) setting.

2.6.5 Linear elliptic systems of order 2m, m ≥ 1

We essentially study weak forms of systems of differential operators of order 2m.We start with coercivity and the related existence and Fredholm results. Again thefollowing results are valid for Wm,p(Ω,Rn), p ≥ 2, in the sense of Corolary 2.44. Butwe essentially restrict the discussion here to the Hilbert space setting and extendit to Banach spaces in the next subsection. For the necessary higher order Fouriertransforms we need Hilbert spaces anyway. For the case of second order systems,Theorem 2.89 shows that the coercivity is a consequence of the strong Legendrecondition (2.343). This implies the strong Legendre–Hadamard condition, see (2.349).This simple situation does not seem to remain correct for higher order 2m. In fact, thestrong Legendre–Hadamard condition is needed to prove the coercivity of the principalpart. We combine

V = Hm0 (Ω,Rq) ⊂ W = L2(Ω,Rq) = W ′ ⊂ V ′ = H−m

0 (Ω,Rq), (2.388)

with the scalar products and norms in (2.337) and L�u in divergence form.We recall and update the notation, cf. (2.73), (2.79), (2.102), (2.336), (2.335).

Instead of �ϑ in (2.335) the η, ϑ are here more appropriate:

�u(x) = (u1, . . . , uq)(x), η = (η1, . . . , ηq) ∈ Rq, ϑ = (ϑ1, . . . , ϑn) ∈ Rn, (2.389)

ϑα = (ϑ1)α1 . . . (ϑn)αn ∈ R, ϑα = 1 for |α| = 0, ∂αui =∂|α|

∂xα11 . . . ∂xn

αnui,

∂α�u = (∂αu1, . . . , ∂αuq), with ∂αui(x) ∈ R, ∂α�u(x) ∈ Rq, ∂α�u = �u for |α| = 0.

Similarly to our Lemma 2.26 above, see (2.82), and the beginning of Subsection2.6.3, generalizations of Dirichlet boundary conditions to Dirichlet system are possible.Several cases are studied by Taylor [620], Chapter 5, Section 11. For avoiding the manytechnicalities we have restricted the discussion to homogeneous Dirichlet boundaryconditions and choose V = Hm

0 (Ω,Rq) instead of a possible Hm0 (Ω1,Rq),Ω1 ⊂ Ω. For

smooth enough Ω we get the equivalence

∂i�u(x)∂νi

= 0 ∀0 ≤ j ≤ m− 1 ⇔ ∂α�u(x) = 0 ∀|α| ≤ m− 1 ∀x ∈ ∂Ω, (2.390)

in the trace sense; the usual weak form below and the strong form are well defined

L : H2m ∩ V → W = L2(Ω,Rq), L�u :=∑

|α|,|β|≤m

(−1)|α|∂α(Aαβ∂β�u) = �f (2.391)

for smooth enough Aαβ . For the weak form we take the standard scalar product with�v = (v1, . . . , vq) ∈ V and integrate by parts. A weak solution �u0 =

(u0

1, . . . , u0q

)∈ V for


(2.391) is defined by equating the corresponding linear and bilinear forms, 〈�f,�v〉 anda(�u0, �v), compare (2.146), (2.274), (2.275),

�u0 ∈ V = Hm0 (Ω,Rq) : 〈�f,�v〉 =

⟨⎛⎝ ∑|α|≤m

f iα∂

α

⎞⎠q

i=1

, �v

⟩=∫

Ω

∑|α|≤m

(�fα, ∂α�v)qdx

= a(�u0, �v) := 〈L�u0, �v〉V′×V :=∫

Ω

∑|α|,|β|≤m

(Aαβ∂β�u0, ∂α�v)qdx (2.392)

=q∑

i,j=1

∑|α|,|β|≤m

∫Ω

(aij

αβ∂βu0

j∂αvi

)dx, ∀ �v ∈ V, Aαβ(x) ∈ Rq×q, aij

αβ ∈ L∞(Ω),

�f =(f i

α

)qi=1,|α|≤m

= (�fα)|α|≤m, f iα ∈ L2(Ω)∀i, j, α, β, or �f ∈ V ′. (2.393)

We generalize the above strong Legendre–Hadamard condition for m = 1, cf. (2.348),to order 2m and will combine it with variable coefficients aij

αβ , see Theorem 2.43.

Definition 2.102. A system of linear differential operators of order 2m is calledelliptic, if its principal part, Lp, satisfies the strong Legendre–Hadamard condition

ap(�u,�v) :=∫

Ω

Lp�u�vdx :=∫

Ω

∑|α|=|β|=m

(Aαβ∂β�u, ∂α�v)qdx, and for (2.394)

Lp(x, ϑ) :=

⎛⎝ ∑|α|=|β|=m

aijαβ(x)ϑβϑα

⎞⎠q

i,j=1

=

⎛⎝ ∑|α|=|β|=m

Aαβ(x)ϑβϑα

⎞⎠ let ∃λ > 0 :

ηTLp(x, ϑ)η =q∑

i,j=1

∑|α|=|β|=m

aijαβ(x)ϑβηjϑ

αηi ≥ λ|ϑ|2m|η|2 ∀ϑ ∈ Rn, η ∈ Cq, x ∈ Ω.

For 1 = m < q the condition (2.343) was required, cf Remark 2.90.

For proving existence and uniqueness results for (2.392) we reinterpret the bilinearform as a(�u,�v) =: ap(�u,�v) + b(�u,�v). As above we prove that a(·, ·), ap(·, ·) : V × V → R

and �f ∈W−m,p′(Ω,Rq) = V ′ → R are bounded bilinear and linear forms and ap(·, ·) is

V-coercive. We summarize

Theorem 2.103. Strong and weak solutions of (2.392): For a system of theform (2.392) define the bilinear and linear forms a(�u,�v) and 〈�f,�v〉V′×V , andthe principal part ap(�u,�v) as in (2.392), (2.394). Under the conditions (2.393),

these a(�u,�v), ap(�u,�v), 〈�f,�v〉V′×V are continuous in V = Hm0 (Ω,Rq). For

(ajk

αβ

)q

i,j=1

= Aαβ ∈W |β|,∞(Ω,Rq), any solution �u0 ∈ H2m0 (Ω,Rq) of (2.391) necessarily satisfies

the weak equation, cf. (2.392),

�u0 ∈ V : a(�u0, �v) = 〈�f,�v〉V′×V∀�v ∈ V. (2.395)


Vice versa, every smooth solution �u0 ∈ H2m(Ω,Rq) ∩Hm0 (Ω,Rq) of (2.392) satisfies

(2.391).

Our next goal are analogies to Theorems 2.43, 2.89 and Corollary 2.44. The keyresult towards this goal is the V-coercivity of the principal part.

Theorem 2.104. Coercivity, unique solutions, Fredholm alternative: In (2.392)let the coefficients ajk

αβ ∈ L∞(Ω), but for |α| = |β| = m > 1 be continuous in Ω, seeTheorem 2.43. Furthermore, let (2.394), the strong Legendre–Hadamard condition, besatisfied. Then the principal part ap(�u,�v) is V = Hm

0 (Ω,Rq)-coercive.

Under these conditions Corollary 2.44 implies for a V-coercive a(·, ·) the existenceof a unique generalized or weak solution �u0 and a boundedly invertible L. Hence, fora V-elliptic a(·, ·) the Fredholm alternative with its consequences is valid. For a V-coercive, symmetric a(·, ·), the unique �u0 solves the variational problem, the updated(2.61). Finally, all Hm-coercivity based existence, uniqueness and Fredholm results,including numerical stability and index L = 0, remain correct for the Wm,p case andfor p ≥ 2, with the modifications in Corollary 2.44 3., 4..

Proof. We cannot repeat the argument in the proof of Theorem 2.89, see (2.347).The reason is the inequality (∂uj/∂xi)αi �= ∂αiuj/∂x

αii for αi > 1. Instead we use the

standard Fourier transforms, see, e.g. Hackbusch [388], Theorem 7.2.7. We start theproof by assuming constant highest coefficients. But the same ideas as in the proof ofTheorem 2.43 can be used here as well to obtain the coercivity for continuous highestcoefficients. So for constant highest coefficients we get

ap(�u, �u) =∫

Ω

⎛⎝ q∑j,k=1

∑|α|=|β|=m

ajkαβ∂

βuk∂αuj

⎞⎠ dx ∀ �u,�v ∈ V. (2.396)

We extend the �u,�v ∈ V by �u(x) = 0 ∀ x ∈ Rn \ Ω. This �u has compact support Ω.Thus the Fourier transform, see, e.g. [388], Subsection 6.2.3, is well defined21 for this

21 It is defined, with the imaginary unit ι with ι2 = −1, via the components u = uj ∈ C∞0 (Rn) :

F : C∞0 (Rn) → L2(Rn), with (ξ, x)n =

n∑k=1

ξkxk ∀ ξ, x ∈ Rn, as

u(ξ) := (Fu)(ξ) := (2π)−n/2

∫Rn

e−ι(ξ,x)nu(x)dx

:= limR→∞

(2π)−n/2

∫|ξ|∞≤R

e−ι(ξ,x)nu(x)dx. (2.397)

Since C∞0 (Rn) is dense in L2(Rn), this F can be extended to L2(Rn) → L2(Rn) and inverted on

C∞0 (Rn), dense in L2(Rn), and again extended to L2(Rn) → L2(Rn) as

(F−1u)(x) := (2π)−n/2

∫Rn

eι(ξ,x)n u(ξ)dξ ∀ u ∈ C∞0 (Rn), extended to L2(Rn).

Its characteristics are summarized in


�u. For ap(�u, �u) we obtain for constant coefficients

ap(�u, �u) =q∑

j,k=1

∑|α|=|β|=m

ajkαβ

∫Rn

(∂βuk · ∂αuj

)dx ∀ �u ∈ V.

We apply the Fourier transform to each term ∂βuk and ∂αuj . By the invariance ofthe scalar products in (2.398), the strong Legendre–Hadamard condition (2.394), andwith ηk = uk(ξ) = F (u)(ξ) we get

ap(�u, �u) =q∑

j,k=1

∑|α|=|β|=m

ajkαβ

∫Rn

([ι|β|ξβ uk(ξ)][ι|α|ξαuj(ξ)]

)dξ

=∫

Rn

q∑j,k=1

∑|α|=|β|=m

(ajk

αβξβι|β|uk(ξ)ξα[ι|α|uj(ξ)]

)dξ (2.400)

≥∫

Rn

λ

(n∑

�=1

(ξ�)2)m q∑

i=1

|ui(ξ)|2dξ

For a fixed m > 1 there exists an ε > 0, such that(∑n

�=1(ξ�)2)m

> ε∑

|α|=m(ξα)2,see (2.75). So we find with (2.399)

ap(�u, �u) > λε

q∑i=1

∫Rn

⎛⎝ ∑|α|=m

(ξα)2

⎞⎠ |ui(ξ)|2dξ = λε|�u|2Hm(Ω,Rq).

Since |�u|Hm(Ω,Rq) and ‖�u‖Hm(Ω,Rq) are equivalent norms on V = Hm0 (Ω,Rq), the V-

coercivity of ap(�u, �u) is proved. The extension to “continuous coefficients” is obtainedas in Theorem 2.43. �

Theorem 2.105. The Fourier transform and its inverse are linear isometries:

F, F−1 : L2(Rn) → L2(Rn), with

‖F‖L2(Rn)←↩L2(Rn) = ‖F−1‖L2(Rn)←↩L2(Rn) = 1,

(u, v)L2(Rn) = (u, v)L2(Rn) ∀ u, v ∈ L2(Rn), (2.398)

∀ u ∈ Hk(Rn), |α| ≤ k F (∂αu)(ξ) = ι|α|ξαu(ξ), and

‖u‖Hk(Rn) =

∥∥∥∥∥∥√ ∑

|α|≤k

|ξα|2u(ξ)

∥∥∥∥∥∥L2(Rn)

and

the seminorm |u|Hk(Rn) =

∥∥∥∥∥∥√ ∑

|α|=k

|ξα|2u(ξ)

∥∥∥∥∥∥L2(Rn)

. (2.399)

By applying F componentwise, this F can be extended to F : L2(Rn, Rq) → L2(Rn, Rq), noting thatthen (ui, vj)L2(Rn) = (ui, vj)L2(Rn) ∀ i, j = 1, . . . , q.


For an elliptic linear system in the sense of Definition 2.85 6., we give a flavor ofsome results for local existence and regularity, by summarizing a few examples fromTaylor [618]. For regularity results we combine several Propositions in Taylor [618],Chapter 5, Section 11 with our previous results. We formulate an interior estimate,a near boundary “collar” estimate, and a local solvability and regularity result. Wesimplify Taylor [618], Chapter 5, Theorem 11.1 ff., originally for elliptic systems onmanifolds, to our Ω ⊂ Rn.

Taylor [618] discusses regular boundary value problems, cf. Chapter 5, Propositions11.9–11.10. For example, a system (2.392) with Dirichlet boundary conditions isregular, if it satisfies a modified strong Legendre condition, cf. Chapter 5, (11.79),

∃λ > 0 : ∀x ∈ Ω,∀ϑ ∈ Rn :

⎛⎝ ∑|α|=|β|=m

aijαβϑ

βϑα

⎞⎠q

i,j=1

≥ λ|ϑ|2mI, I ∈ Rq×q, (2.401)

with the idendity matrix, I. This implies the strong Legendre–Hadamard condition(2.394) and a Garding inequality, cf. (2.141), [618], Chapter 5, Exercise 1. Other typesof boundary conditions are discussed as well Taylor [618], Chapter 5, Proposition 11.10and Theorem 11.11 apply to our situation:

Theorem 2.106. Interior estimate: Let L, a system of linear differential operatorsof order m, be m-elliptic, cf. Definition 2.85, 6., and �u ∈ Hm(Ω,Rq) (a distributionin [618], Theorem 11.1) and L�u = f ∈ Hs(Ω,Rq), 0 ≤ s ∈ N0. Then �u ∈ Hm+s(Ω,Rq).Furthermore, for any Ω1 ⊂⊂ Ω2 ⊂⊂ Ω and σ < m + s the following estimate holds

‖�u‖Hm+s(Ω1,Rq) ≤ C(‖L�u‖Hs(Ω2,Rq) + ‖�u‖Hσ(Ω2,Rq)).

For the next theorem we modify some properties related to the atlas of a domainΩ, see Proposition 1.32, (1.63). In a first step we choose the Vi, i = 1, . . . , N , with∂Ω ⊂ ∪N

i=1Vi. Let Vi := Ω ∩ Vi, V := ∪Ni=1Vi. For Ω ∈ C0,1 Proposition 1.32, implies

that there exist Ω3 ⊂⊂ Ω4 ⊂⊂ V with Ω3 ∩ ∂Ω = Ω4 ∩ ∂Ω = ∂Ω, such that b3 : ∂Ω×[0, 2] → Ω3, and b4 : ∂Ω× [0, 1] → Ω4 are bijections. Now boundary conditions and“collar” norms are relevant, see [618], Chapter 5, Propositions 11.14, 15, 16. With‖�u(·, y)‖Hs(∂Ω), 0 ≤ y ≤ 2, and ∇j

y, the jth partial derivative into the y-direction, letthe norms

‖�u‖2H0,s(Ω4,Rq) :=∫ 1

0

‖�u(·, y)‖2Hs(∂Ω)dy, s, k ∈ N0,

‖�u‖2Hk,s(Ω4,Rq) :=k∑

j=0

∫ 1

0

∥∥∇jy�u(·, y)

∥∥2

Hk−j+s(∂Ω)dy, (2.402)

define the corresponding Banach spaces Hk,s(Ω4,Rq), and similarly Hk,s(Ω3,Rq) with∫ 2

0replacing

∫ 1

0. Additionally, let L, a system of linear differential operators now of

order 2m, be strongly 2m-elliptic, cf. Definition 2.85, 6. By Taylor [618], Chapter5, Proposition 11.10. a strongly 2m-elliptic system (2.392) or (2.323) with Dirichletboundary conditions is regular. For other types of boundary conditions see [618].


Theorem 2.107. “Collar” estimate, Taylor, [618], Chapter 5, Propositions 11.14:Under these conditions let �u0 ∈ H2m,σ(Ω3) solve the (nonhomogeneous) “collar”Dirichlet boundary problem22

L �u0 = �f ∈ Hk,s(Ω3,Rq), ∂j �u0/∂ν

j = ϕj ∈ H2m+k−j+1/2+s(∂Ω,Rq), j = 0, . . . ,m− 1,

for some 0 ≤ σ < s ∈ R. Then we even obtain �u0 ∈ H2m+k,s(Ω4,Rq) and it satisfiesthe estimate

‖ �u0‖H2m+k,s(Ω4,Rq) ≤ C

⎛⎝‖L �u0‖Hk,s(Ω3,Rq) +m−1∑j=0

‖∂j �u0/∂νj‖H2m+k−j+1/2+s(∂Ω,Rq)

⎞⎠.

Proposition 11.16 of [618] applies to our situation and states, compare Zeidler [676],Section 4.17:

Theorem 2.108. Regularity and Fredholm alternative for order 2m: Let thecoefficients aij

αβ, ∂Ω in (2.392) be in C∞, and let Definition 2.85 6. be satisfied.Then the operator L : H2m+k(Ω,Rq) ∩Hm

0 (Ω,Rq) → Hk(Ω,Rq), q ≥ 1, in (2.392) isFredholm, hence, for f ∈ Hk(Ω,Rq), k ≥ 0, any solution for (2.392), if it exists, is�u0 ∈ H2m+k(Ω).

2.6.6 Divergent quasilinear elliptic systems of order 2m

In this subsection we study strong and weak forms of systems of quasilinear differentialoperators of order 2m. The results for this case are not available in a similarly generalform as before. We summarize parts of Koshelev and Chelkak [450] but start with oneof the most important examples:

von Karman equations

Different types of existence and regularity results for this system are discussed byJohn and Necas [423], John and Naumann [422], Ciarlet [175], Giaquinta et al. [343],Kondrat’ev and Olejnik [447,448], Horn [404], and Bock [100]. In John and Naumann[422], the strong form of the von Karman equations is studied:

Gs(u,w) :=

⎛⎝Δ2u −[u,w]

Δ2w + 12 [u, u]

⎞⎠ =

⎛⎝ f

0

⎞⎠ in Ω ⊂ R2, (2.403)

with Δ2 = ΔΔ and the bracket operator [u, v] := uxxvyy − 2uxyvxy + uyyvxx = [v, u].Specific existence and uniqueness results for the von Karman equations are discussed

in the papers of Hlavacek and Naumann [399, 508], John et al. [422, 423], even withdifferent boundary conditions on different parts of ∂Ω, and Giaquinta et al., [343]. Fora full formulation, many technical details would have to be discussed. So we just refer

22 Or let L with the boundary operators B be regular elliptic, see [620], pp. 379 ff.; essentially L, Bsatisfy a generalized complementary condition.


to [343,399,422,423,508], and only combine John and Naumann [422] in (2.403) withsome of our earlier results.

Existence and uniqueness can also be elaborated numerically. The above (2.331)and (2.403) represent compact perturbations of its strong principal part

Gp(u,w) :=(Δ2u,Δ2w

)T =(f, 0)T in Ω. (2.404)

The weak form of (2.404) is obviously H20 (Ω,R2)-coercive, so the linearized (2.403) is

H20 (Ω,R2)-elliptic. Furthermore, (2.404) satisfies the conditions in Subsection 2.6.5.

Hence, existence and uniqueness for (2.331) and (2.403) can be guaranteed by com-bining the Fredholm alternative for the linearized form of (2.403) with discretizationsand numerical continuation techniques, starting with (2.404) and ending with (2.331)or (2.403).

Equation (2.403) is slightly simplified compared to (2.331). In the first equation the+λuxx is missing. As a consequence of the special form of (2.331), and since the +λuxx

term represents a compact perturbation, the arguments for Theorem 2.56 3. remainvalid for this nonlinear case as well. Hence, we formulate John and Naumann [422]directly for the more general (2.331) case. They discuss two sets of non-Dirichletboundary conditions. Using the unit outer normal vector ν = (νx, νy), the boundarytangent vector σ = (−νy, νx) and the Poisson ratio of the plate material, μ, theyintroduce the operators

Mu := μΔu + (1− μ)(uxxν

2x + 2uxyνxνy + uyyν

2y

)with constant μ ∈ (0, 1/2), or

Tu := −μ ∂

∂νΔu + (1− μ)

∂

∂σ(uxxνxνy − uxy

(ν2

x − ν2y

)− uyyνxνy). (2.405)

This allows us to formulate the two sets of boundary conditions, considered in [422],

u = Mu = 0, w =∂

∂νw = 0 on ∂Ω and (2.406)

Mu = Tu = 0, w =∂

∂νw = 0 on ∂Ω. (2.407)

We denote (2.403), (2.406) as Problem I, and (2.403), (2.407) as Problem II.For weak solutions of (2.403), John and Naumann [422] modify the standard

approach. With

a(u, v) :=∫

Ω

[uxxvxx + 2(1− μ)uxyvxy + uyyvyy + μ(uxxvyy + uyyvxx)]dxdy

and ∃c1, c2 : c1‖u‖2H2(Ω) ≤ a(u, u) ≤ c2‖u‖2H2(Ω),∀u, v ∈ H2(Ω), (2.408)

they get a modified Green’s formula, based upon M,T

a(u, v) =∫

Ω

uΔ2vdxdy +∫

∂Ω

uTvds +∫

∂Ω

∂u

∂νMvds ∀u, v ∈ H4(Ω), (2.409)


or combined as∫Ω

(uΔ2v −Δ2uv)dxdy = +∫

∂Ω

(vTu− uTv)ds +∫

∂Ω

(∂v

∂νMu− ∂u

∂νMv

)ds.

Correspondingly we have to determine the solution

(u0, w0) ∈(W := {u ∈ H2(Ω) : u|∂Ω = 0})×H2

0 (Ω)), (2.410)

by testing (2.403) in the first and second equation as

a(u0, ϕ) =∫

Ω

[u0, w0]ϕdxdy +∫

Ω

fϕdxdy∀ϕ ∈W (2.411)

(w0, ψ)H20 (Ω) :=

∑|α|=2

∫Ω

∂αu0∂αvdxdy = −

∫Ω

[u0, u0]ψdxdy ∀ψ ∈ H20 (Ω). (2.412)

Equation (2.331) has to be tested by slightly changing (2.411) into (2.413) and leaving(2.412) unchanged:

a(u0, ϕ) =∫

Ω

[u0, w0]ϕdxdy −∫

Ω

λu0,xxϕdxdy +∫

Ω

fϕdxdy ∀ϕ ∈W. (2.413)

Obviously with a solution (u0, w0) of Problem II, (2.403), (2.407), the (u0 +p1, w0),∀p1 ∈ Π1, the polynomials of degree 1, is again a solution. We eliminate thenonuniqueness for u0 in (2.403), (2.407) by defining K as H2(Ω) =: K ⊕Π1. So thequotient space H2(Ω)/Π1 is the space of all classes u, such that u, v ∈ u iff u− v ∈ Π1.Each u ∈ u has a unique decomposition u = uK + p1 with uk ∈ K, p1 ∈ Π1 and uK

uniquely determines u.A (u0, w0) ∈W ×H2

0 (Ω), cf. (2.410), is a weak solution of Problem I, (2.403),(2.406), if it satisfies (2.411) ∀ϕ ∈W , and (2.412) ∀ψ ∈ H2

0 (Ω), and (2.406). Similarly,(u0, w0) ∈ H2(Ω)×H2

0 (Ω) weakly solves Problem II, if it satisfies (2.411) ∀v ∈ H2(Ω),and (2.412) ∀ψ ∈ H2

0 (Ω), and (2.406). If we consider (2.331), we have to replace (2.411)by (2.413).

Next, we present regularity results for the solutions of (2.331), (2.403). We startwith the result in [422], for (2.403).

Theorem 2.109. Regular Karman solutions:

1. For f ∈ Lp(Ω), 1 < p <∞ let (mind [f ] �= |f |)

[f ] := ‖f‖Lp(Ω)

(1 + ‖f‖2Lp(Ω)

).

Let (u0, w0) be a solution of Problem I or II. Then

u0 ∈W 4,p(Ω),Mu0 = 0 pointwise on ∂Ω, w0 ∈W 6,p(Ω) ∩H20 (Ω). (2.414)


2. If (u0, w0) solves Problem I, then additionally u0 ∈W 4,p(Ω) ∩W . For ProblemII we get Tu0 = 0 a.e. on ∂Ω in the trace sense. Moreover, for

1 < p < 2 : ‖u0‖W 4,p(Ω) ≤ C[f ], ‖w0‖W 6,p(Ω) ≤ C[f ]2

and for

2 ≤ p <∞ : ‖u0‖W 4,p(Ω) ≤ C(‖f‖Lp(Ω) + [f ]3), ‖w0‖W 6,p(Ω)

≤ C(‖f‖Lp(Ω) + [f ]3)2.

Finally, we combine Theorem 2.91 with Corollaries 3.1 and 3.2 in [422] to get

Corollary 2.110. Enforce the conditions of Theorem 2.109 by requiring f ∈W k,p(Ω).Then the solutions are u0 ∈W 4+k,p(Ω), w0 ∈W 6+k,p(Ω). If even f ∈ C∞(Ω) then weeven get u0, w0 ∈ C∞(Ω). In all these cases, Tu0 = 0 pointwise on ∂Ω for Problem II.These results remain correct for (2.331).

2.6.7 Nemyckii operators and quasilinear divergent systemsof order 2m

Generalized solutions of quasilinear divergent systems of order 2m

Zeidler [678] does not study quasilinear divergent systems. But his results for quasi-linear equations, cf. Zeidler [678], pp. 500 ff., 554 ff., Theorem 25B, Proposition 26.6and pp. 572 ff., are easily generalized to the present case. So we update the necessarydefinitions and results from Subsections 2.5.5 and 2.5.6.

For the Banach space X we use here the product space X = Wm,p(Ω,Rq) =(Wm,p(Ω))q. For 1 < p <∞ this X is reflexive and for

X = Wm,p(Ω,Rq) with 1 < p <∞ let 〈A�u−A�v, �u− �v〉 = 〈A�u−A�v, �u− �v〉X ′×X .

Then equations (2.242)–(2.253) remain unchanged, and we have called

A monotone ⇔ 〈A�u−A�v, �u− �v〉 ≥ 0, (2.415)

A strongly monotone ⇔ 〈A�u−A�v, �u− �v〉 ≥ c‖�u− �v‖2X , c ∈ R+, (2.416)

A stable ⇔ ‖A�u−A�v‖X ′ ≥ a(‖�u− �v‖X ), with (2.417)

a : R+,0 → R+,0 continuous and strictly increasing, cf. (2.418), with

a(0) = 0, limt→∞

a(t) =∞, e.g. a(t) = c|t|p−1, 0 < c, 1 < p. (2.418)

Then Proposition 2.67–Theorem 2.70 remain valid. We only reformulate the mostimportant parts of Theorem 2.68:

Theorem 2.111. Properties of nonlinear operators: Let X be a real Banach spaceand A : X → X ′ a (nonlinear) operator. Then


1. If A is monotone, coercive, and hemicontinuous on a reflexive Banach space, thenA is surjective and, for each f ∈ X ′, the set of solutions of Au = f is bounded,convex and closed, cf. [678], Theorems 25B and 26A(a).

2. If A is additionally strictly monotone, the above solution is unique. Then A−1 :X ′ → X exists and, cf. [678], Theorem 26A(c), (d),

a uniformly monotone A implies a continuous A−1,a strongly monotone A implies a Lipschitz–continuous A−1 ∈ CL.

We have to update the definition of Nemyckii operators, cf. Definition 2.69:

Definition 2.112. For given functions �f and appropriate �u ∈ D(�F ), define �F (�u) as

�f = (fi)qi=1 : D(�f) ⊂ Ω× RNm×q → Rq, �u : Ω→ Rq : �F (�u)(x) := �f(x, ∂≤m�u(x)),

(2.419)

cf. (2.423), with (x, ∂≤m�u(x)) ∈ D(�f). This �F is called a Nemyckii operator.

Now we formulate two conditions guaranteeing that �F (�u)(x) makes sense. TheCaratheodory and the growth condition enforce measurability and appropriate growthof the composite function, �F . More precisely, we assume


�f : (x, �ϑ≤m) ∈ D(�f ⊂ Ω× RNm×q → Rq,∀ |α| ≤ m, cf. (2.423),

�ϑ≤m �→ �f(x, �ϑ≤m) is continuous in RNm×q a.e. for x ∈ Ω, (2.420)

x �→ �f(x, �ϑ≤m) is measurable in Ω ∀ �ϑ≤m ∈ RNm×q, e.g. for continuous �f,

and with 1 < p <∞, 1/p + 1/p′ = 1, p− 1 = p/p′ and ∃ a ∈ Lp′(Ω), 0 ≤ a(·) a.e.

∃ C ∈ R+, s.t. ∀ x ∈ Ω, �ϑ≤m ∈ RNm×q, �ϑγ ∈ Rq :

Growth condition: |�f(x, �ϑ≤m)| ≤ C(a(x) +∑

|γ|≤m

|�ϑγ |p−1) ∀|α| ≤ m. (2.421)

Theorem 2.113. Estimates for nonlinear operators: If (2.420), (2.421) are satisfied,then �F : Wm,p(Ω,Rq) → Lp′

(Ω,Rq) in (2.419) is continuous and bounded (this is notequivalent for nonlinear operators!) such that

‖�F (�u)‖(Lp′ (Ω))q ≤ C

(‖a‖Lp′ (Ω) +

k∑i=1

‖∂γ�u‖p−1(Lp(Ω))q

)∀�u ∈Wm,p(Ω,Rq). (2.422)

The following general quasilinear systems are studied by Koshelev and Chelkak[450]. They consider Sobolev spaces Wm,p(Ω,Rq) ⊂ L2(Ω,Rq) ⊂W−m,p′

(Ω,Rq),


1/p + 1/p′ = 1. We extend their existence results to the case 1 < p <∞, by gene-ralizing our results in Subsections 2.5.5 and 2.5.6, cf. Zeidler [678].

We recapitulate the notation for partials and vectors so, cf. (2.367), (2.389), (2.73),(2.79), (2.102), (2.336), (2.335):

�u ∈Wm,s(Ω,Rq), �u(x), ∂β�u(x), �ϑβ =(ϑβ1

1 , · · · , ϑβqq

), ∈ Rq (2.423)

∂≤m�u(x) = (∂β�u(x))0≤|β|≤m, �ϑ≤m = (�ϑβ)0≤|β|≤m,∈ RNm×q.

We discuss the general form of quasilinear elliptic equations on bounded domainsΩ, compare Theorem 2.50, with Dirichlet boundary conditions, see Skrypnik [590,591]and Zeidler [678], and recall the standard notation in (2.165), (2.166), (2.269), (2.16),(2.361). The strong quasilinear system is

G�u0 = Gs�u0 :=

∑|α|≤m

(−1)|α|∂α �Aα(·, ∂≤m�u0) =

⎡⎣ ∑|α|≤m

(−1)|α|∂αAiα(·, ∂≤m�u0)

⎤⎦q

i=1

= 0, �u0 ∈W 2m,p(Ω,Rq) ∩Wm,p0 (Ω,Rq), �Aα =

[Ai

α

]qi=1

: Ω× RNm×q → Rq.

(2.424)

Here, we consider the weak solution �u0 ∈Wm,p0 (Ω,Rq),

a( �u0, �v) := 〈G�u0, �v〉 =∫

Ω

∑|α|≤m

( �Aα(·, ∂≤m�u0), ∂α�v)qdx = 0 ∀�v ∈Wm,p0 (Ω,Rq).

(2.425)

These integrals are well defined under the following sharper Caratheodory and growthconditions, compare (2.420), (2.384), (2.385).

The �Aα(x, ∂≤m�u) are Nemyckii operators, generalizing Subsection 2.5.5. Now weimpose conditions which guarantee that the G(u) in (2.424) and its weak form (2.425)are well defined. The Caratheodory and growth conditions enforce measurability andappropriate growth of the composite function �Aα, cf. Theorems 2.73, 2.113, and [678],pp. 571 ff. (2.426). The following nonlinear coercivity, monotonicity and growthconditions for these quasilinear operators generalize (2.230)–(2.235), (2.242)–(2.247).They are, with the equivalence of all norms, |�ϑγ | in Rq, essential for the existence ofsolutions.


∀ |α| ≤ m : �Aα : (x, �ϑ≤m) ∈ Ω× RNm×q → Rq,

�ϑ≤m �→ �Aα(x, �ϑ≤m) is continuous in RNm×q a.e. for x ∈ Ω, (2.426)

x �→ �Aα(x, �ϑ≤m) is measurable in Ω ∀ �ϑ≤m ∈ RNm×q, e.g. for continuous �Aα.


For the following conditions, we assume 1 < p <∞, 1/p + 1/p′ = 1, the special form�ϑ≤m = (�ϑγ)|γ|≤m, �ϑ≤m

1 = (�ϑγ1)|γ|≤m ∈ RNm×q and ∃ c, d, C ∈ R+, such that

∃ a ∈ Lp′(Ω), h ∈ L1(Ω), s.t. a.e. ∀ x ∈ Ω, �ϑ≤m, �ϑ≤m

1 ∈ RNm×q, �ϑ0, �ϑα, �ϑα1 ∈ Rq :

Growth condition: | �Aα(x, �ϑ≤m)| ≤ C(a(x) +∑

|γ|≤m

|�ϑγ |p−1) ∀|α| ≤ m. (2.427)

(Nonlinear) coercivity:∑

|α|≤m

( �Aα(x, �ϑ≤m), �ϑα)q ≥ c∑

|γ|=m

|�ϑγ |p − h(x). (2.428)

Monotonicity:∑

|α|≤m

(�Aα(x, �ϑ≤m)− �Aα(x, �ϑ≤m

1 ), �ϑα − �ϑα1

)q≥ 0. (2.429)

Strict monotonicity:∑

|γ|≤m

∣∣∣�ϑγ − �ϑγ1

∣∣∣ > 0 implies (2.430)

∑|α|≤m

(�Aα(x, �ϑ≤m)− �Aα(x, �ϑ≤m

1 ), �ϑα − �ϑα1

)q> 0.

Uniform monotonicity: (2.431)∑|α|≤m

(�Aα(x, �ϑ≤m)− �Aα(x, �ϑ≤m

1 ), �ϑα − �ϑα1

)q≥ C

∑|γ|=m

∣∣∣�ϑγ − �ϑγ1

∣∣∣p .For regularity results, p = 2 and additional conditions have to be imposed, cf.

Theorem 2.116.By [678], Proposition 26.12, the conditions (2.426)–(2.429) imply the conditions of

his Theorem 26A. So we refer in the next theorem to the corresponding results forequations in [678].

Theorem 2.114. Generalized solutions for (2.425): Let (2.426)–(2.427) be satisfiedand choose V = Wm,p

0 (Ω,Rq). Let

�Fα : V = Wm,p0 (Ω,Rq) → Lp′

(Ω,Rq), (�Fα�u)(x) := �Aα(x, ∂≤m�u(x)).

1. Then for all |α| ≤ m, a.e. ∀x ∈ Ω these Nemyckii operators �Fα are well defined,continuous and bounded, see Theorem 2.113 and [678], Proposition 26.12, proof.

2. With the above �Aα(x, ∂≤m�u(x)) define the a(�u,�v) as in (2.425). Then there existsexactly one bounded operator G, cf. Proposition 2.67, [678], Proposition 26.12,

G : V → V ′ : a(�u,�v) = 〈G�u,�v〉V′×V ∀�u,�v ∈ V = Wm,p0 (Ω,Rq) and (2.432)

‖Fα�u‖Lp′ (Ω,Rq), ‖G�u‖W−m,p′ (Ω,Rq) ≤ C

(‖a‖Lp′ (Ω,Rq) +

(‖�u‖W m,p

0 (Ω,Rq)

)p/p′).


Similarly to (2.425) and with 〈�f,�v〉V′×V replacing 0, the original problem (2.424)corresponds to the generalized problem: Determine for a(�u,�v) in (2.432)

�u0 ∈ V = Wm,p0 (Ω,Rq) s.t. a(�u0, �v) = 〈�f,�v〉V′×V∀�v ∈ V ⇐⇒ G�u0 = �f.

(2.433)

3. Additionally let all the �Aα be coercive and monotone, see (2.428) and (2.429).Then G is monotone, coercive, continuous and bounded, see [678], Proposition26.12. So by Remark 2.66, Theorem 2.68 2.–4. are valid for (2.425).

4. In particular, for any �f ∈W−m,p′(Ω,Rq) = V ′, (2.433) has a solution �u0 ∈

Wm,p0 (Ω,Rq) and the set of solutions is bounded, convex and closed. If all the �Aα

are additionally strictly monotone, see (2.430), Theorem 2.68, then G is strictlymonotone as well and the above solution is unique. So the inverse G−1 : V ′ → Vexists and is strictly monotone and bounded, see [678], Theorem 26A. If all the�Aα are, and hence G is uniformly monotone, see (2.431), this G−1 is continuous.

5. If G is strongly monotone, cf (2.416), Theorem 2.111, then G−1 is Lipschitz–continuous.

Koshelev–Chelkak existence and regularity results forquasilinear systems

As mentioned above, these problems are discussed by Koshelev and Chelkak [450].Here the Sobolev spaces Wm,s(Ω,Rq) ⊂ L2(Ω,Rq) ⊂W−m,s′

(Ω,Rq), 1/s + 1/s′ = 1,in (2.388) have to be combined with the scalar products and norms in (2.337)and Gs�u in divergence form. We introduce, similarly to the partials and vectors�ϑβ , �u(x), ∂β�u(x) ∈ Rq in (2.423), (2.367), (2.389), the notation for the components �pβ :

�p := (p1, · · · , pq), �pβ =(pβ11 , · · · , pβq

q

)∈ Rq, �p ≤m := (pβ)0≤|β|≤m ∈ RNm×q.

(2.434)

We consider and determine the solution, �u0 ∈W 2m,s(Ω,Rq) ∩Wm,s0 (Ω,Rq), of the

strong quasilinear system (2.424), see (2.16), (2.361). Koshelev and Chelkak [450]restrict (2.424) essentially to the case 1 < s ≤ 2. In their introduction they intensivelydiscuss other possibilities and aspects of the problem (2.424) and give references.

The weak solution �u0 ∈Wm,s0 (Ω,Rq) is determined from (2.425). The integrals

in (2.425) are well defined under the following modified Caratheodory and growthconditions, compare (2.426)–(2.431), and (2.284), (2.384), (2.385). We impose a strongLegendre condition for the linearized operator, compare (2.384). In fact, compared to(2.378),

∑|α|,|β|=m is replaced by

∑|α|,|β|≤m in (2.437). This appears again in the

growth condition (2.437): For w = x, �p ≤m and with |�p ≤m|2 =∑

|β|≤m

∑qi=1 |p

βi

i |2and λ,Λ = const. > 0, let, for almost all x ∈ Ω,

Aiα(x, �p ≤m) are measurable and continuouslydifferentiable with respect to �p ≤m

∀|α| ≤ m in any domain of boundedvariation of �p ≤m = ∂≤m�u of �u, (2.435)


λ(1 + |�p ≤m|2)(s−2)/2|�ϑ≤m|2 (2.436)

≤q∑

i,j=1

∑|α|,|β|≤m

(∂Ai

α/∂pjβ

)(w)ϑα

i ϑβj ≤ Λ(1 + |�p ≤m|2)(s−2)/2|�ϑ≤m|2,

for 1 < s ≤ 2,

|(∂Ai

α/∂pβj

j

)(w)| ≤ Λ(1 + |�p ≤m|2)(s−1)/2 ∀ i, j, α, β, and for any 1 < q′, (2.437)

domain Ω′ ⊂ Ω and �u ∈W 1,q′(Ω′,Rq) : �Aα(·, ∂≤m�u(x)) ∈ Lq′/(s−1)(Ω′). (2.438)

Koshelev and Chelkak call them systems with bounded nonlinearities. Similarly to ourLemma 2.26, see (2.82), generalizations of Dirichlet boundary conditions to Dirichletsystems are possible. To avoid the many technicalities we do not present the iterationprocess, which plays an important role in [450], for determining the solution. Werestrict the discussion to Dirichlet boundary conditions

∂j�u(x)∂νj

= gj ∀0 ≤ j ≤ m− 1 ⇔ ∂α�u(x) = gα ∀|α| ≤ m− 1 ∀x ∈ ∂Ω. (2.439)

Theorem 2.115. Existence – Theorems 2.1.1 and 2.3.1 in [450]:

1. Assume s = 2, and the above modified Caratheodory and growth conditions(2.435)–(2.438), implying |(∂Ai

α/∂pβj

j )(x, �p ≤m)| ≤ C. Then the quasilinear sys-tem (2.425) with the Dirichlet boundary conditions (2.439) admits a solution�u0 ∈ Hm(Ω,Rq).

2. If, more generally, for 1 < s ≤ 2, a weak solution �u0 for (2.425), (2.439) existsand satisfies the estimate∫

Ω

∑|α|≤m

|Aα(x, ∂≤m�u0(x))|s/(s−1)dx ≤ C1

∥∥�u0∥∥s

W m,s(Ω,Rq)+ C2, (2.440)

then this �u0 is the unique solution of the problem.

To get regularity results, again s = 2 and additional conditions have to be imposed.Whenever conditions for partials, e.g. ∂ �Aα/∂xk are required, the existence of thesepartials is silently assumed. For Ω′ ⊂⊂ Ω we introduce

‖�u‖2Hmδ (Ω′,Rq) := sup

x0∈Ω′

∫Ω′

⎛⎝ ∑|α|<m

|∂α�u(x)|2 +∑

|α|=m

|∂α�u(x)|2|x− x0|δ⎞⎠dx (2.441)

Hmδ (Ω′,Rq) := {�u ∈ Hm(Ω′,Rq) : ‖�u‖Hm

δ (Ω′,Rq) <∞}, L2δ(Ω

′,Rq) := H0δ (Ω′,Rq).

We impose three sets of conditions for the three cases: (2.443) for �u ∈ Hm(Ω′,Rq),(2.444) for �u ∈ Hm

δ (Ω′,Rq), and (2.445) for �u ∈Wm+1,q′(Ω′,Rq) :

for all three cases let Ω′ ⊂⊂ Ω : ∃ C(R) independent of �u, such that (2.442)


∀�u ∈ Hm(Ω′,Rq) with ‖�u‖Hm(Ω′,Rq) ≤ R, (2.443)

∀ 0 ≤ |α| ≤ m : ‖ �Aα(·, ∂≤m�u(·))‖L2(Ω′,Rq) ≤ C(R), and

∀|α| = m, k = 1, . . . , n, let

∥∥∥∥∥∂ �Aα

∂xk(·, ∂≤m�u(·))

∥∥∥∥∥L2(Ω′,Rq)

≤ C(R) or

∀ �u ∈ Hmδ (Ω′,Rq) with ‖�u‖Hm

δ (Ω′,Rq) ≤ R, δ ∈ [2−m− 2γ, 0], (2.444)

0 < γ < 1, ∀ 0 ≤ |α| ≤ m let ‖ �Aα(·, ∂≤m�u(·))‖L2δ(Ω′,Rq) ≤ C(R) and

∀|α| = m, k = 1, . . . , n, let

∥∥∥∥∥∂ �Aα

∂xk(·, ∂≤m�u(·))

∥∥∥∥∥L2

δ(Ω′,Rq)

≤ C(R), finally

∀�u ∈Wm+1,q′(Ω′,Rq),m < q′,

∂ �Aα

∂xk(·,∇≤m�u(·)) ∈ Lq′

(Ω′,Rq). (2.445)

Theorem 2.116. Regularity, Theorem 4.3.1 in [450]: Assume the conditions (2.442)–(2.445), those in Theorem 2.115 (note s = 2), except (2.440), and some highly tech-nical conditions, see (1.3.6), (1.3.7), (1.3.18), (1.3.20), (1.3.46), (4.1.11), (4.3.3)in [450]. Let the boundary values in (2.439) be defined by one function g as

g ∈ Hm(Ω,Rq) : gj =∂jg(x)∂νj

∀0 ≤ j ≤ m− 1.

Then the weak solution �u0, for (2.425), (2.439) belongs to �u0 ∈ Cm,γ(Ω′,Rq).

The techniques for proving higher regularity as in Theorem 2.91 3. do not apply.

2.6.8 Fully nonlinear elliptic systems of orders 2 and 2m

In contrast to the earlier subsections, only few results for fully nonlinear systems havefound their way into the monographs consulted here. Formally, they are obtained bythe obvious generalizations of (2.174), (2.306), (2.319) as c.f. Remark around (2.306),

G(·) : D(G) ⊂ U := H10 (Ω,Rq) ∩H2(Ω,Rq) → V := L2(Ω,Rq), with

G(�u(·)) := Gw(·, �u(·), ∂�u(·),∇2�u(·)), G(�u0) = 0 (2.446)

for systems of order 2. The corresponding equations are updated by replacing theoriginal U ,V,W . For systems of order 2m we obtain

G(·) : D(G) ⊂ U := Hm0 (Ω,Rq) ∩H2m(Ω,Rq) → V := L2(Ω,Rq), with

G(�u) := Gw(·, �u(·), . . . ,∇2m�u(·)), G(�u0) = Gw(·, ∂γ�u0(·), |γ| ≤ 2m) = 0. (2.447)

Often H2m(Ω,Rq) is replaced by C2m,γ(Ω,Rq) or C∞(Ω,Rq), disregarding boundaryconditions. In Taylor [620], pp. 108 ff., some “local existence results” are proved: if ata point x0 ∈ Ω the equation Gw(·, ∂γ�u1(·), |γ| ≤ 2m)(x0) = 0 is satisfied for a �u1, then

2.7. Linearization of nonlinear operators 147

there exists a solution �u in a neighborhood of x0. For second order some results canbe found, e.g. in [79,80,311,475,671].

One of the few exceptional results is proved by Taylor [620], Chapter 14, Theorem3.1, for nonlinear systems of PDEs of order m, see Definitions 2.51 and 2.85. It extendsverbatim (2.319) and Theorem 2.84 to systems, by simply replacing u by �u. Nonlinearsystems of PDEs of order m have the form

G(�u) := G(x, �u,∇�u, . . . ,∇m�u) and G(�u)(x) = g(x) ∀ x ∈ Ω. (2.448)

Theorem 2.117. Local solvability result for nonlinear elliptic systems: Let g ∈C∞(Ω), and let �u1 ∈ C∞(Ω) satisfy (2.448) at x = x0, hence

G(�u1)(x) = g(x) for x = x0

and let G be m-elliptic at �u1. Then for any � there exists a �u ∈ C�(Ω) such that

G(�u)(x) = g(x) in a neighborhood of x0.

In this situation with only few known results, a combination of the linearizedoperators in Subsection 2.7.3, our FE or other discretization methods below andcontinuation techniques, allows us to study existence of solutions of these problems.So we give some details in Subsections 2.7.3 and 2.7.4 below.

2.7 Linearization of nonlinear operators

2.7.1 Introduction

Linearization of operators including systems plays an essential role in our approach.We are mainly interested in numerical methods for these PDEs and their bifurcationscenarios. The stability of numerical methods can be proved via the stability ofthe method for the linearized operator. For the numerical approximation of theirbifurcation scenarios we do need the Fredholm alternative, again formulated for thelinearized operator. For this reason we have not included the many Fredholm typeresults, directly available for nonlinear differential equations, see, e.g. Zeidler [676–678].

In many contexts, nonlinear parameter-dependent operators G(u, λ) are studied. Itis important to start with a parameter value λ0 such that the existence and (local)uniqueness of a solution u0 of G(u0, λ0) = 0 is guaranteed. The results in Sections 2.4–2.6 yield this information for solutions u0 for a fixed parameter λ0. Based upon thisknowledge, the Implicit Function Theorem 1.46 defines a uniquely determined solutioncurve of the nonlinear problem G(u, λ) = 0 through (u0, λ0). Note that, e.g. for anoperator G : V × Rp → V ′ the derivative G′(u, λ) = (Gu(u, λ), Gλ(u, λ)) always mapsV × Rp → V ′, whenever it exists as a bounded operator. The corresponding continua-tion process again is based upon the Frechet derivative G′(u, λ) = (Gu(u, λ), Gλ(u, λ))at a solution point (u, λ). The continuation breaks down and the solution curveusually bifurcates, whenever G′(u, λ) is not surjective in a solution point (u, λ), cf.Bohmer [120]. Then usually Gu(u, λ) has a nontrivial kernel. So results from theFredholm alternative for the linearized operator Gu(u, λ) apply in combination withthe Liapunov–Schmidt methods studied in [120]. For both aspects we have to verify


that for the different types of nonlinear operators G(u), studied in the last Sections2.5, 2.6, the linearized operators G′(u0) or the Gu(u0, λ1) are well defined and thatFredholm alternatives are available for them. By appropriate combinations of the cor-responding Banach spaces, linearizations for a wide class of nonlinear elliptic equationsand systems make sense. In this monograph we use discretization concepts differentfrom the Brezzi–Rappaz–Raviart [147–149] ideas. So, most of Santos’s [566] negativearguments concerning numerical methods as in [147–149], related to linearization, donot apply.

Prima vista, linearization does not seem to be appropriate for some of the quasilinearelliptic equations and systems. In fact the results for quasilinear equations fromSubsection 2.5.6 strongly depend upon monotony and nonlinear coercivity arguments.But we show in Subsection 2.7.3 that the linearization approach works for mostquasilinear equations. Noncoercive equations, e.g. of the form

Au0 + Nu0 = b, with A ∈ L(U ,U ′) linear, N nonlinear lim‖u‖→∞

‖Nu‖/‖u‖ = 0,

require specific handling. However, some type of Fredholm alternative still holds. Thisequation has a solution u0∀b ∈ U ′ if Au = 0 has the unique solution u = 0, otherwisesolutions u0 exist only for special b ∈ U ′, see Zeidler [678], Chapter 29.

The situation for systems is different, since for orders 2 and 2m the ellipticity of(the linearized) operator plays a dominant role compared to analogous monotonyarguments. The situation changes if viscosity solutions are admitted instead, as studiedin an impressive series of papers by Crandall et al. [215, 217–228], which we did notconsider here.

Above we have documented mainly two sets of results for nonlinear equations, eitherbased upon (linear) coercivity arguments and the corresponding Hm(Ω) or Wm,p(Ω)setting, combined with Schauder results for second order equations. Or they are basedupon monotony arguments in the Wm,p(Ω) setting. In both cases the linearizationusually yields operators as discussed in Section 2.4 and Subsections 2.6.3, 2.6.5, andsatisfies the Fredholm alternative. An essential fact here is the strong ellipticity orthe Legendre condition of the linearizations and, for systems of order 2m,m > 1, thestrong Legendre–Hadamard condition.

In this section we will show that the linearization of all nonlinear problems studiedbefore in the Hm(Ω) or Wm,p(Ω) setting, 2 < p ≤ ∞, yields bounded linear operators,with the corresponding bounded bilinear forms a(u, v). They are compact pertur-bations of their principal parts. For 2 < p ≤ ∞, we use the compact embedding ofthe Wm,p(Ω) into Hm(Ω). Then the principal parts are Hm(Ω)-coercive, and theconvergence for numerical methods is measured with respect to a discrete Hm(Ω)norm, cf. Corollary 2.44. For 1 ≤ p < 2 corresponding bilinear forms a(u, v) arenot even defined. So we have to exclude this case. Finally, for the fully nonlinearsystems in Subsection 2.6.8 no existence results are available. However, linearizationsmay be combined with the Fredholm alternative and discretization and continuationtechniques to obtain existence results.

In Section 4.5 we will apply the general discretization theory from Chapter 3 toall our general quasilinear problems. This is formulated for FEMs, but is valid for all


other methods as well. It is based upon monotony and related properties and allowsthe previously excluded cases 1 < p <∞.

For all the following cases we do not repeat, but only refer to, the conditions forthe defining coefficients and the domain imposed for the original nonlinear problem.We formulate the stronger conditions where they are necessary for the linearization.To simplify the notation we always omit parameters λ and use the abbreviations

∂G

∂u(u) :=

∂G

∂u(u, λ),

∂f

∂u(u) :=

∂f

∂u(u, λ), etc., often for (u, λ) = (u0, λ0). (2.449)

We summarize the main result of this section as

Theorem 2.118. For the different types of nonlinear equations and systems G(u),studied in Sections 2.5, 2.6, the corresponding linearized operators are well definedand bounded under the conditions in the following theorems, throughout assumingexisting derivatives. Then Corollary 2.44 implies for 2 ≤ p ≤ ∞ and the Hm-coerciveessentially principal parts, the existence and uniqueness of solutions and a boundedlyinvertible A, for a Hm

0 -elliptic a(·, ·), the Fredholm alternative and, in Chapter 3,the numerical stability, for the discertization methods studied in our books, includingbifurcation and the related dynamics with respect to discrete Hm norms. For naturallimitations see the Introduction in Subsection 2.7.3.

2.7.2 Special semilinear and quasilinear equations

In contrast to the general quasilinear differential equations in Subsection 2.5.6 the lin-earization of the equations of the special cases of semilinear and quasilinear differentialequations in Subsection 2.5.3 is straightforward.

A: Special semilinear operators of order 2m

For the equations of order 2m in (2.177), (2.191) with boundary conditions (2.178)and the conditions in (2.179) and (2.190) for f we get the strong form (thelinear boundary operators are unchanged and satisfy the same complementarycondition),

Gu = A− f(u),∂G

∂u(u0)v := Av − ∂f

∂u(u0)v : C2m,γ(Ω) → Cγ(Ω),

Av =∑

|α|≤2m

aα∂αu, unif. elliptic of index 0, fu =

∂f

∂u∈ C∞(R2). (2.450)

For23 order 2 we only have to replace 2m by 2 in (2.450), reduce the smoothness offu, and choose Au, Bu and the conditions by the special form with

Av :=n∑

i,j=0

aij(x)∂i ∂j u, with Bu as in (2.185), and fu ∈ Cγ(R2). (2.451)

23 According to (2.449), fu ∈ C∞(R2) often means, here and below, fu(u, λ) ∈ C∞(R3).


These are exactly the linear operators in Theorems 2.37 ff. with (2.83), (2.88), (2.93),(2.117), (2.181). They are studied in Subsections 2.4.2 and 2.4.3. In Theorems 2.32,2.41, 2.52, and Proposition 2.54, the corresponding results and Fredholm alternativesare formulated.

B: Special quasilinear operators of order 2

For the linearization of the special quasilinear operators in (2.193) and (2.452) wecombine (2.194) and (2.452a), (2.453) of order 2m = 2 and n = 2. We recall thenotation in (2.269), ∂v/∂ϑk, k ≥ 0 with = v for k = 0, and ∂aij/∂ϑ

0 = ∂aij/∂u. Theoriginal form and its linearization are

n = 2 : G(u) =2∑

i,j=1

aij(x, u,∇u)∂i∂j u,∂G

∂u(u0) : C2,γ(Ω) → Cγ(Ω), (2.452)

∂G

∂u(u0)v =

2∑i,j=1

(aij(x, u0,∇u0)∂i∂j v +

2∑k=0

∂aij

∂ϑk(x, u0,∇u0)∂i∂j u0∂

kv), (2.452a)

see Theorem 2.57, but with stronger condition for coefficients, than (2.194),

aij ,∂aij

∂ϑk∈ Cγ(Ω× R3),∀i, j = 1, 2, k = 0, 1, 2. (2.453)

C: Special semilinear operators in nonlinear spaces

For the last case in Subsection 2.5.3, (2.197), (2.198), we require (2.196), (2.199)–(2.202) and (2.455). Then we consider

G(u) = Au− f(u)− g =∑

|α|,|β|≤m

(−1)|β|∂β(aαβ∂αu)− f(u)− g (2.454)

in (2.196) and (2.199) in nonlinear spaces. The linearized strong form is now defined,with appropriate fu(u0), as (∂G/∂u)(u0) : H2m(Ω) → L2(Ω). Applied to v we get

∂G

∂u(u0)v =

(A− ∂f

∂u(u0)

)v, with fu(u0) ∈ L∞(Ω). (2.455)

The conditions for the weak form are listed in (2.201), (2.202) and in Theorem 2.58.

D: Semilinear nonautonomous operators

For this (2.296), (2.297) we replace the previous f(u) + g(x) by f(x, u) and obtain,with A as in (2.454), the linearized strong form

G,A,∂G

∂u(u0) : H2m(Ω) → L2(Ω), with fu(·, u0(·)) ∈ L∞(Ω) (2.456)

G(u) = Au− f(x, u),∂G

∂u(u0)v =

(A− ∂f

∂u(·, u0(·))

)v, fu ∈ L∞(R2),

with Dirichlet boundary conditions. The conditions for the weak form are listed in(2.201), (2.202), (2.298)–(2.300) and Theorem 2.76. f(x, u) has to be monotone in


u ∀x ∈ Ω for existence and uniqueness (or, only for existence, f(x, u)u ≥ 0 ∀x ∈ Ω, u ∈R for large enough |u|) to get the linearization in (2.456).

We summarize the conditions for linearization of the special cases of semilinear andquasilinear differential equations in Subsection 2.5.3 and the last example (2.297) inSubsection 2.5.6.

Theorem 2.119. Linearization for different problems: We consider the special semi-and quasilinear equations in the above cases A, B, C, D and the necessary conditionsfor their elliptic and bounded linearizations listed above. Then the original problemsand their linearizations have the form

G(u) = Au− f(u) = g,∂G

∂u(u0)v = Av − ∂f

∂u(u0)v, for A, C, and (2.457)

G(u) = Au− f(·, u) = g,∂G

∂u(u0)v = Av − ∂f

∂u(·, u0)v for D, and (2.452) for B.

Under the listed conditions all the linearizations in (2.457) are bounded linear opera-tors, induced by compact perturbations of Hm

0 -coercive bilinear forms, hence Corollary2.44 holds, e.g. implying the Fredholm alternative.

2.7.3 Divergent quasilinear and fully nonlinear equations

We begin with the general divergent quasilinear equations studied in Subsections2.5.4 and 2.5.6, see, e.g. (2.206)–(2.208), (2.228)–(2.229), (2.270)–(2.275). Here thelinearization for these equations makes sense if we exclude some cases for the followingreasons: Since many of the existence, uniqueness and regularity results are only knownfor the case Hm(Ω), generalizations to the Wm,p(Ω,Rq) situation would be worthwhile.But for the appropriate Wm,p(Ω) setting and for 1 ≤ p < 2, the bilinear forms, a(u, v),for the linearized operator, G′(u0), usually are not even defined, cf. (2.459). So weexclude 1 ≤ p < 2. For 2 ≤ p ≤ ∞, and a satisfied ellipticity condition (2.461), thebilinear form, ap(u, v), for the principal part of the linearized operators turns out tobe only Hm

0 -coercive. The compact embedding of Wm,p(Ω) into Hm(Ω) allows usto formulate numerical results for a discrete Hm(Ω) norm, cf. Corollary 2.44. But,the above (linear) Hm

0 -coercivity arguments and the corresponding C2,γ(Ω) for aSchauder type approach do not always apply to m > 1. The original conditions forthese quasilinear elliptic equations are related, but not identical with those garanteeingthe Hm

0 -coercivity of their linearized principal part, see Lemma 2.77 and the followingdiscussion. So, for some cases, based upon monotony arguments, we impose modifiedconditions, cf. Proposition 2.121 and Theorem 2.122. We discuss the relations betweenthese conditions.

We had assumed Ω as a bounded nonempty measurable domain in Rn. The weakform, see (2.274), for general m is defined for u, v ∈ V = Wm,p

0 (Ω), as

a(u, v) = 〈Gu, v〉V′×V =∫

Ω

∑|α|≤m

Aα(x, u, . . . ,∇mu)∂αvdx = 〈f, v〉V′×V (2.458)


with the exact solution u0. We linearize analogously to (2.455) and get

∂G

∂u(u0) : V := Wm,p

0 (Ω) → V ′ = W−m,q0 (Ω), 1 < p <∞, 1/p + 1/q = 1, with (2.459)

⟨∂G(u0)∂u

w, v

⟩V′×V

=∫

Ω

∑|α|≤m

⎛⎝ ∑|β|≤m

∂Aα

∂ϑβ(x, u0, . . . ,∇mu0)∂βw

⎞⎠ ∂αvdx, u0, v,

w ∈ V.Although the condition G, ∂G/∂u(u0) : V := Wm,p

0 (Ω) → V ′ = W−m,q0 (Ω), makes

sense, the previous 〈(∂G(u0)/∂u)w, v〉V′×V needs ∂Aα/∂ϑβ(x, u0, . . . ,∇mu0) ∂βw∂αv

∈ L1(Ω) ∀u0, v, w ∈Wm,p0 (Ω), requiring p ≥ 2.

The usual ellipticity condition, and sometimes the not quite usual ellipticity condi-tion (2.301) is imposed, with d ∈ RNm−1 , Θm ∈ Rnmλ(d),Λ(d), μ(Θm), defined there.This implies the usual ellipticity. Recalling (2.75), (2.79) with

|ϑ|2mn,α :=

∑|α|=m

ϑ2α and |ϑ|2mn /C ≤ |ϑ|2m

n,α ≤ |ϑ|2mn ≤ C |ϑ|2m

n,α we have (2.460)

λ(d)μ(Θm) |ϑ|2mn /C ≤ λ(d)μ(Θm) |ϑ|2m

n,α

≤∑

|α|=|β|=m

∂Aα

∂ϑβ(x, d,Θm)ϑβϑα ≤ Λ(d)μ(Θm) |ϑ|2m

n,α .

We have to linearize in the exact solution u0. The preceding regularity and bound-edness results show that the special values for the d = (ϑβ = ∂βu0, |β| < m), Θm =(ϑβ = ∂βu0, |β| = m), and hence the λ(d), μ(Θm),Λ(d) are bounded. Then (2.460)has the form of a standard ellipticity condition for the linearized operator, eval-uated in u0, formulated either with |ϑ|n or |ϑ|n,α, cf. (2.75) and the followingDefinition 2.25:

λ |ϑ|2mn /C ≤ λ |ϑ|2m

n,α ≤∑

|α|=|β|=m

∂Aα

∂ϑβ(x, u0(x), . . . ,∇mu0(x))ϑβϑα ≤ Λ|ϑ|2m

n,α.

(2.461)

For the above Example 2.72 the derivatives do exist. Similarly to the Monge–Ampereequation, the ellipticity of G′(u0) depends upon u0.

Example 2.120.

1. For n = 1, 2, 3, and u ∈ H2(Ω) ∩H10 (Ω) the linearization of

Gu = −Δu + α

n∑i=1

sin(u)∂iu = f ∈ L2(Ω) on Ω,


yields

G′(u0)v = −Δv + αn∑

i=1

⎛⎝ n∑j=1

cos(u0)∂iu0∂jv + sin(u0)∂iv

⎞⎠.

2. For 2 ≤ p <∞, 1/p + 1/q = 1, s ∈ R, s ≥ 0, we obtain

Gu = −n∑

i=1

∂i(|∂iu|p−2∂iu) + su = f ∈ Lq(Ω) on Ω,

G′(u0)v = −n∑

i=1

∂i(|∂iu0|p−2

(1 + (p− 2)(∂iu0)−1

)∂iv)

+lower order derivatives for v, with a singularity in ∂iu0(x) = 0.

3. The linearization of the minimal surface equation,

Gu = (1 + |∇u|2)Δu +n∑

i,j=1

∂iu∂ju∂i∂ju = 0 on Ω

yields

G′(u0)v = (1 + |∇u0|2)Δv +n∑

i,j=1

∂iu0∂ju0∂

i∂jv

+lower order derivatives for v.

�

Some of the results for the general divergent quasilinear equations of order 2 inSubsection 2.5.4 are based upon the ellipticity conditions in the form of (2.460) withm = 1, see (2.209), (2.239), for the linearized operators. In Theorems 2.61 and 2.62existence and uniqueness results are summarized under appropriate growth conditionslisted there. Theorems 2.63 and 2.64 yield existence and regularity results in theCm,γ(Ω) setting, combining appropriate growth and Hm

0 -coercivity conditions.General divergent quasilinear equations of order 2m are studied in Subsection

2.5.6, see, e.g. (2.270)–(2.275). Conditions (2.284)–(2.289) are chosen in differentcombinations. Comparing condition (2.461), see Lemma 2.77, its evaluation in (2.461)is related to a modified Hm

0 monotony (2.294) in Theorem 2.75. This is relevant forthe existence of a solution u0 for (2.295), but not for the finer results of Theorem 2.73based upon the nonlinear Hm

0 -coercivity in (2.286). Hence, if we impose the conditionsof Theorem 2.75 and Lemma 2.77, in particular (2.301), we obtain an Hm

0 (Ω)-coercivelinearization of the principal part. In fact we find

Proposition 2.121. The monotonicity conditions (2.287), (2.288) and (2.289) forquasilinear equations imply, for p = 2, Θ≤m = (d,Θ≤m), and Aα(x, d,Θm) ∈ C1(Rnm)


with respect to Θm, in a neighborhood of Θm, classical and strong and uniform Hm0 (Ω)-

ellipticity. This implies, for uniform ellipticity, an Hm0 (Ω)-coercive linearization of the

principal part.

Proof. We only consider (2.289); the other cases are similar. In (2.289) we choose theabove Θ≤m and Θ≤m

1 =(d,Θ≤m

1

), and obtain∑

|α|≤m

=∑

|α|=m

(Aα (x, d,Θm1 )−Aα(x, d,Θm)) (ϑα

1 − ϑα) ≥ C∑

|γ|=m

|ϑγ1 − ϑγ |p .

This implies, by Theorem 1.43 and for sufficiently small (ϑγ)|γ|=m := Θm1 −Θm,

∑|α|=|β|=m

(∂Aα(x, d,Θm)/∂β)ϑαϑβ ≥ C/2∑

|γ|=m

|ϑγ |p ≥ C2,pC/2

⎛⎝ ∑|γ|=m

|ϑγ |2⎞⎠p/2

,

since∑

|γ|=m |ϑγ |p ≥ C2,p

(∑|γ|=m |ϑγ |2

)p/2 for the equivalent norms in Rnm . Skippingthe middle term in this equation and dividing it by

(∑|γ|=m |ϑγ |2

)yields

∑|α|=|β|=m

(∂Aα(x, d,Θm)/∂β)ϑαϑβ/

⎛⎝ ∑|γ|=m

|ϑγ |2⎞⎠ ≥ C2,pC/2

⎛⎝ ∑|γ|=m

|ϑγ |2⎞⎠p/2−1

.

For p �= 2 the right-hand side in this inequality can become arbitrarily small or large.So (2.289) implies uniform ellipticity only for p = 2. �

We have excluded the case Wm,p0 (Ω), 1 ≤ p < 2. The continuous embedding of

Wm,p0 (Ω), 2 ≤ p ≤ ∞, into Hm

0 (Ω), allows, under slightly enforced conditions, aHm

0 (Ω)-coercive linearization and full convergence results with respect to discreteHm(Ω) norms later on for the different discretization methods.

For quasilinear equations of order 2m,m ≥ 1, in Subsection 2.5.6 the results referto the nonlinear Hm

0 -coercivity in (2.286) and the monotonicity conditions (2.287)–(2.289), but not to ellipticity. In particular, linearization via the standard (2.461) iscertainly not equivalent to (2.289). If the derivatives of the Aα are bounded andΘNm

1 −ΘNm is sufficiently small, then (2.461) implies with the Taylor formula, amodified (2.289) with

∑|γ|≤m |ϑγ − ϑγ

1 |p on the right-hand side, instead of the usual∑

|γ|=m |ϑγ − ϑγ1 |

2, cf. the proof of Proposition 2.121. In this sense, (2.461) implies

uniform monotonicity (2.294), and Hm0 -coercivity for the principal part of G′(u0).

With the standard techniques and Corollary 2.14 this shows that (2.59) is satisfied,hence G′ is boundedly invertible.

We have seen that for quasilinear problems of order 2, ellipticity conditions asin (2.461) are required anyway. For orders 2m,m ≥ 1, they are pretty close to thestandard conditions. We now prove that for 2 ≤ p ≤ ∞, the derivatives G′(u0) of thenonlinear G(u) in Wm,p

0 (Ω) are bounded and the principal parts are Hm0 (Ω)-coercive.

Theorem 2.122. Bounded linearization in Wm,p0 (Ω), 2 ≤ p ≤ ∞, Hm

0 (Ω)-coerciveprincipal part, Fredholm alternative: In addition to the conditions (2.284), (2.285),


for the original quasilinear operator in (2.290), we impose for its linearization (2.459)the Legendre condition (2.461). Either for the solution u0 ∈Wm,p(Ω) of (2.295) orclose to it,24 cf. (1.49),

let for 2 ≤ p ≤ ∞ : h :=∂Aα

∂ϑβ(·, u0, . . . ,∇mu0) ∈ L∞(Ω), or h ∈ Lp/(p−2)(Ω),

or let∥∥∥∥∂Aα

∂ϑβ(·, ϑβ′

, |β′| ≤ m)∥∥∥∥

L∞(Ω)

≤ C ∀ϑβ′ ∈ R, |ϑβ′ − ∂β′u0| ≤ ρ > 0. (2.462)

Then the linearization in (2.459) is bounded.We use the compact embedding of Wm,p(Ω), 2 ≤ p ≤ ∞, into Hm(Ω), and obtain

G′(u0) and its principal part as Hm(Ω)-elliptic and Hm0 (Ω)-coercive, respectively.

Remark 2.123.

1. Any types of estimates based upon linearization for u ∈W k,p(Ω), 2 ≤ p ≤ ∞,are therefore still possible, however with respect to the Hk(Ω) norm. This isparticularly important for discretization methods. Then stability, consistency, andconvergence are valid with respect to the discrete, e.g. Hk(Ωh) norm.

2. The results of Corollary 2.44 imply, e.g. existence and uniqueness of solutionsand a boundedly invertible A, for the Hm

0 -coercive and the Fredholm alternativefor the elliptic case. Regularity results as indicated in Subsections 2.5.4, 2.5.7,and 2.5.6, are valid under the corresponding conditions.

Proof. With ∂βw, ∂αv ∈ Lp(Ω), ∂Aα/∂ϑβ(·, u0, . . . ,∇mu0) ∈ L∞(Ω) or, for p ≥ 2,

in Lp/(p−2)(Ω) and the continuous embedding of Lp(Ω) into L2(Ω), the following∫Ω

are defined. By (1.49),

h =∂Aα

∂ϑβ(x, u0, ..,∇mu0) ∈ Lp/(p−2)(Ω), ∂βw, ∂αv ∈ Lp(Ω) ⇒ h∂βw∂αv ∈ L1(Ω)

and∣∣∣∣∫

Ω

∂Aα

∂ϑβ(x, u0, ..,∇mu0)∂βw∂αvdx

∣∣∣∣≤ C

∥∥∥∥∂Aα

∂ϑβ(x, u0, ..,∇mu0)

∥∥∥∥Lp/(p−2)(Ω)

‖∂βw‖Lp(Ω)‖∂αv‖Lp(Ω) hence,

∑|α|,|β|≤m

∣∣∣∣∫Ω

∂Aα

∂ϑβ(x, u0, ..,∇mu0)∂βw∂αvdx

∣∣∣∣≤ C

∑|α|,|β|≤m

‖∂βw‖Lp(Ω)‖∂αv‖Lp(Ω) ≤ C‖w‖W m,p(Ω)‖v‖W m,p(Ω), (2.463)

hence (∂G/∂u)(u0) = G′(u0) is bounded for 2 ≤ p ≤ ∞, but not for 1 ≤ p < 2.

24 Since the W m,p(Ω) are continuously embedded into W j,s(Ω) for 1/p − 1/s ≤ (m − j)/n theconditions for ∂Aα/∂ϑβ can be relaxed for |α|, |β| ≤ m − 1.


Theorem 2.43 with (2.461) for m ≥ 1 imply the Hm0 (Ω)-coercivity of the principal

part, G′p(u0), and hence the Hm

0 (Ω)-ellipticity of the operator. The continuous embed-ding of Wm,p(Ω) ↪→ Hm(Ω), p ≥ 2, for bounded Ω allows considering u ∈Wm,p(Ω) asu ∈ Hm(Ω), with the Hm(Ω) norm. These Hm

0 -coercivity results imply all the otherclaims. �

In other cases the Fredholm alternative can be directly verified for the quasilinearproblem as formulated by Zeidler [677,678].

For fully nonlinear equations in (2.306), the G(u) and G′(u0)u have the form

G,G′(u0) : U = W 2,p(Ω) ∩W 1,p0 (Ω) → V ′ = Lq(Ω), u, u0 ∈ D(G) ⊂ U s.t. (2.464)

w = (x, z, p, r), w0(x) := (x, u0(x),∇u0(x),∇2u0(x)) ∈ D(Gw) ∀x ∈ Ω

G(u0(·)) = Gw(·, u0(·),∇u0(·),∇2u0(·)) = 0, hence,

G′(u0)u =∂Gw

∂z(w0)u +

n∑i=1

∂Gw

∂pi(w0)∂iu +

n∑i,j=1

∂Gw

∂rij(w0)∂i∂ju,

for∂Gw

∂z(w0(·)),

∂Gw

∂pi(w0(·)),

∂Gw

∂rij(w0(·)) ∈ Lp/(p−2)(Ω) ⊂ L∞(Ω). (2.465)

The term∑n

i,j=1 in (2.464), its principal part, is used in (2.307) defining the uniformellipticity of G: For w ∈ Uw0 a neighborhood of w0, we required for 0 �= ϑ ∈ Rn

0 < λ|ϑ|2 ≤n∑

i,j=1

∂Gw

∂rij(w)ϑiϑj ≤ Λ|ϑ|2∀w = (x, u, p, r) ∈ Uw0 ⊂ D(Gw). (2.466)

We do not explicitly formulate the case of nondivergent quasilinear equations. Thisis a special case of (2.306) or (2.464) and, compared to (2.459), G′(u0) requires onlyslight changes.

For our later studies in discretization methods it is important that, in contrast tothe original nonlinear problem, it is possible to transform the linearization in (2.464)and in (2.467) below into the standard weak form.

For the Skrypnik [591] setting, the derivative G′(u0)u for the fully nonlinear operatorG(u) = G(0, u) of order 2m is obtained. In a Hilbert space, evaluated at u0 applied tou, see (2.316) and the notation in (2.269), it has the form

u, u0 ∈ D(G), u, u0 ∈ U with w0(x) := (0, x, ∂αu0(x), |α| ≤ 2m) (2.467)

hence,

G′(u0) : U = H2m(Ω) ∩Hm0 (Ω) → V ′ = L2(Ω), G′(u0)u =

∑|α|≤2m

∂α(Gw(w0)

)∂αu

and we need, for the weak form,

∂α(Gw(w0(·)) ∈ L∞(Ω) ∀α : |α| ≤ 2m. (2.468)


We recapitulate the modified ellipticity condition (2.318): There exist constants 0 <k, 0 < μ < 1, 0 < Λ <∞ such that

∀t ∈ [0, 1], v ∈ U : G(t, v) = 0 implies ‖v‖C2m,γ(Ω) ≤ k, and in Ω the (2.469)

ellipticity condition k−1|ϑ|2m ≤∑

|α|=2m

∂α(Gw(t, ·, v, . . . ,∇2mv)(x)

)ϑα ≤ C|ϑ|2m.

A combination of the last proof with these results and Theorem 2.33 yields

Theorem 2.124. Elliptic bilinear form, Fredholm alternative:

1. For the linearization (2.464) of the fully nonlinear differential operator of order2m = 2 in (2.306) we require: In addition to the conditions for the data of theoriginal problem, listed in Theorems 2.79 and 2.80, e.g. (2.469), assume 2 ≤ p ≤∞, and (2.465), for any u0 ∈ U in (2.464) with w0(x) ∈ D(Gw)∀x ∈ Ω. Then thelinearization in (2.464) is bounded in Wm,p

0 (Ω,Rq) with an Hm0 (Ω,Rq)-coercive

principal part for 2 ≤ p ≤ ∞.2. This remains correct for order 2m,m > 1, p ≥ 2 in (2.467) with the correspond-

ing (2.468), (2.469) for ∂α(Gw)(w0(·)) ∈ C(Ω) for |α| = 2m. In both cases,

Corollary 2.44 is valid.

2.7.4 Quasilinear elliptic systems of orders 2 and 2m

Here we are in a situation similar to Subsection 2.7.3, now starting in the Hm0 (Ω,Rq)

setting. Most of the existence, uniqueness and regularity results are known only for thiscase. Generalizations to the Wm,p

0 (Ω,Rq) situation are possible and worthwhile. In factresults as in Theorem 2.122 form the basis for numerical methods for the Wm,p

0 (Ω,Rq)setting, opening the possibility for research into new problems beyond the previousscope. The same extensions are possible for fully nonlinear equations and systems.

We have studied quasilinear elliptic systems as Euler equations of a variationalproblem and the general case. We start with the Euler equations and formally linearizethis system (2.362) in a generalized solution �u0 using �w0(x) := (x, �u0(x),∇�u0(x)) or�w(x) to get, compare (2.377) ff.,

〈G(�u), �v〉 =∫

Ω

[Fp(�w(x))∇�v + Fz(�w(x))�v

]dx ∀�v ∈ H1

0 (Ω,Rq)

〈G′(�u0)�u,�v〉 =∫

Ω

[Fpz(�w0(x))�u + Fpp(�w0(x))∇�u

]∇�v

+[Fzz(�w0(x))�u + Fzp(�w0(x))∇�u

]�vdx (2.470)

=∫

Ω

q∑i,j=1

n∑k=1

[(ai

k

)zj

(�w0(x))uj +n∑

l=1

(ai

k

)pl

j

(�w0(x))∂luj

]∂kvi

+[ (

ai0

)zj

(�w0(x))uj +n∑

l=1

(ai0

)pl

j

(�w0(x))∂luj

]vidx = 0.


We have to impose, similarly to (2.462), slightly stronger conditions than the original(2.372)–(2.374) or (2.372), (2.375)–(2.376), and (2.377)–(2.379) or (2.377), (2.378),(2.380) for the differentiability of J and the regularity of solutions of (2.362). Forgeneral �u1 or our solution �u0 and the above �w1(x), �w0(x) we require for p = q = 2 andthe usual ∂0u = u, ϑβ , ϑl

j ∈ R, �ϑ ∈ Rnq: Assume there exists, similarly to (2.462) and(2.371), a neighborhood Uw0 of �w0, for the solution, such that

�u0 ∈ H1(Ω,Rq) : Fpp, Fpz, Fzz(�w0(x)) ∈ L∞(Ω), and for �w1 ∈ Uw0∃λ > 0 (2.471)

s.t.q∑

i,j=1

n∑k,l=1

Fpki pl

j(�w(x))ϑk

i ϑlj ≥ λ|�ϑ|2nq ∀ �ϑ ∈ Rnq (ellipticity condition). (2.472)

Theorem 2.125. Quasilinear operators of order 2 with H10 -coercive linearization:

For the quasilinear operator G(�u) and its linearization, G′(�u0), cf. (2.470), assume(2.471). Then G′(�u0) : H1

0 → H−10 is a bounded linear operator. Its principal part

is uniformly H10 (Ω,Rq)-coercive if (2.472) is satisfied, hence G′(�u0) is uniformly

elliptic. As in Theorem 2.122 this remains correct for W 2,p0 (Ω,Rq), 2 ≤ p ≤ ∞. The

full machinery of Corollary 2.44 applies, e.g. with respect to existence and uniqueness.

Similarly as before we now differentiate the weak form (2.425) of a general quasi-linear elliptic system. This yields

〈G′(�u0)�u,�v〉 =∫

Ω

∑|α|,|β|≤m

(∂|β|�aα

∂ϑβ(·, �∂≤m�u0(x))∂β�u(x), ∂α�v(x)

)q

dx. (2.473)

This is the linear form (2.392) of the system of order 2m, now restricted to �u, �u0 ∈Hm(Ω) instead of the above �u ∈Wm,p(Ω) for m = 1. The differentiation is correct iffor the solution

�u0 ∈ Hm(Ω) : Aα,β(·) :=∂�aα

∂ϑβ(·; �∂≤m�u0(·)) ∈ L∞(Ω), e.g. |Aα,β(·)| < Λ.

Theorem 2.126. Quasilinear operators of order 2m with Hm0 -coercive linearization:

Under these conditions, the operator G of the quasilinear system (2.425) of order2m is differentiable in �u0 ∈ Hm(Ω). The weak linearized operator G′(�u0) : Hm

0 (Ω) →H−m

0 (Ω) has the form (2.473). Let the strong Legendre–Hadamard condition (2.394)be satisfied for the above (∂�aα/∂ϑ

β)(·, �∂≤m�u0(x)) ∈ C(Ω) for |α| = |β| = m. Then theprincipal part of G′(�u0) is Hm

0 (Ω)-coercive and G′(�u0) is elliptic. As in Theorem 2.125this remains correct for Wm,p

0 (Ω,Rq), 2 ≤ p ≤ ∞, and allows us the full machinery ofCorollary 2.44.

2.7.5 Linearizing general divergent quasilinear and fully nonlinearsystems

In Subsection 2.6.8 we have already indicated that for systems of q fully nonlinearequations no results seem to be listed in the consulted monographs. We had onlyhinted at Yang [671], Fanciullo [311], Belopolskaya [79, 80], Chen and Wu [170] andLi and Liu [475].


In this situation, linearization cannot be used for theoretical results. However,it might be the only possibility, for obtaining existence and bifurcation resultsby combining the FEMs later on for these problems with numerical continuationtechniques. In fact, the linearized operators do fit the linear elliptic systems studiedin Subsection 2.6.5

For convenience, we repeat the necessary generalizations, compare (2.446),(2.447). For �v, �u = (u1, . . . , uq) ∈ W = Hm

0 (Ω,Rq) we have to apply the ∂l, l =0, . . . , n, ∂α, |α| ≤ 2m, to the �u, uj , vi, i, j = 1, . . . , q and use the notation in (2.389).Accordingly, (2.306) and (2.317) have to be reinterpreted as q equations in strong formfor the q components of �u, defined in

U := Hm0 (Ω,Rq) ∩H2m(Ω,Rq),W := Hm

0 (Ω,Rq),V := L2(Ω,Rq) (2.474)

with the obvious norms. For systems of order 2 with m = 1 in (2.474), we obtain

G : D(G) ⊂ U → V, G(�u(·)) := Gw(·, �u(·),∇�u,∇2�u(·)), G(�u0) = 0. (2.475)

The linearization G′(�u0) : U → V ′ in �u0 applied to �u has the form, with �w0(x) :=(x, �u0(x),∇�u0(x),∇2�u0(x)) ∈ D(Gw) ∀x ∈ Ω, p = (pk ∈ Rq)n

k=1, r = (rkl ∈ Rq)nk,l=1,

G′(�u0)�u =∂Gw

∂z(�w0)�u +

n∑k=1

∂Gw

∂pk(�w0)∂k�u +

n∑k,l=1

∂Gw

∂rkl(�w0)∂k∂l�u, (2.476)

and we require

∂Gw

∂z(�w0(·)), ∂Gw

∂pk(�w0(·)), ∂G

w

∂rkl(�w0(·)) ∈ L∞(Ω, . . .).

Again, in contrast to the original nonlinear problem, it is possible to transform thelinearization in (2.476), with the usual partial integration, into the standard weakform. For the strong operator G′(�u0) : U → V ′ we introduce new coefficients Akl.This yields the strong bilinear form as(·, ·), and conditions for the partials of Gw:

G′(�u0) : U → V = V ′, as(·, ·) : U × V → R, and with δ0k, (−1)k>0 we find

G′(�u0)�u :=n∑

k,l=0

(−1)k>0∂k(Akl∂

l �u)

withAkl = Akl(�w0(·)) ∈W 1−δk,0,∞(Ω,Rq×q),

Alk = Akl = −∂Gw

∂rkl(�w0) for k, l = 1, . . . , n,

A0l =∂Gw

∂pl(�w0) +

n∑k=1

∂k ∂Gw

∂rkl(�w0),

Al0 = 0 for l = 1, . . . , n, A00 =∂Gw

∂z(�w0), then the (2.477)

G′(�u0) : U → V = V ′ and as(�u,�v) := (G′(�u0)�u,�v)V ∀ �u ∈ U , �v ∈ V are bounded.


With (·, ·) := (·, ·)V and 〈·, ·〉 := 〈·, ·〉W×W′ the strong and weak solutions �u0 ∈ U and�u0 ∈ W, respectively, of the second order linearized systems are determined by, cf.(2.476),

(G′(�u0)�u,�v) =( n∑

k,l=0

(−1)k>0∂k(Akl∂

l�u), �v)

q= (�f,�v) ∀�v ∈ V, (2.478)

a(�u,�v) := 〈G′(�u0)�u,�v〉 :=∫

Ω

n∑k,l=0

(Akl∂l�u, ∂k�v)qdx = 〈�f,�v〉,∀�v ∈ W

and with Akl = Akl(w0) ∈ L∞(Ω,Rq×q) for the weak form. (2.479)

The uniform Legendre condition defines a uniformly elliptic system:

λ|�ϑ|2 ≤n∑

k,l=1

(Akl(x)ϑl)ϑk ≤ Λ|�ϑ|2 ∀�ϑ ∈ Rnq, |�ϑ|2 =q∑

i=1

n∑k=1

(ϑk

i

)2. (2.480)

This implies the H10 (Ω,Rq×q)-coercivity of the principal part ap(�u,�v).

For order 2m,m > 1, we again reinterpret, now (2.317), as a nonlinear system of qequations in U := Hm

0 (Ω,Rq) ∩H2m(Ω,Rq). We update the above notations in (2.335)as:

�w := (x, ϑγ , |γ| ≤ 2m) ∈ D(Gw), �u, �u0 ∈ D(G), �w0(x) := (x, ∂γ�u0(x), |γ| ≤ 2m).

Then a systems of order 2m has the form G : U → V

G(�u(·)) := Gw(·, �u(·), · · · ,∇2m�u(·)) = Gw(·, ∂γ�u(·), |γ| ≤ 2m), G(�u0) = 0. (2.481)

The linearization G′(�u0) : U → V = V ′ in �u0 applied to �u has the form,

G′(�u0)�u =∑

|γ|≤2m

∂Gw

∂ϑγ(�w0)∂γ�u, with

∂Gw

∂ϑγ(�w0(·)) ∈ L∞(Ω,Rq) ∀|γ| ≤ 2m. (2.482)

Introducing new coefficients Aαβ yields the strong and weak bounded linear operatorsand bilinear forms G′

s(�u0), G′(�u0), and as(·, ·), a(·, ·), and conditions for the partials

of Gw:

as(·, ·) : U × V → R, a(·, ·) : W ×W → R, G′(�u0) :W →W ′,

G′s(�u

0)�u :=∑

|α|,|β|≤m

(−1)|α|∂α(Aαβ∂β �u) :=

∑|γ|≤2m

∂Gw

∂ϑγ(�w0)∂γ�u, with

Aαβ = Aαβ(�w0(·)) ∈W |α|,∞(Ω,Rq×q),∀|α|, |β| ≤ m, e.g., (2.483)

Aαβ = (−1)m ∂Gw

∂ϑαβ(�w0) = Aβα for |α| = |β| = m, and with

(·, ·) := (·, ·)V , 〈·, ·〉 := 〈·, ·〉W×W′ we get the strong solution �u1 ∈ U from


(�f,�v) = as(�u1, �v) := (G′s(�u

0)�u1, �v) ∀ �v ∈ V or the weak solution �u1 ∈ W, from

a(�u1, �v) := 〈G′(�u0)�u1, �v〉 =∫

Ω

∑|α|,|β|≤m

(Aαβ∂β�u1, ∂α�v)dx = 〈�f,�v〉,

∀�v ∈ Hm0 (Ω,Rq) and with Aαβ ∈ L∞(Ω,Rq×q) for this weak form. (2.484)

Here we have to impose the uniform Legendre–Hadamard condition: ∃Λ > λ > 0 :

λ|η|2|ϑ|2m ≤n∑

|α|,|β|=m

(ηAαβ(�w0(x))η

)ϑβϑα ≤ Λ|η|2|ϑ|2m ∀ϑ ∈ Rn, η ∈ Cq. (2.485)

To enforce the Hm0 -coercivity of the principal part, see Theorem 2.104, we need, in

addition to the above Aαβ ∈W |α|,∞(Ω,Rq×q), ∀|α|, |β| ≤ m,

Aαβ(�w0(·)) ∈ C(Ω,Rq×q) ∀|α| = |β| = m > 1. (2.486)

A combination of Theorems 2.104 and 2.33 yields the following

Theorem 2.127. Bounded linearization, Fredholm alternative for fully nonlinearsystems in W 2m,p(Ω,Rq×q), 2 ≤ p ≤ ∞:

1. For the linearizations in (2.478), and (2.483), (2.484) of the fully nonlinearsystems in (2.475), and (2.481), respectively, we require, for the solution �u0 in(2.478) and (2.481), (2.483), the conditions in (2.477),(2.483). Furthermore, let

Aij ∈ L∞(Ω,Rq×q) for m = 1, and Aαβ ∈ L∞(Ω,Rq×q) for |α| = |β| ≤ m, (2.487)

and assume the conditions for the data of the original problem in Subsection2.6.8. Then the above linearizations are bounded.

2. Additionally, we assume the strong Legendre condition (2.480) for m ≥ 1 = q andq ≥ 1 = m with (2.486) for m > 1 and the strong Legendre–Hadamard condition,(2.485), and (2.486) for m, q > 1. Then the above linearizations are even ellipticand the principal parts are Hm

0 (Ω,Rq)-coercive. Thus Corollary 2.44 is validfor p ≥ 2. The strong and weak bilinear forms coincide in u ∈ H2m(Ω,Rq) ∩Hm

0 (Ω,Rq),m ≥ 1, and �v ∈ Hm0 (Ω,Rq) if furthermore Akl ∈W 1−δk0,∞(Ω,Rq×q)

for m = 1 and Aαβ ∈W |α|,∞(Ω,Rq×q) for m > 1.

We discuss the problem of how nonexactly satisfied differential and boundary equa-tions perturb the exact solution. This is an important problem for discretization meth-ods. In particular, the mesh-free methods in [120] usually aim for a discrete solution ofan approximate original problem. Similarly to Section 5.2, we introduce an operatorF := (G,B) : D(G) ∩ D(B) ⊂ U → V × Z with differential and boundary operators,G amd B as components. We restrict the discussion here to, cf. (2.158), linear specialDirichlet systems. In Theorems 1.29, 2.50 we have extended functions defined onthe boundary to the whole domain, Ω. In Lemmas 2.26, 2.27, this was used forreducing the inhomogeneous to homogeneous boundary conditions. Similar strategiesare possible for some more general, in particular nonlinear boundary conditions as well.We leave that to the interested reader. Note that for fully nonlinear problems (in strong


form) we have U = H2m(Ω,Rq) and V = L2(Ω,Rq), Z = Πmj=1H

m−j+1−1/2(∂Ω,Rq).The solutions are denoted for all cases q ≥ 1, as u0. For all the other problems(in weak form) we have U = Wm,p(Ω,Rq) and V = W−m,p′

(Ω,Rq), 1/p + 1/p′ = 1,Z = Πm

j=1Wm−j+1−1/p,p(∂Ω,Rq). We combine Theorems 1.48 and 1.43 with our

results for boundedly invertible linearizations.Therefore we present, in the next theorem, estimates for the norm of the difference

‖u0 − u0‖U , with the exact and the perturbed solutions, hence for

F (u0) = (G(u0), B(u0)) = (0, 0) and F (u0) = (G(u0), B(u0)) = (g, ϕ). (2.488)

For many applications, e.g. for discretization methods, results for boundedly invert-ible G′(u0), and bounds for its norm, are more directly available than for F ′(u0).Therefore we present in the next theorem estimates for ‖u0 − u0‖U , in terms of‖(F ′(u0))−1‖U←↩V×Z and of ‖(G′(u0))−1‖U←↩V .

Theorem 2.128. Error estimates for perturbed elliptic problems: With the differen-tial and boundary operators, G and B, let the operator F := (G,B) : D(G) ∩ D(B) ⊂U → V × Z have a locally unique solution u0 for F (u0) = 0, and let F ∈ C1 in a neigh-borhood of u0, hence F ′(u0) = (G′(u0), B′(u0)) : U → V × Z exists and is boundedlyinvertible. Let u0 and u0 be the solutions of the exact and the perturbed problems in(2.488) with small enough (g, ϕ) ∈ V × Z. Then

‖u0 − u0‖U ≤ 2‖(F ′(u0))−1‖U←↩V×Z‖(g, ϕ)‖V×Z . (2.489)

G(u) = 0, B(u) = 0 has the same locally unique solution u0, and G′(u0) : U ∩ N (B) →V exists in a neighborhood of u0, and is also boundedly invertible. Then

‖u0 − u0‖U ≤ 2‖(G′(u0))−1‖U←↩V(‖g‖V + C|ϕ‖Z(1 + 2‖G′(u0)‖V←↩U‖)

). (2.490)

Proof. F ′ is continuous in a neighborhood of u0, hence equi continuous in a smallerclosed neighborhood of u0. Therefore, (2.489) is an immediate consequence of Theo-rems 1.48 and 1.43.

We employ the lemmas and theorems mentioned above. In the first step, we extendϕ ∈ Z, defined along ∂Ω, to a u, defined on Ω. By lemma 2.26 this yields

Bu = ϕ on ∂Ω =⇒ ‖u‖U ≤ C‖ϕ‖Z . (2.491)

Next we estimate for the linear B, and with (2.491) and O(???) as defined by (2.493),

F (u0 − u)− F (u0) =(G(u0 − u)−G(u0), B(u0 − u)−B(u0)

)(2.492)

= (G(u0 − u)− 0, 0) = (G′(u0)(u0 − u− u0) +O(???), 0) with

‖G(u0 − u)‖V = ‖G(u0)−G′(u0)u +O(???)‖V ≤ (‖g‖V + 2‖G′(u0)‖V←↩U‖C‖ϕ‖Z).

We have modified (1.78), e.g. with x0 = u0 near u0u0, for the last inequality as

‖G(u0)−G(u0)−G′(x0)(u0 − u0)‖ ≤ ‖u0 − u0‖ supx∈u0u0

‖G′(x)−G′(x0)‖ =:‖O(???)‖V .

(2.493)

2.8. The Navier–Stokes equation 163

The right-hand term (2.493) is indicated in (2.492) as O(???). It gets very small andimplies the previous factors 2 in the last two lines in (2.492) and in (2.494). Thecombination of the last two lines in (2.492) yields

‖u0 − u− u0‖U ≤ 2‖(G′(u0))−1‖U←↩V(‖g‖V + 2‖G′(u0)‖V←↩U‖C|ϕ‖Z

). (2.494)

The triangle inequality

‖u0 − u0‖U ≤ ‖u0 − u− u0‖U + ‖u‖Ufinally yields the claim. �

2.8 The Navier–Stokes equation

2.8.1 Introduction

This is one of the most stimulating equations for research in mathematics, physics,and engineering. It models fluid mechanics, e.g. the flow of water around a vessel, andthe air around a wing of an airplane. In this context (2.495) with small values of 0 < νare interesting. Then boundary layers and turbulence develop, modeling the up-windproperties carrying an airplane. Mathematically, it was one of the first extremelyimportant systems of nonlinear differential equations. It required an update of thedefinition of elliptic or parabolic equations, and for small 0 < ν the borders betweenparabolic and hyperbolic problems are no longer well defined. Furthermore, by varying0 < ν and other parameters, extremely interesting regimes of bifurcation and thecorresponding dynamics and chaotic phenomena originate. Still, for a lot of theseproblems satisfactory answers are missing. Some answers can be found in [120].

We only consider incompressible Navier-Stokes equations. From the huge numberof publications we have to choose a very narrow selection of results.

Further results can be found in, e.g. Kreiss and Lorenz [454], von Wahl and Grunau[378, 658, 659], Kotter [453], Bensoussan and Frehse [81], and in Ladyzenskaja andUralceva [463–466]. Further regularity results for Navier-Stokes equations are due,e.g. to Cannone and Karch [157], Ladyzhenskaya [465], and Zubelevich [683].

Models of Navier-Stokes equations as Newtonian compressible fluids in the steadyand unsteady regime are studied, e.g. by Feistauer et al. [313, 315], and Novotny andStraskraba [515]. The steady regime is discussed in Subsection 7.15.2 as well.

2.8.2 The Stokes operator and saddle point problems

We start with the Stokes operator and its properties, see Hackbusch [386, 387]. It isclosely related to the stationary Navier-Stokes equation, cf. (2.320),

G(�u, p) :=

⎛⎝−νΔ�u +n∑

i=1

ui∂i�u +∇ p

− div �u

⎞⎠ =(�f0

)in Ω, (2.495)

�u = 0 on Γ = ∂Ω,∫

Ω

pdx = 0, Ω satisfies (2.5)


and �u, p and �f denote the velocity, pressure and forcing terms of an incompressiblemedium, �v a test function with div �u =

∑ni=1 ∂ui/∂xi, and ν = 1/R with the Reynolds

number, R. The Stokes operator is obtained from (2.495) for ν = 1 and by deletingthe nonlinear term as

S(�u, p) :=(−Δ�u +∇ p− div �u

)=(�f0

)in Ω, �u|∂Ω = 0,

∫Ω

pdx = 0. (2.496)

We introduce Vb,W, see Subsection 2.6.2, as

Vb := H10 (Ω,Rn),V := H1,W := L2

∗(Ω) :=

⎧⎨⎩p ∈ L2(Ω) :∫Ω

p(x)dx = 0

⎫⎬⎭. (2.497)

Now we take the scalar product of the Stokes operator with �v ∈ Vb and apply theGreen’s formula for smooth enough �u,�v, p. With the Euclidean norm | · |n and innerproduct (·, ·)n in Rn and the product norm ‖�v‖V := |

(‖vi‖H1(Ω)

)ni=1|n the first line in

(2.496) yields

(−Δ�u +∇p,�v)L2(Ω,Rn) :=∫

Ω

(−Δ�u +∇p,�v)ndx = (∇�u,∇�v)L2 − (p, div �v)L2(Ω).

We define the bilinear and linear forms a(·, ·), b(·, ·) and (�f, ·) as

a(�u,�v) :=∫

Ω

(∇�u(x),∇�v(x))ndx for �u,�v ∈ Vb,

b(p,�v) := −∫

Ω

p(x) div �v(x)dx for p ∈ L2∗(Ω), (2.498)

(�f,�v) :=∫

Ω

(f,�v)ndx for �f ∈ L2(Ω,Rn) or ∈ V ′ = H−1(Ω,Rn), cf. (2.110).

This yields for the two equations in (2.496) and for u ∈ H2(Ω,Rn) ∩ Vb

(−Δ�u +∇p,�v)L2(Ω,Rn) = a(�u,�v) + b(p,�v) = (�f,�v), ∀ �u,�v ∈ Vb, p ∈ W and

−∫

Ω

q(x) div �u(x)dx = b(q, �u) = 0 ∀ q ∈ W, �u ∈ Vb.

This requires testing the first equation by �v and div �u = 0 by q ∈ W. For our boundedΩ in (2.5), the a(·, ·), b(·, ·), and (�f, ·) are continuous, cf. Theorem 2.43 and Proposition2.34:

|a(�u,�v)| ≤ Ca‖�u‖V · ‖�v‖V , |b(p,�v)| ≤ Cb‖�v‖V · ‖p‖W , (2.499)

|(�f,�v)| ≤ ‖�f‖L2(Ω,Rn) · ‖�v‖V , with ‖�f‖L2(Ω,Rn) or ‖�f‖V′ .


In the weak formulation of (2.496), we replace �f ∈ L2(Ω,Rn) and 0 by f1 ∈ V ′ andf2 ∈ W ′, respectively, and obtain

for given �f = (f1, f2) ∈ V ′b ×W ′ determine �u0 ∈ Vb, p0 ∈ W such that

a(�u0, �v) + b(p0, �v) = (f1, �v) ∀ �v ∈ Vb, (2.500)

b(q, �u0) = (f2, q) ∀ q ∈ W.

Saddle point problems

To obtain a situation similar to Section 2.3, we reinterpret (2.500): We replacethe Vb,W, a(·, ·), b(·, ·), (�f , ·), 0 in (2.497)–(2.500) by general Hilbert spaces Vb,W,continuous bilinear forms a, b and linear forms f1 ∈ V ′

b, f2 ∈ W ′, respectively. We call

the Stokes problem as in (2.498), (2.500) and this generalized version a saddle pointproblem. Furthermore, we introduce

X := Vb ×W and �xT := (�u, p), �yT := (�v, q) ∈ X to obtain

c(�x, �y) := c

((�u

p

),

(�v

q

)):= a(�u,�v) + b(p,�v) + b(q, �u), (2.501)

(�f, �y) := (f1, �v) + (f2, q), �f = (f1, f2) ∈ X ′ = V ′b ×W ′.

Then c(·, ·) and (�f, ·) are continuous bilinear and linear forms on X × X and X . LetA ∈ L (Vb,V ′

b) , B ∈ L (W,V ′b) and C ∈ L(X ,X ′) be the linear operators induced by

a(·, ·), b(·, ·) and c(·, ·), respectively. Then

C :=(

ABd

B0

)∈ L(X ,X ′), C�x =

(ABd

B0

)(�up

)=(A�u + BpBd�u

),

c(�x, �y) = 〈C�x, �y〉X ′×X =⟨(

A�u + BpBd�u

),

(�vq

)⟩X ′×X

(2.502)

= 〈A�u,�v〉V′b×Vb

+ 〈Bp,�v〉V′b×Vb

+ 〈Bd�u, q〉W′×W = a(�u,�v) + b(p,�v) + b(q, �u).

Obviously, c(·, ·) is bounded, but it is not coercive, since c(�x, �x) = 0 for all �x = (0, p).We formulate this saddle point problem in three equivalent forms as

for given�f, = (f1, f2) ∈ X ′ determine �x0 = (�u0, p0) ∈ X such that

c(�x0, �y) = 〈�f, �y〉X ′×X for all �y ∈ X ⇔ C�x0 = �f ∈ X ′ (2.503)

⇔ a(�u0, �v) + b(p0, �v) = (f1, �v) ∀ �v ∈ Vb, b(q, �u0) = (f2, q) ∀ q ∈ W,

compare (2.55). The following theorem explains why we call this a saddle pointproblem. Let

J(�x) := J(�u, p) = a(�u, �u) + 2b(p, �u)− 2(f1, �u)− 2(f2, p) (2.504)

= c(�x, �x)− 2(�f, �x) for �x = (�u, p) ∈ Vb ×W.


This J(�u, p) is neither bounded from above nor from below. So a solution �x0 =(�u0, p0) ∈ X of (2.503) cannot be characterized by a minimum as in Theorem 2.16,but only as a saddle point, Hackbusch [387], Theorem 12.2.4.

Theorem 2.129. Saddle point problem: Let a(·, ·) be symmetric and Vb-coercive.Then �x0 = (�u0, p0) ∈ Vb ×W solves one of the equivalent problems in (2.503), if andonly if

J(�u0, p) ≤ J(�u0, p0) ≤ J(�u, p0) for �x = (�u, p) ∈ X = Vb ×W⇔ J(�u0, p0) = min

u∈Vb

J(�u, p0) = maxp∈W

minu∈Vb

J(�u, p). (2.505)

Existence and uniqueness of solutions �x0 = (�u0, p0) ∈ X for (2.503) cannot beobtained with our previous methods. So we introduce a closed V0 and its orthogonalcomplement V⊥ such that

V0 = kerBd = {�v ∈ Vb : Bd�v = 0} = {�v ∈ Vb : b(q,�v) = 0∀q ∈ W},Vb = V0 ⊕ V⊥.

Conditions for the unique solvability of (2.503) require a slight reinterpretation. Let

A :=(A00

A⊥0

A0⊥A⊥⊥

)with (2.506)

A00 ∈ L (V0,V ′0) , A0⊥ ∈ L (V⊥,V ′

0) , A⊥0 ∈ L (V0,V ′⊥) , A⊥⊥ ∈ L (V⊥,V ′

⊥) ,

and an analogous B∗ = (B∗0 , B

∗⊥) = (0, B∗), since B∗

0 = 0.

Theorem 2.130. Unique solution for a saddle point problem: Let a(·, ·) : V × V → R,b(·, ·) : W ×V → R,, and f1 ∈ V ′, f2 ∈ W ′ be bounded and V0,V⊥ be defined as above.Then (2.503) or equivalently (2.505) is uniquely solvable if and only if the inverses ofA00 and B exist, hence

V0 = kerBd = {�v ∈ Vb : Bd�v = 0} = {�v ∈ Vb : b(q,�v) = 0∀q ∈ W},Vb = V0 ⊕ V⊥, V0,V⊥ ⊂ Vb. (2.507)

Now we are able to modify Theorem 2.12 to this new situation. With the aboveA,B,C the C is boundedly invertible, if and only if the following modified Brezzi–Babuska condition or Ladyzenskaja–Brezzi–Babuska condition is fulfilled:

C ∈ L(X ,X ′) is boundedly invertible ⇔∃ α, β > 0 s.t. sup

0�=�v0∈V0

|a(�v0, �v1)|/‖�v0‖Vb> 0 ∀0 �= �v1 ∈ V0, (2.508)

sup0�=�v1∈V0

|a(�v0, �v1)|/‖�v1‖Vb≥ α‖�v0‖Vb

∀�v0 ∈ V0, (2.509)

sup0�=�v∈Vb

|b(p,�v)|/‖�v‖Vb≥ β‖p‖W ∀p ∈ W. (2.510)

We do not always need all these conditions. If a(·, ·) is symmetric or Proposition 2.19holds, then (2.508) can be skipped. The second line (2.509) implies the V0-coercivityof a(·, ·). For other combinations, see Hackbusch [387], Lemma 12.2.9 ff.


Solvability and regularity of the Stokes problem

Hackbusch’s Lemmas 12.2.12 and 13 [387], yield (2.508), (2.509), (2.510). He refersto Necas [509] for n ≥ 2, Ladyzenskaja [461], and Kellog and Osborn [443], see [387],Theorems 12.2.14, 18 and 19, for the following

Theorem 2.131. Unique and regular solution for the Stokes problem:

1. Let Ω ∈ C0,1 be bounded, see (2.5), C be defined by (2.502) for the Stokes casewith A,B as induced by (2.498). Then C is boundedly invertible, C−1 ∈ L(X ′,X ).So the Stokes problem

−Δ�u +∇p = f1 ∈ H−1(Ω),− div �u = f2 ∈ L2∗(Ω) on Ω, �u = 0 on ∂Ω

has a unique weak solution (�u0, p0) ∈ Vb ×W = H10 (Ω,R3)× L2

∗(Ω) for (2.500)satisfying

‖�u0‖H1(Ω) + ‖p0‖L2(Ω) ≤ CΩ(‖f1‖H−1(Ω) + ‖f2‖L2(Ω)). (2.511)

2. Smoother data imply smoother solutions: If, in the above Stokes problem, Ω issmooth enough and f1 ∈ Hk(Ω), f2 ∈ Hk+1(Ω) ∩ L2

∗(Ω), k ∈ N0, we get for thesolution �u0 ∈ Hk+2(Ω) ∩H1

0 (Ω), p0 ∈ Hk+1(Ω) ∩ L2∗(Ω). There exists a constant

C > 0, only depending upon Ω, such that

‖�u0‖Hk+2(Ω) + ‖p0‖Hk+1(Ω) ≤ CΩ(‖f1‖Hk(Ω) + ‖f2‖Hk+1(Ω)).

For n = 2, k = 0 a bounded and convex Ω ⊂ R2 suffices.

2.8.3 The Navier–Stokes operator and its linearization

We want to study the linearization of the stationary Navier–Stokes operator in (2.495).Its derivative, the linearized Navier-Stokes operator, is

G′(�u, p)(�w, r) =

⎛⎝−νΔ�w +n∑

i=1

(wi∂i�u + ui∂

i �w) +∇r− div �w

⎞⎠. (2.512)

The special cases �u ≡ 0, ν = 1, yields the above Stokes operator, S, see (2.496), and

�u ≡ 0, ν �= 1, Sν := G′(0, p) we call the ν-Stokes operator. (2.513)

The results of the last subsection remain correct for this ν-Stokes operator, if through-out, the a(�u,�v) are replaced by νa(�u,�v) and ν is not too small.

We present the stationary Navier-Stokes operator following Temam et al. [349,623,624]. They discuss the local existence of solutions for the nonstationary Navier-Stokesequations and many open problems, e.g. related to nonlocal existence. As in thetransition from (2.497) to (2.498), we multiply the first equation in (2.495) with thetest function �v = (v1, . . . , vn) and the second with q and integrate. We use Green’sformula to obtain, with a(·, ·), b(·, ·) in (2.498),


for given (f1, f2) ∈ V ′b ×W ′ determine �u0 ∈ Vb, p0 ∈ W such that∫

Ω

(−νΔ�u0 +

n∑i=1

(�u0)i∂i�u0 +∇ p0, �v

)n

dx (2.514)

= νa(�u0, �v) + d(�u0, �u0, �v) + b(p0, �v) = (f1, �v) ∀ �v ∈ Vb for the first, and

b(q, �u0) = (f2, q) ∀ q ∈ W for the second equation;

here we use the previous a(�u,�v), b(q,�v), (�f,�v) in (2.498) and introduce the trilinear

d(�u, �w,�v) :=n∑

i=1

∫Ω

(ui(∂i �w), �v)ndx =n∑

i,j=1

∫Ω

ui(∂iwj)vjdx. (2.515)

For bounded Ω, and n ≤ 4, see Temam [624] Lemma 1.2, Chapter II, Section 1.

d(�u,�v, �w) is a bounded trilinear form on Vb × Vb × Vb (2.516)

with Vb ↪→W∗ := L2(Ω,R3) compactly embedded.

To linearize (2.514) with respect to �u, we consider, for fixed �u,�v and small �w,

d(�u + �w, �u + �w,�v)− d(�u, �u,�v) = d(�u, �w,�v) + d(�w, �u,�v) +O(‖�w‖2V ‖�v‖V

)=⇒((

(G′(�u, p)(�w, r)), (�v, q))n+1

)L2(Ω)

(2.517)

=(νa(�w,�v) + d(�u, �w,�v) + d(�w, �u,�v) + b(r,�v)

b(q, �w)

)∀(�v, q) ∈ Vb ×W = X .

Now we consider the d(�u, �w,�v), d(�w, �u,�v) in (2.515). For fixed �u, both terms d(�u, �w,�v),d(�w, �u,�v) are continuous bilinear forms in �w and �v, corresponding to the operators∫Ω

∑ni,j=1

(ui∂

i �w +∑n

i,j=1 wi∂i�u)dx in (2.515). Thus, the d(�u, �w,�v) and d(�w, �u,�v) ∈

R define elements in V ′b :

d(�u, �w, ·), d(�w, �u, ·) ∈ V ′b for variable �w ∈ Vb and for fixed �u ∈ Vb.

Hence, by Section 2.3 they induce linear continuous operators, see (2.514),

D1, D2 ∈ L (Vb,V ′b) as D1 �w := d(�u, �w, ·), D2 �w := d(�w, �u, ·), �u fixed (2.518)

with 〈D1 �w,�v〉V′b×Vb

= d(�u, �w,�v), 〈D2 �w,�v〉V′b×Vb

= d(�w, �u,�v)∀�v ∈ Vb. (2.519)

The embedding I : H10 (Ω,Rn) → L2(Ω,Rn) is continuous and compact, see Theorem

1.26. So (2.519) shows that

D1 �w = D1I �w,D2 �w = D2I �w ∀ �w ∈ H10 (Ω,Rn).

Hence, as a product of a compact and a continuous operator, D1 = D1I and D2 = D2Iare compact operators.


Similarly to the transformation from (2.500) to (2.503) we use here, with the factorν and the same A,B as in (2.502),

Cν :=(νABd

B0

), D :=

(D1 + D2

000

). (2.520)

Consequently, D1 + D2 is a compact perturbation of νA and D of Cν . As indicatedabove, this observation is only appropriate if ν is not too small, we call it moderateν. Then (Cν)−1(D1 + D2) is compact and allows us to apply Theorem 2.20. Thisshows that for the linearized Navier-Stokes operator Corollary 2.44, implying theFredholm alternative, is valid. For 0 < ν ≈ 0 the convection terms D1 + D2 dominatethe behavior of the equation.

We apply Cν + D to the above �x = (�u, p)T, multiply by �y = (�v, q)T and integrate:

so (Cν + D)�x =(νA�u + Bp + D1�u + D2�u

Bd�u

)yields

〈(Cν + D)�x, �y〉X ′×X = ν〈A�u,�v〉V′b×Vb


(2.521)

+ 〈D1�u + D2�u,�v〉V′b×Vb

+ 〈Bd�u, q〉W′×W .

The exact solution �x0 = (�u0, p0) has to be determined from

for given �f ∈ X ′ determine �x0 = (�u0, p0) such that

〈(Cν + D)�x0, �y〉X ′×X = 〈�f, �y〉X ′×X ∀ �y ∈ X

⇔ (Cν + D)�x0 = �f ∈ X ′ (2.522)

⇔ Cν(I + (Cν)−1D)�x0 = �f ∈ X ′

⇔ (I + (Cν)−1D)�x0 = (Cν)−1f ∈ X .

Applying Theorems 2.20 and 2.131, we can guarantee the Fredholm alternative forCν + D.

Theorem 2.132. Unique existence, Fredholm alternative:

1. For ν > 0 not too close to 0 the linearized Navier-Stokes operator Cν + D in(2.512) is a compact perturbation of the ν-Stokes operator Cν in (2.513). Thisimplies the spectrum of results of Corollary 2.44.

2. These statements remain correct if the Navier-Stokes operator is replaced by ageneralized nonlinear saddle point problem, according to (2.521), where the non-linearity satisfies (2.516) with the special Vb = H1

0 (Ω,Rn),W∗ := L2(Ω) replacedby general compactly embedded Vb ↪→W∗.

So whenever −1 is not an eigenvalue of the compact operator C−1ν D, or equivalently

N (Cν + D) = {0}, the linearized Navier-Stokes operator Cν + D is boundedly invert-ible, and thus excludes bifurcation phenomena. On the other hand, this allows us touse continuation techniques for determining solutions of the nonlinear Navier-Stokesequations (2.495) and localizing bifurcation.


Finally we inherit the regularity from the Stokes to the Navier-Stokes operator bycombining Theorem 2.56 with the second part of Theorem 2.131 to yield:

Theorem 2.133. Smooth data imply smooth solutions: If, in the above Navier-Stokes problem, Ω is smooth enough and f1 ∈ Hk(Ω), f2 ∈ Hk+1(Ω) ∩ L2

∗(Ω), k ∈ N0,we get for the solution �u0 ∈ Hk+2(Ω) ∩H1

0 (Ω), p0 ∈ Hk+1(Ω) ∩ L2∗(Ω). There exists a

constant C > 0, only depending upon Ω, such that

‖�u0‖Hk+2(Ω) + ‖p0‖Hk+1(Ω) ≤ CΩ(‖f1‖Hk(Ω) + ‖f2‖Hk+1(Ω)).

Part II

Numerical Methods


3

A general discretization theory

3.1 Introduction

In this chapter we combine general Petrov–Galerkin methods, the standard approachfor many elliptic discretizations, with a modification of one of the classical generaldiscretization methods, cf. Stetter [596]. This is a new way of proving stability for thegeneral class of elliptic equations discussed in Chapter 2, cf. Bohmer and Sassman-nshausen [128] and Bohmer [116]. It probably applies to all up-to-date discretizationmethods. As examples we demonstrate this for six different types of discretizationmethods: we choose the still very alive grandfather, the difference methods, themost popular, the FEM including variational crimes, and their adaptive version, theDCGMs, then two methods still strongly in the process of development, the waveletsand the mesh–free methods, and finally, the spectral methods. The two last methodsare particularly useful for bifurcation problems with their underlying symmetries.So they are described by Bohmer [120]. The proof of the convergence of mesh–freemethods applied to general nonlinear problems was an open problem for two decadesand is solved for the first time in [120]. Proofs of different versions of linear problemsare given by Schaback and his group, e.g. [327, 328, 401, 568, 569]. Specific properties,e.g. the adaptivity for wavelets, have to be studied additionally.

We combine compact perturbations of coercive linear, hence boundedly invertibleoperators with additional approximation properties, see Bohmer [116] and BohmerSassmannshausen [128]. This technique applies to Navier–Stokes problems and to thebordered systems, required for numerical bifurcation. Or we directly apply this theoryto nonlinear via monotone operators.

This mix of different concepts is strongly influenced by many earlier papersand books. Discrete convergence has been studied by Stummel [607–610], Rein-hardt [547] and Vainikko [644]. Stetter [596] studies admissible discretization meth-ods; the (inner and outer) admissible approximation schemes due to Petryshyn[528–530, 532] are nicely collected, presented and extended by Zeidler [677]. Keller[441] has combined, for special problems, linearization with stability and consis-tency for proving convergence. Anselone and Ansorge [28–31] strongly elaborate thefunctional analytic side of general discretizations, based in the concept of compactperturbations. In Ansorge’s iterated discretization [32] additional iteration steps inimproved projection methods for more general discretization procedures, not onlyprojection methods, and more general classes of problems, sometimes even yieldsuperconvergence.

174 3. A general discretization theory

Brezzi, Rappaz and Raviart have applied a related version to finite dimensionalapproximations of nonlinear problems, including limit and simple bifurcation points,however excluding many operator equations and, necessarily, missing the new dis-cretization methods and higher singularities [147–149]. Rappaz and Raugel extendedthis approach to special cases of finite dimensional approximation of bifurcationproblems at a multiple eigenvalue [542–544], and to symmetric problems. Crouzeixand Rappaz gave a short survey book on numerical approximation in bifurcationtheory [229]. Griewank and Reddien [366–370] prove convergence for general bifur-cation scenarios under the conditions of [147–149]. Jepson and Spence [418] studybifurcation for perturbed operators, not covering discretization methods. Calos andRappaz presented numerical approximations in nonlinear and bifurcation problemsin [156]. Here, in contrast to the earlier [147–149, 542, 543], two projectors for theanalysis of FEMs for nonlinear problems are appropriately introduced, but are notemployed for the bifurcation analysis. Finally, Cliffe, Spence and Tavener [180] andCalos and Rappaz [156] give a well–written introduction to the numerical analysis ofbifurcation problems with application to fluid mechanics.

All these papers, starting with [147], do not prove convergence for higher bifurcationsand sometimes of more complicated problems such as porous media nor the newmethods. Often the Navier–Stokes equation is studied. Extensions to higher bifur-cation scenarios and the previously omitted other methods are discussed by Bohmeret al. [122, 124, 128]. Approaches to these more complicated cases use the conceptsof consistent differentiability and modified or bordered stability, introduced in [111],cf. [8, 41,42,113,114,114,115,127–129].

We have discussed many different elliptic equations and systems. We consider linearand nonlinear, often parameter–dependent, problems of the form

G : D(G) ⊂ U → V ′, G(u0) = 0, U ,V Banach spaces, with duals U ′,V ′. (3.1)

For some problems solutions with special features can be obtained by analyticalmethods. For general equations or interesting parameters in the problem this is usuallyimpossible. So we introduce discretization methods. Let

Gh : D(Gh) ⊂ Uh → V ′h, Gh(uh) = 0, Uh,Vh discrete Banach spaces, (3.2)

represent a corresponding discrete problem. The classical convergence results for(nonlinear) problems (3.1), (3.2) are obtained by combining the well-known conceptsof stability and consistency. Consistency requires small errors in

0 = G(u0) ≈ Gh(uh

0

)+ errors, with uh

0 an “approximation” for u0.

The following Petrov–Galerkin and difference methods have to be included in the gen-eral discretization methods. Stability and consistency, cf. Subsections 3.3 and 3.4,are the basic concepts. They guarantee, with some technical conditions, the existenceand uniqueness of the discrete solutions for uniquely existing original solutions andconvergence for the discrete compared to the exact solution. Variational consistencyfor linear operators is more familiar in the FE and Petrov–Galerkin community, butclassical consistency is more appropriate in combination with stability and nonlinearproblems. It is unavoidable for fully nonlinear problems.

3.2. Petrov–Galerkin and general discretization methods 175

For the many different discretization methods in the following chapters we choosethe neutral notation of U ,V and Uh,Vh for the Banach spaces and their discreteapproximations. Ub,Vb and Uh

b ,Vhb is used for contrasting the full space to the subspace

of functions satisfying the boundary conditions, e.g. U = H1(Ω),Ub = H10 (Ω). Often

we work with the weak form. The strong form is important for all kinds of variationalcrimes, e.g. in FEMs or DCGMs or difference or mesh–free methods, and for fullynonlinear elliptic problems. Boundary conditions are imposed for unique existence. Incontrast to many other books, we maintain this U versus Ub form, since nonconformingFE, DCG, and mesh–free methods violate boundary conditions and, e.g. continuity,hence Uh

b �⊂ Ub. The linear and nonlinear operators and their discretizations aredenoted as A, G and Ah, Gh. In the previous chapter we indicated, for most classesof problems, appropriate conditions for higher regularity of the solutions, which willallow better convergence.

We will formulate the following Examples 3.1–3.4 as FEMs in Petrov–Galerkin formor as difference methods in classical form. We are aiming for a definition of a generaldiscretization theory, applicable to all intended analytical problems. It will includethese examples and all the discretization methods, considered in both books, and veryprobably most of the future methods as well. In addition, we will be able to avoid thedichotomy between interior and exterior approximation methods.

3.2 Petrov–Galerkin and general discretization methods

Our (stationary) nonlinear problems are usually formulated in weak, and sometimes instrong form. For unique existence, we have to impose boundary conditions, indicatedby u ∈ Ub ⊂ U . So we determine u0 ∈ Ub by testing with v ∈ Vb :

G : Ub → V ′b, determine u0 ∈ Ub s.t. G(u0) = 0 ∈ V ′

b ⇐⇒ G(u0) ⊥ Vb

e.g. Vb = H10 (Ω) ⇐⇒ < G(u0), v >V′×V= 0 ∀ v ∈ Vb. (3.3)

In Remark 2.3, we have summarized the fact that for the problems in weak form inChapter 2, V ′ = V ′

b. However, here we have to use both duals for the discrete V ′hb �= V ′h,

and sometimes use V ′,V ′b as well, thus maintaining U versus Ub.

Reducing the technical difficulties we start with a bounded linear operator, e.g. thederivative A := Gu(u1), in either weak or strong form. Nonlinear operators will bediscussed below. For a second order linear A and f,

A ∈ L(U ,V ′), determine u0 ∈ Ub s.t. Au0 − f = 0 ∈ V ′b (3.4)

⇔ Au0 − f ⊥ Vb ⇐⇒ a(u0, v) = 〈f, v〉V′×V =∫

Ω

n∑j=0

f j∂jvdx ∀ v ∈ Vb,

with the standard a(u, v) = 〈Au, v〉V′×V ∀ v ∈ V, ∀u ∈ U . The boundary conditionsUb and Vb, necessary for unique solvability of (3.3) and (3.4) are realized by continuous


trace operators, e.g. B(u) = 0.

u ∈ Ub := {u ∈ U : B(u) = 0} ⊂ U , v ∈ Vb := {v ∈ V : B2(v) = 0} ⊂ V,with closed subspaces Ub,Vb, and usually V ′

b = V ′, often B(u) = B2(u).

So, A ∈ L(U ,V ′) is a bounded linear operator, defined, but usually only invertible asA ∈ L (Ub,V ′

b).Before we give some general definitions and properties for the intended discretization

methods, we formulate as motivation the simplest cases of the finite element and finitedifference method for the Laplacian with Dirichlet boundary conditions. We choose,for the weak bilinear form, Ub,Vb = H1

0 (Ω), U = H1(Ω), and for the strong form,Ub = H2(Ω) ∩H1

0 (Ω),V = L2(Ω), similarly for natural boundary conditions.

Example 3.1. Finite element method for the LaplacianIts weak and strong form, A and As, respectively, with the same boundary conditionsfor Ub = Vb, are

u0 ∈ Ub = Vb = H10 (Ω) : 〈Au0, v〉V′×V = a(u0, v) =

∫Ω

(∇u0,∇v)ndx (3.5)

= 〈f, v〉V′×V∀v ∈ Vb, and for f ∈ L2(Ω), u0 ∈ H2(Ω) ∩ Ub : Asu0 = −Δu0 = f.

Our goal is the reduction of the original problems, defined and tested on infinitedimensional spaces Ub,Vb, to related discrete problems, defined and tested on spacesUh

b ,Vhb of the same finite dimension. We choose two grids Ωh and Ωh

0 in Ω = [0, 1]2:

Ωh0 := {Pi,j := (xi, yj), xi := i/N, yj := j/N, 0 ≤ i, j ≤ N} and (3.6)

Ωh := {Pi,j , 1 ≤ i, j ≤ N − 1}, ∂Ωh := Ωh0 \ Ωh, cf. F igure 3.1.

The following approximating spaces, Uhb , are defined as the set of piecewise linear hat

functions, φi,j , with 1 and 0 in the grid points, hence

Uhb :=

{uh :=

N−1∑i,j=1

Ci,jφi,j : Ci,j ∈ R, φi,j(Pk,l) = 1 for (i, j) = (k, l), (3.7)

or Pi,j ∈ Ωh, and = 0 else, so ∀0 ≤ k, l ≤ N, (i, j) �= (k, l)}⊂ H1

0 (Ω).

Since Uhb ⊂ H1

0 (Ω), we call the uh conforming FEs. Then the weak form in (3.5) isreduced to determining the Ci,j for uh

0 ∈ Uhb in (3.7), such that⟨

Auh0 , v

h⟩V′×V = a

(uh

0 , vh)

=∫

Ω

(∇uh

0 ,∇vh)ndx = 〈f, vh〉V′×V∀vh ∈ Vh

b ,

⇐⇒ a(uh

0 , φk,l

)=∫

Ω

(∇uh

0 ,∇φk,l

)ndx = 〈f, φk,l〉V′×V ∀1 ≤ k, l ≤ N − 1. (3.8)

We expect this uh0 to be a good approximation for u0. �


: points in ∂Ωh

: points in Ωh

Figure 3.1 Two-dimensional grid for Dirichlet boundary conditions.

Example 3.2.

Nonconforming finite element methods are characterized by Uhb �⊂ H1

0 (Ω). In modifiedforms this ansatz is used in DCGMs as well. The basic case are the discontinuousCrouzeix–Raviart elements, cf. Figure 4.10. They are defined as follows: For all Pi,j ∈Ωh

0 let

Ωhm := {Qk,l are the midpoints between all neighboring Pi,j ∈ Ωh

0} (3.9)

Uhb := span

{ψi,j : ψi,j(Qk,l) = 1 for (i, j) = (k, l), and Qi,j ∈ Ω,

and = 0 else and, in particular, ∀Qk,l ∈ ∂Ω}�⊂ H1

0 (Ω).

Then obviously the a(uh

0 , φk,l

)in (3.8) are no longer defined. So we define a triangu-

lation, T h, of Ω: through all points Pi,j ∈ Ωh0 the parallels to the x- and y-axes and

to the diagonal x = y are drawn. Then let

T h := {all these constructed open rectangular triangles T}.

We introduce broken Sobolev spaces of functions and functionals on T h by

UT h : = VT h := H1(T h) := {u : Ω→ R : ∀T ∈ T h : u|T ∈ H1(T )} ⊃ U , (3.10)

U ′T h : = V ′

T h := H−1(T h) := {fh ∈ L(VT h ,R) : ∀T ∈ T h : fh|T ∈ H−1(T )} ⊂ U ′.


Then we generalize A, a(uh, vh), and∫Ωfvhdx in (3.8) by defining

Ah : UT h → U ′T h , ah(·, ·) : UT h × UT h → R, fh(·) : UT h → R, (3.11)

A = Ah|U , a(·, ·) = ah(·, ·)|U×U , ah(·, ·) := ah(·, ·)|Uh×Uh ,

〈Ahuh, vh〉U ′

T h×UT h:= ah(uh, vh) :=

∑T∈T h

∫T

(∇uh,∇vh)ndx,

fh ∈ U ′T h : 〈fh, v

h〉 := 〈fh, vh〉U ′

T h×UT h:=∑

T∈T h

∫T

n∑j=0

f jh∂

jvdx

with f(·) = fh(·)|U , fh(·) := fh(·)|Uh .

Finally, determine uh0 ∈ Uh

b = Vhb = span {ψk,l, Qk,l ∈ Ω} ⊂ UT h , cf. (3.9), from

ah(uh

0 , ψk,l

)=∑

T∈T h

∫T

(∇uh

0 ,∇ψk,l

)ndx = 〈fh, ψk,l〉U ′

T h×UT h∀ψk,l ∈ Uh

b . (3.12)

Note that for conforming FEMs all these new forms coincide, hence, for

conforming FEMs A = Ah, a(·, ·) = ah(·, ·) = ah(·, ·), f(·) = fh(·) = fh(·). (3.13)

Again this uh0 in (3.12) should be a good approximation for u0. �

Example 3.3. Finite difference method for the Laplacian in its strong form,Asu = Au = Δu, on Ω = [0, 1]2

We consider the strong form as A : Ub = C2(Ω) ∩H10 → V ′ = C(Ω). This unusual

notation V ′ = C(Ω) is a consequence of our aim, keeping the general discretizationapproach as close as possible to the usual Petrov–Galerkin methods, cf. below.25 Weevaluate Au in (3.5) in Pi,j ∈ Ωh, cf. (3.6), and approximate −(uxx + uyy)(Pi,j) bythe usual five-point star. This yields the difference approximation Ah, defining anapproximate solution uh

0 ∈ Uhb with uh

i,j := uh0 (Pi,j) as indicated in Table 3.1. Let

Uh : = {uh : Ωh → R} ⊃ Uhb := {uh ∈ Uh : uh|∂Ωh = 0},Vh = V ′h := {uh : Ωh

0 → R}

and ‖uh‖Uhb

:= maxPi,j∈Ωh

0

{|uh(Pi,j)|}, ‖fh‖V′hb

:= maxPi,j∈Ωh

0

{|(fh)(Pi,j)|}, (3.14)

uh0 ∈ Uh

b :(Ahuh

0

)(Pi,j) :=

1N2

(4uh

i,j − uhi−1,j − uh

i+1,j − uhi,j−1 − uh

i,j+1

)= f(Pi,j), 1 ≤ i, j ≤ N − 1, h := 1/N, uh(Pi,j) = 0 for i = 0, N or j = 0, N.

25 Then V would be {f : Ω → R, f of bounded variation}. At the same time, this example providesthe possibility of indicating Stetter’s [596] approach in the original form: He avoids testing by v ∈ Vb

and studies directly A : Ub → V = C(Ω), with the norms in (3.14).


Table 3.1: Stencil for the five-point star around Pi,j ∈ Ωh ⊂ R2 for−h2Δh.

0 −uhi,j+1 0 0 −1 0

−uhi−1,j 4uh

i,j −uhi+1,j abbreviated as −1 4 −1

0 −uhi,j−1 0 0 −1 0

Note that (3.14) directly defines an Ah : Uhb → V ′h without any detour as for

nonconforming FEMs. It will turn out to be obtainable as the strong form of a weakanalogue by a less familiar testing process in Chapter 8. To this end, we will discussdiscrete Sobolev spaces of grid functions. This will yield the more or less standardapproach.

A similar construction as for FEMs is possible: In the first step, we reduce Ω to Ωr :=[h, 1− h]2. Next, we approximate the previous A for functions u ∈ Ub = C2(Ω) ∩H1

0

by applying the five-point star in (3.14), thus defining

(Ahu)(P ) :=(five point star for u

)(P )∀P ∈ Ωr ⇒ Ahu(Pi,j) = (Ahu)(Pi,j), (3.15)

again well defined for uh ∈ Uhb . This will be modified in Example 3.8 and in

Chapter 8. �

Example 3.4. Summary of these discretizationsFor all three cases the original A : Ub → V ′

b, tested by elements of the space Vb, istransformed into Ah : Uh

b → V ′hb . All equations are tested with elements of Vh

b andspaces Uh

b ,Vhb with the same finite dimension. For the above difference method this is

realized by evaluation in Pi,j ; a more natural setting is presented in Chapter 8. Thereare still essential differences between Examples 3.1–3.3.

As in (2.20), (2.22), the a(u, v) in Example 3.1, (3.5) is Ub-coercive, and, by Uhb ⊂ Ub,

the a(u, v) in (3.8) is Uhb -coercive as well. This will imply the stability of the discrete

Ah with respect to the ‖ · ‖H1(Ω). For conforming FEMs the variational consistencyerror will be shown to vanish, and the classical consistency error tends to 0. This willimply the convergence in the sense of

∥∥u0 − uh0

∥∥H1(Ω)

→ 0 for h→ 0.

For the nonconforming FEMs in Example 3.2, and with H10 (T h) := {u ∈ H1(T h) :

u(Qi,j) = 0∀Qi,j ∈ ∂Ω} we show similarly as above that the a(u, v) in (3.11) is nowUh

b -coercive. However the proof for variational or classical consistency is much morecomplicated, cf. Subsections 5.5.2, 5.5.6, and 5.5.7. Still, we get the same convergence,now with respect to the discrete ‖ · ‖H1(T h).

For the difference methods in Example 3.3, (3.14), the matrix for Ah is a weaklydiagonal dominant matrix, a special case of an M -matrix. This implies the stability,cf. Definition 3.19, of Ah with respect to the discrete ‖ · ‖L∞(Ωh). The classical consis-tency is an immediate consequence of the local errors of the difference approximations,cf. Lemma 8.2. This allows convergence results with respect to the max norm, so


maxPi,j∈Ωh

{∣∣u0(Pi,j)− uh0 (Pi,j)

∣∣}→ 0 for h→ 0, cf. Definition 3.18. In contrast toExamples 3.1 and 3.2, with the optimal discrete ‖ · ‖H1(T h) norm, we use here,as in [596], the discrete ‖ · ‖L∞(T h) norm. However, we will prove in Chapter 8convergence for difference methods even in a discrete H1 Sobolev norm, ‖ · ‖H1(Ωh),obtaining a much better

∥∥u0 − uh0

∥∥H1

h(Ωh)→ 0. This is possible by transforming the

strong form (3.14) into a corresponding weak form, similar to (3.8), combined withgeneral discretization methods. �

Examples 3.1–3.4 motivate the definition of a general discretization theory, applica-ble to all intended analytical problems and their discretization methods. The previousforms of FEMs are special cases of the so-called Petrov–Galerkin methods.

Definition 3.5. Approximating spaces: Let U be a Banach space, and Ub ⊂ U , itsclosed subspace. Let the sequence {Uh}h∈H assign to h, 0 < h ∈ H ⊂ R, infh∈H h = 0the26 spaces Uh. Often we use the short notation of Uh. Then we call Uh (a sequenceof) approximating spaces for U if for well defined ||u||Uh , e.g. for conforming andnonconforming FEMs, and for DCGMs,

dist(u,Uh) := infuh∈Uh

||u− uh||Uh → 0 ∀u ∈ U for h→ 0. (3.16)

This is sometimes only required for u ∈ Ub and hence uh ∈ Uhb . The (sequence of)

approximating spaces Uhb for Ub, with b indicating the boundary conditions, are called

conforming spaces, if Uhb ⊂ Ub for all h ∈ H. Otherwise they are called nonconforming

spaces. In the latter case, either the Uhb violate the boundary conditions, or Uh �⊂ U .

The modifications for difference methods are discussed in Proposition 8.15 andRemark 8.16.

For difference methods ||u||Uh is not defined, so infuh∈Uh ||u− uh||Uh → 0 does notmake sense. Nevertheless, the grid functions there do approximate Sobolev spaces inan appropriate sense as well, cf. Subsection 8.5.2. This problem motivates introducingthe concept of external in contrast to internal approximation schemes, cf. Petryshyn[528–530,532], Temam [621] and Zeidler [677]. These concepts only play a minor rolefor the approach chosen here.

Definition 3.6. Admissible, conforming, nonconforming Petrov–Galerkin approxi-mating spaces and general Petrov–Galerkin methods:

1. A pair of admissible approximating spaces Uhb ⊂ Uh and Vh

b ⊂ Vh for Ub

and Vb, respectively, has to have the same finite dimensions

dim Uhb = dim Vh

b <∞ ∀ h. (3.17)

2. For admissible approximating spaces, the terms conforming and Galerkin andPetrov–Galerkin approximating spaces are used if Uh

b ⊂ Ub, Vhb ⊂ Vb and Uh

b =Vh

b and Uhb �= Vh

b , respectively. We denote Uhb �⊂ Ub, Vh

b �⊂ Vb or Uhb �⊂ U , Vh

b �⊂ Vas nonconforming (Petrov–Galerkin) approximating spaces. Sometimes, all these

26 We do not want to over-formalize the notation and have chosen h ∈ H, instead of the possiblehn = (h1, · · · , hn) ∈ Hn.


cases are denoted as generalized Petrov–Galerkin approximating spaces. For adiscretization method, see Definition 3.12, we use the same name as for its pairof approximating spaces.

3. As in Examples 3.1, 3.2, let the weak form of a linear, or more general quasilinearproblem be given as in (3.5). Discretize it by replacing the Ub,Vb in (3.5)by its approximating spaces Uh

b ,Vhb as in (3.8) or (3.12). Then we call this

transformation a Petrov–Galerkin (discretization) method.4. If, in addition to (3.16) for Ub, the Vh satisfy (3.16) ∀ v ∈ Vb and ∀ v′′ ∈ V ′′,

the bi–dual space of V, then we choose Vh = V ′′h and call Uh, Vh a bi–dual pairof conforming or nonconforming (Petrov–Galerkin) approximating spaces.

Remark 3.7. We will need the bi–duality only in Section 3.6. There we prove that fora bi–dual pair of approximating spaces, defining a stable Ah, this stability and technicalconditions imply the invertibility of A. This bi–duality condition is essentially satisfiedin separable reflexive spaces.27 The most important reflexive examples are Sobolevspaces, Wm

p (Ω), with 1 < p <∞. In these separable spaces the bi–duality is correct forall discretization methods studied here.

Example 3.8. Structure of generalized Petrov–Galerkin methods

1. The form of Petrov–Galerkin methods in Examples 3.1, 3.2 are not directlyapplicable to finite difference methods, e.g. in Example 3.3. However, we observethat the arguments and images of the original A and their discrete counterpartsAh are related by two types of projectors: Ph : Ub → Uh

b map functions ontotheir FE approximations or restrict them to their grid values on Ωh. Similarly,we reinterpret the right-hand sides of the conforming and nonconforming FEequations (3.8) and (3.12) as an orthogonal projection of Au or f onto, e.g.

Q′h : V ′

b → V ′hb : 〈Q′hf − f, vh〉V′h×Vh = 0∀vh ∈ V ′h

b , generally, (3.18)

Q′h : V ′

b ∪ V ′T h → V ′h

b : 〈Q′hf − f, vh〉V′T h×VT h

= 0∀vh ∈ V ′hb .

2. For the difference methods in ( 3.14), we use restriction operators, cf. Definition8.13,

Ph : U → Uh : Phu := {u(Pi,j) : 0 ≤ i, j ≤ N,h := 1/N} ∈ Uh, (3.19)

Q′h : V ′ → V ′h

b : Q′hf := {f(Pi,j) : 1 ≤ i, j ≤ N − 1} ∈ V ′h

b .

3. Then the discrete operators Ah for Examples 3.1–3.3, can be formulated as

Ah = Q′hAh|Uh

bcf. (3.8),(3.12), and Ah = Q

′hAh|Uhb

for (3.15). (3.20)

This Ah could, as well, be obtained by quadrature formulas for A.

27 Certainly, separability is not enough. This is shown by the well-known example of spaces ofinfinite sequences: c0 ⊂ l1 ⊂ l∞ = (c0)′′, where a countable basis for c0 is not a basis for l∞. However,a statement for W m

p (Ω), analogous to the c0 ⊂ l1 ⊂ l∞ = (c0)′′ situation does not seem to be known.


4. In a similar way, we define the nonlinear discrete operators, Gh, replacing A byG, as

Gh = Q′hGh|Uh

bcf. (3.8),(3.12), and Gh = Q

′hGh|Uhb

for (3.15). (3.21)

Here Ah : Uh ∪ UT h → V ′T h is either an extension, or Ah : Uh ∪ U → Vh ∪ V ′ is an

approximation for A : U → V ′. We will denote ΦhA := Ah as the discrete operatorsgenerated by Petrov–Galerkin methods as general discretization methods. Similarly,Gh : Uh ∪ UT h → V ′

T h is an extension, or Gh : Uh ∪ U → Vh ∪ V ′ is an approximationfor G : U → V ′, or both. �

The above Q′h : V ′

b → V ′hb or V ′ → V ′h

b will be shown to satisfy the necessaryconditions for a general discretization theory, cf. Definition 3.12, (3.24), even fornonconforming FEMs. However, for discussing their variational consistency and withAh : UT h → V ′

T h we have extended these Q′h into Q

′h : V ′b ∪ V ′

T h → V ′hb .

For Petrov–Galerkin methods we sometimes obtain useful properties for the projec-tors and approximating spaces.

Lemma 3.9. Let V = V ′′and Qh

inf : V → Vh be uniquely defined by∥∥v −Qhinfv∥∥V = inf

vh∈Vh‖v − vh‖V = min

vh∈Vh‖v − vh‖V . (3.22)

Then Q′h = Qh

inf .

Proof. For V = V ′′we have ‖v‖ = sup

v′∈V′〈v, v′〉Vh×V′ . Now, we choose in addition to

V ′h, annihilating Vh, an additional nontrivial v′v ∈ V ′ annihilating span{v,Vh} andlet V ′

v := span{V ′h, v′v

}. Then V ′

v annihilates Vhv := span{Vh, v}. This implies, with

V ′ = V ′v ⊕ V ′

comp where 〈v − v′, v′〉 = 0 ∀v′ ∈ V ′comp,∥∥v −Qh

infv∥∥V = min

vh∈Vhsup

v′∈V′|〈v − vh, v′〉|/‖v′‖V′

= minvh∈Vh

supv′∈V′

v

|〈v − vh, v′〉|/‖v′‖V′

= minvh∈Vh

supv′∈V′h

|〈v − vh, v′〉|/‖v′‖V′ = ‖v −Q′hv‖V ,

hence, the claim. �

Lemma 3.10. Let U be a separable Banach space, e.g. our Sobolev spaces. Let Uh

be approximating spaces for a dense subspace Ud of U with respect to the norm ‖ · ‖U .Then Uh are approximating spaces for U as well.

Proof. Let u ∈ U be given. Then ∀ε > 0 we can find ud ∈ Ud with ‖u− ud‖U < ε/2.For this ud there exists a uh ∈ Uh such that ‖uh − ud‖U < ε/2. �

We can extend this to the dual spaces and define the corresponding projectors:usually, the dual space V ′ of V consists of functions which admit approximatingsubspaces, V ′h. This is the background for


Lemma 3.11. Let V be a separable Banach space and V ′h be approximating spacesfor V ′. Finally, define Ph : V → Vh := (V ′h)′ by

for f ∈ V let 〈P hf − f, v′h〉V×V′ = 0 ∀ v

′h ∈ V ′h. (3.23)

Then Vh are approximating spaces for V, Ph is a projector and

limh→0

‖Phf − f‖V = 0 ∀f ∈ V.

Proof. With V the V ′ is separable as well. As a consequence of the separable V ′, wecan and do choose a basis ϕ′

i ∈ V ′ such that

V ′h = span{ϕ′

i, . . . , ϕ′i(h)

}.

Furthermore, define a dual basis ϕi for V, hence

ϕ′i ∈ V ′, ϕj ∈ V and

⟨ϕi, ϕ

′j

⟩V×V′ = δi,j .

Obviously, Vh := span{ϕi, . . . , ϕi(h)}. Then

f ∈ V ⇒ f =∑

i

⟨f, ϕ′

i

⟩ϕi, i.e.

∀ ε > 0 ∃ j = j(ε) and a h(ε) : ε >∥∥f − ∑

i≤j(ε)

⟨f, ϕ′

i

⟩ϕi

∥∥V

=∥∥ ∑

i>j(ε)

⟨f, ϕ′

i

⟩ϕi

∥∥V ≥

∥∥ ∑i>j(ε)

⟨f, ϕ′

i > ϕi : ϕi �∈ Vh(ε)⟩∥∥

V .

This Vh(ε) includes at least the ϕi, i ≤ j(ε), but usually some more. The above (3.23)implies

0 =⟨Phf − f, ϕ′

i

⟩∀ ϕ′

i ∈ V′h ⇔ ϕi ∈ Vh,hence we obtain

‖Phf − f‖V ≤ ‖∑⟨

f, ϕ′i

⟩ϕi : ϕi �∈ Vh‖V → 0 for h→ 0. �

Summarizing our examples, we have defined an operator Φh such that the original Ais discretized into Ah = ΦhA. This process has to be generalized to linear and nonlinearoperators A and G, respectively. Formalizing, we introduce general discretizationmethods. We modify Stetter’s [596] approach to fit the testing as in Petrov–Galerkinmethods and to the finite difference methods.

Definition 3.12. A discretization method M, applicable to a problem G : D(G) ⊂Ub → V ′

b is defined by a sequence of quintuples, M :={Uh

b ,Vhb , P

h, Q′h,Φh

}h∈H

:

1. Let U ,V be Banach spaces, Ub ⊂ U , Vb ⊂ V closed subspaces. To h, 0 < h ∈ H ⊂R, infh∈H h = 0 assign sequences {Uh,Vh}h∈H , of finite dimensional28 spaces.Often we use the short notation of Uh. The relation between the spaces is realizedby the bounded approximation and projection operators, Ph and Q

′h, usually

28 We do not want to over-formalize the notation and have chosen h ∈ H.


A,G tested by

(approximately) tested by

V ′b

V ′hb

A h,G h

U ⊃

P h (3.27)Q′h

Ubh Vb

h

Vb ⊂ VΦh

Ub

Figure 3.2 General discretization methods.

linear and generalized approximation and projection operators, respectively, suchthat, cf. Proposition 4.50, We use the notation V ′h = Vh′

, Q′h = Qh′

equivalentlythroughout the book.

Ph : Ub → Uhb , and Q

′h : V ′b → V ′h

b , s.t. Ph0 = 0, Q′h0 = 0, and (3.24)

limh→0

‖Phu‖Uh = ‖u‖U∀u ∈ U , limh→0

‖Q′hf‖hV′h

b

= ‖f‖V′b∀f ∈ V ′

b,

2. We need the mappings Φh defined for nonlinear mappings denoted asNL(Ub,V ′

b) =(Ub → V ′

b

). They might be defined on subsets of Ub.

Φh : (Ub → V ′b) →

(Uh

b → V ′hb

). (3.25)

If G ∈ D(Φh) ∀ h ∈ H,h < h0, then Φh or M is called a discretization applicableto G, and Gh := ΦhG the discretization or discrete operator for G.

3. Let the restriction of Φh to linear operators L be linear and satisfy

Φh : NL(Ub,V ′b)→NL

(Uh

b ,V′hb

),Φh|L :L (Ub,V ′

b)→L(Uh

b ,V′hb

)with (3.26)

Φh(cA) = cΦhA,Φh(A + A1) = ΦhA + ΦhA1,

Φh(A + f) = ΦhA + Φhf and Φh(G + G1) = ΦhG + ΦhG1

∀c ∈ R, A,A1 ∈ L (Ub,V ′b) , f ∈ V ′, G,G1 ∈ NL (Ub,V ′

b) , all in D(Φh)

Then we call Φh a linear discretization method.4. Often we use the short notation (for the infinite sequence of) Φh for indicating

the discretization method M.5. Whenever in the next chapters a problem is transformed and solved only in its

weak form we will choose U = V, Ub = Vb. This is no longer correct, e.g. for fullynonlinear problem, cf. Subsections 2.5.7, 2.6.8 and 5.2.3 ff.

Summarizing, we obtain the diagram in Figure 3.2 with uniformly bounded Ph, Q′h,

cf. e.g. Proposition 4.50.

Remark 3.13.

1. It has to be pointed out that the choice of Ph, Q′h for the given Uh

b ,V′hb

and problem (3.3) or (3.4) is certainly not unique. However the combinationUh

b ,V′hb , Ph, Q

′h,Φh should be chosen appropriately to yield consistency, see

3.3. Variational and classical consistency 185

below, with the highest possible order p. Otherwise the results are no longeroptimal, see, e.g. (3.32).

2. In Example 3.8, we have formulated the discrete operators Ah for Examples 3.1–3.3 with appropriate Ph, Q

′h,Φh, Ah, Ah, as

ΦhA = Ah = Q′hAh|Uh

bor ΦhA = Ah = Q

′hAh|Uhb. (3.28)

3. We have to show that the Ph, Q′h satisfy (3.24) for the case of generalized

Petrov–Galerkin methods. Then these ΦhA = Ah,ΦhG = Gh define general dis-cretization methods, in the sense of Definition 3.12.

Obviously, the generalized discretization methods and some of the following resultsin this chapter include nonlinear methods. The later applications to bifurcation numer-ics, however, require linear methods. All the generalized discretization methods, studiedhere, applicable to linear and nonlinear elliptic problems, are linear discretizationmethods. It is important that Φh acts on the different u dependent terms in A orG by applying linear operators. This holds even for more complicated cases, e.g. theFourier collocation derivative in spectral methods, cf. [114,115,120,123,124,158,631]for details. The above linearity condition (3.26) is violated for the (nonlinear) Runge–Kutta methods. They are general discretization methods in the above sense, but arenot relevant for our space discretizations in bifurcation and the small dimensionalcenter manifolds. Therefor we avoid all the difficulties with parabolic problems. Ourdefinition allows all forms of finite difference, finite element, discontinuous Galerkin,spectral, wavelet, and mesh–free methods, and approximating the discrete operatorsby quadrature formulas or difference approximations. These methods are necessary forstudying effects in space discretization for numerical bifurcation and center manifoldsin partial differential equations.

Definition 3.12 is a modification of [596] and is related to [607–609,677]. Differentlyfrom these, we study operators G : Ub → V ′

b and require, in some form, approximatingspaces Uh, Vh for U , V. This emphasizes the testing with the Vb, Vh

b and avoids,appropriately for all our methods, a direct consideration of the dual space V ′. Sowe do not have to distinguish the inner and outer discretization schemes, importantin [528,531,532,677].

Next, we have to study the relations between operators and their discretizations.

3.3 Variational and classical consistency

We want to know how well the discrete solution, uh0 , approximates the exact solution,

u0. One tool is an estimate of how well uh0 satisfies the original equation. We motivate

by our previous examples, two different, however, closely related concepts, the so-calledvariational and the classical consistency (errors) for linear and nonlinear operators.

For conforming FEMs we have Uhb ⊂ Ub, Vh

b ⊂ Vb; for nonconforming FEMs and fordifference methods the Uh

b , Vhb have to be modified as in Example 3.8. Accordingly,


we have defined the exact and discrete solutions u0 and uh0 , respectively,

u0 ∈ Ub : 〈Au0, v〉V′×V = a(u0, v) = 〈f, v〉V′×V ∀ v ∈ Vb and (3.29)

uh0 ∈ Uh

b :⟨Auh

0 , vh⟩V′×V = a

(uh

0 , vh)

=⟨f, vh

⟩V×V ∀v

h ∈ Vhb , conform. FE

uh0 ∈ Uh

b :⟨Ahu

h0 , v

h⟩V′h×Vh = ah

(uh

0 , vh)

=⟨fh, vh

⟩··· ∀v

h ∈ Vhb , nonc. FE, so

uh0 ∈ Uh

b : Q′h(Auh

0 − f)

= 0 for conf. FEs or Q′h(Ahu

h0 − f

)= 0 for nonconf. FEs

uh0 ∈ Uh

b : Q′h(Ahu

h0 − f

)= 0, for diff. meth., with the Ph, Q

′h in (3.18), (3.19).

Hence, the second equation represents a subset of the conditions in the first equationfor conforming FEs, the third and fourth equations for nonconforming Petrov–Galerkinand difference methods don’t.

For conforming FEs with a(u, v) = ah(u, v)∀u ∈ U , v ∈ V, we obtain

a(u0 − uh

0 , vh)

= 0 ∀ vh ∈ Vhb , Q

′h(Au0 −Auh

0

)= 0, (3.30)

hence, these variational consistency errors vanish, see below.For nonconforming FEs a direct comparison of Au0 and Ahuh

0 is impossible. Sowe have to employ the extended ah

(u0 − uh

0 , vh)

or Ahu0 −Ahuh0 . This variational

consistency error is nontrivial for nonconforming methods, and with Ah = Q′hAh|Uh

b

sup0�=vh∈Vh

b

{|ah

(u0 − uh

0 , vh)|/‖vh‖Vh

}=∥∥∥Q′hAhu0 −Ahuh

0

∥∥∥V′h

b

. (3.31)

This variational consistency error is the usual concept in the FE community. For thelinear case the results in Lemma 5.52 will allow estimates of the form∥∥uh

0 − u0

∥∥Uh ≤ C

(dist

(u0,Uh

b

)+ sup

0�=vh∈Vhb

|ah

(u0 − uh

0 , vh)|

‖vh‖Vh

). (3.32)

However, this variational consistency error differs from the classical consistency error,Q

′hAu0 −AhPhu0, cf. Definition 3.14, (3.33), (3.35), and [596]. The latter is moreappropriate for using the whole nonlinear machinery. We aim for the relation of thevariational consistency to the classical consistency error. Again, we use the generalnotation of Ah for the different cases, unless the difference has to be emphasized.

To compare both “consistencies”, we modify (3.31) in operator form as Q′hAhu0

−Ahuh0 . This can be generalized to arbitrary (smooth enough) u, if we introduce

fu := Au and the discrete solution uh of Ahuh = Q′hfu := Q

′hAu:

Q′hAhu−Ahuh︸︷︷︸var.cons.error

= Ah(Phu− uh) + Q′hAhu−Q

′hAhPhu (3.33)

= AhPhu−Q′hfu + Q

′hAhu−Q′hAhP

hu

= AhPhu−Q′hAu︸︷︷︸

class.cons.error

+Q′hAhu−Q

′hAhPhu︸︷︷︸

related to interpol.error

.

3.3. Variational and classical consistency 187

As indicated, the term Q′hAhu−Ahuh left in the first line in (3.33) and the first

term in the last line, AhPhu−Q′hAu, are called the variational and classical consis-

tency error for the linear operator A. For linear Q′h we find Q

′hAhu−Q′hAhP

hu =Q

′hAh(u− Phu), hence ‖Q′hAh(u− Phu)‖ ≤ C‖u− Phu‖ is bounded by the inter-polation error. For difference methods we simply replace the Ah in (3.33) by Ah,however, the variational consistency error, Q

′hAhu−Ahuh, is only defined for smoothenough u. There is no problem with the classical consistency error, AhPhu−Q

′hAu =Q

′hAhPhu−Q

′hAu.This (3.33) has to be slightly modified for nonlinear operators. Again, we introduce

fu := Gu and assume uh, a discrete solution of Ghuh = Q′hfu = Q

′hGu. From thepossibly many solutions uh for the nonlinear G we choose that with small ‖uh − u‖for an appropriate norm. In general notation we find with Gh = Q

′hGh|Uhb

in (3.21),

Q′hGhu−Ghuh︸︷︷︸var.cons.error

= (GhPhu−Q′hGu)︸︷︷︸

class.cons.error

+ (Q′hGhu−Q

′hGhPhu)︸︷︷︸

related to interpol.error

(3.34)

Again, for difference methods the Q′hGhu only makes sense for smooth enough u.

Similarly, for the general discretization methods in Definition 3.12, with Gh = ΦhG,the classical discretization error, GhPhu−Q

′hGu = (ΦhG)Phu−Q′hGu, is always

defined, but the other terms are usually not defined. So we introduce in the followingdefinition the classical discretization error for general discretization methods. Next,we discuss, for generalized Petrov–Galerkin methods, the variational and classicaldiscretization errors in Example 3.15.

Definition 3.14. Classical consistency, local discretization error:

1. Under the conditions of Definition 3.12 for a general discretization method

(ΦhA)Phu−Q′hAu = AhPhu−Q

′hAu and (ΦhG)Phu−Q′hGu (3.35)

or its norm ‖AhPhu−Q′hAu‖V′h

band ‖GhPhu−Q

′hGu‖V′hb,

for A and G, respectively, are called the (classical) consistency error or localdiscretization error in u ∈ D(G) ∩ Ub . Note that for the exact solution u0 ofG(u0) = 0 the classical consistency error reduces to ‖GhPhu0‖V′h

b.

2. A method is called classically consistent for A or G in u, if

‖AhPhu−Q′hAu‖V′h

b→ 0 or ‖GhPhu−Q

′hGu‖V′ → 0 for h→ 0.

3. If even

‖AhPhu−Q′hAu‖V′h

bor ‖GhPhu−Q

′hGhu‖V′ = O(hp),

we call it consistent of order p in u. These O(hp) are usually estimated byChp‖u‖W k,q(Ω) with appropriate p, k, q.

Example 3.15. Consistency errors

1. As in Example 3.8 we choose ah(·, ·) and the discrete operators Ah = Q′hAh|Uh

b

and Gh = Q′hGh|Uh

bby applying a generalized Petrov–Galerkin method to A and


G. Let u and uh be related as

Ahuh = Q′hfu := Q

′hAu and Ghuh = Q′hfu := Q

′hGu

for the linear and nonlinear operator A and G, respectively. Then

sup0�=vh∈Vh

b

{|ah(u− uh, vh)|/‖vh‖} = ‖Q′hAhu−Ahuh‖V′hb

and (3.36)

‖Q′hGhu−Ghuh‖V′hb, for A and G, respectively,

is called the variational consistency error in u.29 A method is called variationallyconsistent for A in u, if

sup0�=vh∈Vh

b

{|ah(u− uh, vh)|/‖vh‖} = ‖Q′hAhu−Ahuh‖V′hb→ 0, (3.37)

similarly for the nonlinear G, e.g. ‖GhPhu−Q′hGu‖V′h → 0 for h→ 0.

2. Note that for all these cases the variational errors vanish for conforming FEMs.

�

The variational consistency error usually is only considered and estimated forthe exact solution, u0, or if necessary for an approximant Phu0, see Section 5.5for estimates. A generalization to arbitrary u is formulated in (3.33). The classicalconsistency errors are often studied for general u. There is the simple relation for thedifference of variational and classical consistency errors in (3.33) and (3.34). In fact, seeProposition 5.47 and Theorem 3.17, this difference is, under appropriate conditions,bounded as O(‖u− Phu‖Uh) → 0 ∀u ∈ Ub ∀h→ 0. Hence, they even have the samesize if both dominate or have the same size as the interpolation error ‖u0 − Phu0‖U .Therefore it suffices to estimate the more convenient of the two consistency errors.

The general case needs the continuity or even Lipschitz-continuity of the Gh, acondition satisfied for all the examples studied here, simultaneously with G. We assumeequicontinuous and equibounded Q

′h, cf. Theorem 4.54.

Theorem 3.16. Conforming classical consistency, and interpolation errors:

1. Let ΦhG = Gh = Q′hG|Uh

b: D(Gh) = Uh

b ∩ D(G) be a conforming FEM and let

uh ∈ D(Gh) be given. Assume G is Lipschitz-continuous in Br(u) and Q′h

equicontinuous and equibounded. Then the classical consistency error satisfies

‖GhPhu−Q′hGu‖V′h

b= ‖Q′hGPhu−Q

′hGu‖V′hb≤ L‖(Phu− u)‖Uh . (3.38)

2. If Ph : Ub → Uhb is chosen as the interpolation or approximation operator, then

the classical consistency error vanishes as, e.g. the interpolation error.

29 We sometimes use the notation ‖Q′hAhu − Ahuh‖V′hb

instead of ‖Q′hAhu − Ahuh‖V′hb

to

emphasize that this Q′hAhu − Ahuh is tested by the vh ∈ Vh

b . This notation in (3.36) is used similarlyfor the isoparametric variants.

3.4. Stability and consistency yield convergence 189

Theorem 3.17. Nonconforming variational, classical consistency, interpolationerrors:

1. For the case of nonconforming FEs we extend the original linear A : U → V ′

and the nonlinear G : D(G) ⊂ U → V ′ as Ah : UT h → V ′T h and Gh : D(Gh) ⊂

UT h → V ′T h . Assume Ah and Q

′h to be equibounded and equicontinuous and(G ∈ CL(D(G)) usually implying) Gh ∈ CL(D(Gh)), with Lipschitz constant L.Furthermore, let Ah = Q

′hAh|UT h, Gh = Q

′hGh|UT hand u ∈ D(G), uh, Phu ∈

D(Gh), and uh be the discrete approximation for u from Ghuh = Q′hGhu

h =Q

′hfu = Q′hGu or its analogue for the linear A,Ah.

2. Then the difference between the variational and classical discretization errors

(Q′hGhu−Ghuh)− (GhPhu−Q

′hGu) = (Q′hGhu−Q

′hGhPhu) (3.39)

∀ u ∈ D(G), uh, Phu ∈ D(Gh)

can be estimated ∀ u ∈ U as

‖Q′hGhu−Q′hGhP

hu‖V′hb, ‖Q′hAhu−Q

′hAhPhu‖V′h

b≤ O(‖Phu− u‖UT h

).

3. If only G ∈ C(D(G)), Gh ∈ C(D(Gh)), then

‖Q′hGhu−Q′hGhP

hu‖V′hb≤ ωGh

(‖Phu− u‖UT h),

with the modulus of continuity of Gh,

ωGh(t) := sup

‖x−y‖UT h≤t, x,y∈D(Gh)

‖Ghx−Ghy‖V′hb.

4. For nonconforming FEs and DCGMs the estimates for variational and classicaldiscretization errors are rather technically involved, cf. Section 5.5 and Chapter 7.

Proof. With the equibounded and equicontinuous Q′h and Gh ∈ CL

(Ub,c + Uh

b

),

u ∈ D(G), uh, Phu ∈ D(Gh), the first estimate is obvious. The ωGh(‖Phu− u‖Uc

)modification is straightforward. �

3.4 Stability and consistency yield convergence

We want to unfold this well-known fact for the general discretization methods intro-duced in Definition 3.12. We combine consistency introduced in Section 3.3 with themissing convergence and stability to be defined now. An appropriate interplay ofthe different concepts of consistency, convergence, stability, the operators Ph, Q

′h,the norms, the different approximations by spaces and operators is the basic conditionfor the success of this approach. These choices are by no means unique. Stetter [596]gives some examples for failing combinations. In Theorem 3.21 we prove the existenceof a discrete solution and its convergence towards the exact solution essentially fora stable and consistent discretization as in Definition 3.12. This includes our generalPetrov–Galerkin methods, cf. Remark 3.13. Except in Section 3.7, we skip, in the


remainder of this chapter, the condition of a “linear” Φh, see (3.25), and study thegeneral discretization methods.

Convergence is expected to describe a discrete solution, uh0 , well approximating the

exact solution, u0. In some methods we can directly compare uh0 and u0 since for both

‖ · ‖Uh , and hence∥∥u0 − uh

0

∥∥Uh is well defined with hopefully

∥∥u0 − uh0

∥∥Uh → 0 for

h→ 0. Theoretically, two other measures are possible as well. We might consider∥∥Phu0 − uh0

∥∥Uh with our known Ph : U → Uh, e.g. for difference methods. Or we

might study∥∥u0 − Phu

h0

∥∥U and need an additional Ph : Uh → U . We apply this Ph,

essentially defined by an anticrime transformation, cf. Lemma 5.77, to nonconformingFEs for proving stability of linearized elliptic operators.

Definition 3.18. Global discretization error, convergence:

1. Let G : D(G) ⊂ Ub → V ′b and its discretization Gh = ΦhG : D(Gh) ⊂ Uh

b → V ′hb

have the (locally) unique exact and approximate solutions u0 and uh0 , respectively.

Then (the sequence)

uh0 − u0 or uh

0 − Phu0 (3.40)

is called the (global) discretization error of Φh for G.2. The uh

0 are called convergent to u0 and convergent of order p (sometimes, Gh =ΦhG is called convergent to G in u0) if for h→ 0, h ∈ H, respectively∥∥u0 − uh

0

∥∥Uh → 0 or

∥∥Phu0 − uh0

∥∥Uh → 0 and (3.41)∥∥u0 − uh

0

∥∥Uh = O(hp) or

∥∥Phu0 − uh0

∥∥Uh = O(hp).

3. As in Definition 3.5, modifications are necessary for difference methods.

We want to prove convergence. This will not be possible directly. So we try toderive it from the local discretization error. Consistency alone will not do it, cf. [596]for counterexamples. Let us motivate what is missing: G(u0) = 0 and Gh

(uh

0

)= 0

have to be solved and Phu0 and uh0 have to be compared. The classical consistency,

see (3.35), in u0 with Gu0 = 0 implies

GhPhu0 −Q′hGu0 = GhPhu0 −Q

′h0 = GhPhu0 → 0 ∀ h → 0, h ∈ H. (3.42)

If (Gh)−1 is defined in and close to GhPhu0, then with Ghuh0 = 0,

Phu0 − uh0 = (Gh)−1GhPhu0 − (Gh)−10.

For a vanishing consistency error in u0 and a continuous or even Lipschitz-continuous(Gh)−1 this implies∥∥Phu0 − uh

0

∥∥Uh → 0 or ≤ L‖Gh(Phu0)‖V′h

b→ 0 ∀ h → 0, h ∈ H. (3.43)

This kind of perturbation insensitivity is called stability.

Definition 3.19. Stability: Under the conditions of Definition 3.12 we assume a non-linear and linear operator Gh : D(Gh) ⊆ Uh

b → V ′hb , and Bh : Uh

b → Vhb , respectively,


to be assigned to every h ∈ H (usually defined by a discretization method M). For uh

and h0 > 0 let

Br(uh) = {vh ∈ Uh : ‖vh − uh‖Uh < r} ⊂ D(Gh) ∀h < h0, h ∈ H.

Furthermore, let h0, r, S ∈ R+ be fixed constants, such that, uniformly in h ∈ H,h <h0:

uhi ∈ Br(uh), i = 1, 2,=⇒

∥∥uh1 − uh

2

∥∥Uh ≤ S

∥∥Gh(uh

1

)−Gh

(uh

2

)∥∥V′h

b

. (3.44)

For Bh or Bh ·+fh this reduces to (for any r and uh, but for h < h0)

(Bh)−1 ∈ L(V ′h

b ,Uhb

)exists and ||(Bh)−1||Uh

b ←V′hb≤ S.

Then Gh and Bh are called stable in uh and stable, respectively. The S and r arecalled stability bound and stability threshold, respectively.

The proof of stability is simple for some cases; for general problems it is usuallyrather complicated. For example, for the conforming FEs in Example 3.1 it is animmediate consequence of the coercivity and the Brezzi–Babuska condition in Theorem2.12. For more general cases we will discuss in Theorems 3.23–3.29, conditions whichguarantee stability. For example, it allows reducing stability for the nonlinear problemto that of the linearized problem, and from a linear coercive operator to its compactperturbations. In Theorem 2.122 we discuss the implication of Hm

0 (Ω) ⇒Wm,p0 (Ω)-

coercivity for p < 2 and �⇒Wm,p0 (Ω)-coercivity for 2 < p. In modified form these

arguments are applicable to the (linearized) Navier–Stokes operator as well.In [596] the first inequality in (3.44), uh

i ∈ Br(uh), is replaced by the weaker

‖Gh(uh

i

)−Gh(uh)‖V′h

b< R, i = 1, 2. (3.45)

Since for nonlinear G (and hence Gh) this (3.45) may be satisfied for large∥∥uh − uh

i

∥∥Uh

we have chosen the stronger assumption in (3.44), cf. [596], Corollary 1.2.2.

Proposition 3.20. In the discretization Gh : D(Gh) ⊂ Uhb → V ′h

b let the Gh be con-tinuous in Br(uh) ⊂ D(Gh) and satisfy (3.44). Then these uh

i satisfy (3.45) withR = r/S, and thus are stable in this weaker sense.

Assuming continuity, consistency and stability and some technicalities, we will nowprove the existence of a discrete solution uh

0 for Gh(uh

0

)= 0, defined by the general

discretization methods in Definition 3.12, cf. [596].

Theorem 3.21. Let the original problem (3.3) have the exact solution u0. Let Gh :D(Gh) ⊂ Uh

b → V ′hb , h ∈ H, be its discretization, see Definition 3.12 (here nonlinear

Φh are allowed) and satisfy the following conditions:

1. Gh = ΦhG : D(Gh) ⊂ Uhb → V ′h

b is defined and continuous in Br(Phu0) ⊂ D(Gh)with r > 0 independent of h;

2. Gh is (classically) consistent with G in Phu0;3. Gh is stable for Phu0.


Then the discrete problem Gh(uh) = 0 possesses the unique solution uh0 ∈ D(Gh) ⊂ Uh

b

near u0 for all sufficiently small h and uh0 converges to u0. If Gh is consistent and

consistent of order p, then uh0 converges and converges of order p, hence∥∥uh

0 − u0

∥∥Uh ≤ S‖Gh(Phu0)‖V′h and ≤ O(hp) for h→ 0, h ∈ H, respectively (3.46)

As essential tool for the proof is the following.

Proposition 3.22. For a fixed r > 0 and a sequence uh, let the discrete

Gh : Br := Br(uh) → V ′hb , Gh ∈ C(Br(uh)) be continuous.

Furthermore for S ∈ R+ let

uhi ∈ Br, i = 1, 2, imply

∥∥uh1 − uh

2

∥∥Uh ≤ S

∥∥Ghuh1 −Ghuh

2

∥∥V′h

b

. (3.47)

Then Gh is invertible on

B′R =

{v

′h ∈ V ′hb : ‖v′h −Ghuh‖V′h

b< R

}, with R := r/S. (3.48)

This (Gh)−1 : B′R → Uh

b is Lipschitz-continuous with Lipschitz-bound S.

Proof. By (3.47) the Gh are invertible on every set M ⊂ Gh(Br). In fact, if uh1 , u

h2 ∈ Br

have the same images Gh(uh

i

)= v

′h ∈M, i = 1, 2, then (3.47) implies∥∥uh2 − uh

1

∥∥Uh ≤ S

∥∥Ghuh2 −Ghuh

1

∥∥V′h

b

= S · 0, such that uh2 = uh

1 . (3.49)

Hence, it suffices to prove

B′R ⊂ Gh(Br). (3.50)

If this were incorrect, a v′h1 ∈ B′

R\Gh(Br) would exist. We define

v′h(λ) := (1− λ)Ghuh + λv

′h1 , λ ≥ 0 for v

′h1 ∈ B′

R\Gh(Br). (3.51)

This implies v′h(0) = Ghuh ∈ Gh(Br), v

′h(1) = v′h1 �∈ Gh(Br). With

Λ := {λ′ > 0 : v′h(λ) ∈ Gh(Br) for λ ∈ [0, λ′)}

we define

λ :=

{supλ∈Λ λ for Λ �= ∅,

0 for Λ = ∅. (3.52)

The inclusion (3.50) would be proved if we could show that λ > 1. We construct acontradiction to λ ≤ 1 in two steps: first we show vh := v

′h(λ) ∈ Gh(Br). This allowsconstructing the contradiction in step 2.

For λ = 0 we automatically have vh = v′h(0) = Ghuh ∈ Gh(Br). For the comple-

mentary case 0 < λ ≤ 1 we have by definition v′h(λ− ε) ∈ Gh(Br), see (3.52) for any

0 < ε < λ and, see (3.51),

v′h(λ− ε) ∈ B′

R, since ‖v′h(λ− ε)−Ghuh‖V′hb

<∥∥∥v′h

1 −Gh(uh)∥∥∥V′h

b

< R.


Therefore an uhε ∈ Br exists such that Ghuh

ε = v′h(λ− ε), see (3.52). A combination

with (3.47) and (3.51) shows the property of a Cauchy sequence∥∥uhε − uh

ε

∥∥Uh ≤ S

∥∥Ghuhε −Ghuh

ε

∥∥V′h

b

= S‖v′h(λ− ε)− v′h(λ− ε)‖V′h

b→ 0

for ε, ε > 0 and ε→ ε.

Hence the (Gh)−1(v′h(λ− ε)) exist and we have convergence for the uh

ε , v′h(λ− ε),

and, by (3.47), for the (Gh)−1(v′h(λ− ε)) as well for ε→ +0. This yields

uh0 := lim

ε→+0uh

ε = limε→+0

(Gh)−1(v′h(λ− ε)).

This uh0 is ∈ Br: in fact,

‖(Gh)−1(v′h(λ− ε))− uh‖Uh ≤ S‖v′h(λ− ε)−Ghuh‖V′h

b

= S‖(λ− ε)(v

′h1 −Ghuh

)‖V′h

b.

and, see (3.51), ‖v′h1 −Ghuh‖V′h

b< R, independent of ε. With 0 < λ− ε < 1, we obtain

the following inequality

‖(Gh)−1(v′h(λ− ε))− uh‖Uh < S(λ− ε)R ≤ (λ− ε)r ≤ r, hence

∥∥uh0 − uh

∥∥Uh < r.

Finally Gh is continuous in Br and we have reached our first goal

vh := v′h(λ) = Ghuh

0 ∈ Gh(Br).

This allows the contradiction: since uh0 ∈ Br and Gh is continuous in Br, there exists

a closed ball B = Bδ

(uh

0

)⊂ Br with Gh(B) ⊂ B′

R. For this B the Gh : B → Gh(B) ⊂B′

R is injective, as we have seen already in (3.49).Since dimUh <∞ then B is compact. Now Gh|B is an injective continuous

mapping with compact domain B. Then (Gh)−1 : Gh(B) → B is defined and con-tinuous as well, see e.g. [560], [74], p. 56. In particular, with compact S :={uh ∈ Uh

b :∥∥uh − uh

0

∥∥Uh = δ

}the injectivity implies

ρ := minuh∈S

∥∥Ghuh0 −Ghuh

∥∥ > 0,

and hence

Ghuh0 = vh ∈ B0

ρ = B0ρ(vh) ⊂ Gh(B) ⊂ Gh(Br) with open B0

ρ.

Then the above λ cannot be the supremum defined in (3.52). This is our contradiction.So either λ > 1 or v

′h1 ∈ Gh(Br).

The Lipschitz-continuity for (Gh)−1 and the Lipschitz constant S are immediateconsequences from (3.47). �

Proof of Theorem 3.21. The conditions 1, 3 imply by Proposition 3.22 the exis-tence of a unique discrete solution uh

0 ∈ Br(Phu0): in fact we use uh = Phu0 andGhuh = GhPhu0 in (3.48), the local discretization error GhPhu0, see Definition 3.14.


Furthermore, R is independent of h since r, S are independent of h. The consistencyof the method implies

‖GhPhu0‖V′hb→ 0 for h→ 0.

Therefore 0 ∈ B0R(GhPhu0) for h < h1 sufficiently small and uh

0 = (Gh)−10 is uniquelydetermined for h < h1.

In the next step we show convergence and convergence of order p for this discretesolution. For h < h1 let uh

0 be the unique solution of Ghuh0 = 0. Now we use the global

discretization error εh = uh0 − Phu0, see Definition 3.18, obtaining

Ghuh0︸︷︷︸

=0

−GhPhu0 = Gh(Phu0 + εh)︸︷︷︸=0

−GhPhu0 = −GhPhu0.

Gh is stable for Phu0 hence

‖εh‖Uh =∥∥uh

0 − Phu0

∥∥Uh ≤ S‖GhPhu0‖V′h

bfor ‖GhPhu0‖V′h

b< R.

Since Gh is consistent with G in Phu0, then ‖GhPhu0‖V′hb→ 0. This implies, for

h < h2, that ‖GhPhu0‖V′hb

< R and hence

‖εh‖Uh ≤ S‖GhPhu0‖V′hb→ 0 for h→ 0.

This shows the convergence and convergence of order p for this method. �

3.5 Techniques for proving stability

The following theorems formulate known and well applicable criteria for stability. Westart discussing the relation between stability of a nonlinear operator and its linearizedform and with perturbed nonlinear problems. Most importantly, we combine a coerciveprincipal part with its compact perturbations, cf. Theorem 3.29. This allows, for allproblems and methods treated in both books, the proof of stability for (nonlinear)operators. Their linearizations have to be boundedly invertible in the exact solution,implying locally unique solutions. Natural extensions to the bifurcation situation arevalid as well. Here we prove Theorem 3.29 only for the case of conforming FEMs.Generalizations to nonconforming methods require specific additional information,available in the corresponding chapters.

Theorem 3.23. Under the conditions of Definition 3.12 let a discretization Φh,Gh = ΦhG : D(Gh) ⊂ Uh

b → V ′hb and uh ∈ D(Gh) ⊂ Uh be given. Assume Gh ∈

C1(Br(uh)), Br(uh) ⊂ D(Gh), with fixed r > 0. Furthermore, for all vh ∈ Br(uh) letthe inverse of the derivative, ((Gh)′(vh))−1, exist for h < h0 and

‖((Gh)′(vh))−1‖Uhb ←V′h

b≤ S ∀vh ∈ Br(uh) uniformly for h0 > h ∈ H. (3.53)

1. Then the discretization Gh is stable at uh with stability bound S and stabilitythreshold r0 = r/S.

3.5. Techniques for proving stability 195

2. The above ((Gh)′(vh))−1 in Br(uh) can be replaced by ((Gh)′(uh))−1 inBs(uh) with s < r and, for bounded Φh, the Gh ∈ C1(Br(uh)) by G ∈C1(Br(u)), Br(u) ⊂ D(G) and S by S/2 in (3.53).

Proof. The existence of an inverse operator (Gh)−1 : B′R → Uh

b in the balls B′R =

B′R(Ghuh), R = r/S, is shown as for Proposition 3.22. B′

R ⊂ V′hb is an open

neighborhood of Ghuh with respect to V ′hb . Then Theorem 1.48 guarantees the

differentiability of this inverse operator (Gh)−1. Moreover it yields, uniformly forh0 > h ∈ H,

‖((Gh)−1)′(Ghvh)‖Uhb ←V′h

b= ‖((Gh)′(vh))−1‖Uh

b ←V′hb≤ S (3.54)

implying

‖((Gh)′)−1(δ)‖Uhb ←V′h

b≤ S ∀ δ ∈ B′

R(Ghuh), h0 > h ∈ H. (3.55)

Now B′R(Ghuh) is convex, hence, Ghuh

2 + λ(Ghuh

1 −Ghuh2

)∈ B′

R(Ghuh) for 0 ≤ λ< 1 ∀ Ghuh

1 , Ghuh

2 ∈ B′R(Ghuh). For these uh

i and with the existence of ((Gh)−1)′

Theorem 1.43 yields∥∥uh1 − uh

2

∥∥Uh

b

=∥∥(Gh)−1

(Ghuh

1

)− (Gh)−1

(Ghuh

2

)∥∥ |Uhb

≤ maxδ∈B′

R(Ghuh)‖((Gh)−1)′(δ)‖Uh

b ←V′hb

∥∥(Ghuh1

)−Ghuh

2

∥∥ |V′hb

≤ S∥∥Ghuh

1 −Ghuh2

∥∥V′h

b

by (3.55),

hence the claim. Theorem 1.46 and bounded Φh imply the modifications. �

Theorem 3.24. Stability of a perturbed nonlinear operator: Under the conditions ofDefinition 3.12 let G,F,G + F ∈ D(Φh), the discretization Gh := Gh + Fh : D(Gh) ∩D(Fh) ⊂ Uh

b → V ′hb and a sequence uh ∈ Uh

b be given. Assume S, r, L > 0 exist, inde-pendent of h, such that Br(uh) ⊂ D(Gh) ∩ D(Fh) and

1. Gh are continuous and stable in Br(uh) with the stability bound S;2. Fh are Lipschitz-continuous in Br(uh), with∥∥Fh

(uh

1

)− Fh

(uh

2

)∥∥V′h

b

≤ L∥∥uh

1 − uh2

∥∥Uh and L · S < 1.

Then Gh is stable in uh, with bound S/(1− LS) and threshold (1− LS)r/S.

Proof. For all h ∈ H and uhi ∈ Br(uh) we have∥∥Ghuh

1 − Ghuh2

∥∥V′h

b

=∥∥(Ghuh

1 −Ghuh2

)∥∥V′h

b

−∥∥Fhuh

1 − Fhuh2

∥∥V′h

b

≥ (1/S − L)∥∥uh

1 − uh2

∥∥Uh ,

hence the claim. �

Stetter [596] proves the following modification:


Theorem 3.25. Under the conditions in Theorem 3.24 replace 1., 2. there by

1. Gh : Br(uh) → Gh(Br(uh)) are differentiable, one-to-one and Gh(Br(uh)) areconvex with

‖((Gh)′(vh))−1‖Uhb ←V′h

b≤ S ∀ vh ∈ Br(uh). (3.56)

2. For all vh, uh1 , u

h2 ∈ Br(uh) and with L′S < 1 let

‖((Gh)′(vh))−1(Fh(uh

1

)− Fh

(uh

2

))‖Uh ≤ L′ ∥∥uh

1 − uh2

∥∥Uh . (3.57)

Then again Gh is stable in uh, with bound S/(1− L′) and threshold (1− L′)r/S.

In the previous sections two related conditions compete with each other fornonlinear operators, G. We required well defined continuous G for consistencyresults and Gh for convergence. The continuity of G : D(G) ⊂ U → V ′ can often beavoided, since consistency requires smoother functions u ∈ D(G) ∩ Us, where G isin fact well defined and continuous, cf. Theorems 4.53 and 5.4 We will discuss somepossibilities in Example 3.26 below. The continuity of Gh : D(Gh) ⊂ Uh → V ′h ismuch less restrictive than that of G, since the uh ∈ Uh have a very special structure,e.g. piecewise polynomials for FE, DCG and wavelet methods on appropriatetriangulations, or algebraic or trigonometric polynomials of a finite degree on the wholeor domain decomposed Ω. For these concepts we refer to the corresponding chaptersbelow.

The condition of continuous Gh : D(Gh) ⊂ Uh → V ′h is unavoidable in the contextof piecewise polynomial methods for fully nonlinear elliptic operators G on Ω. Butit is applicable as well for other nonlinear operators G. E.g. a combination with thestability results in Section 3.6, implying boundedly invertible derivatives G′(u0), allowsextended search for discrete and related exact solutions of quasilinear problems. Theseresults might relax the conditions imposed, e.g. in Chapters 4 and 5.

Example 3.26. The G,Gh dilemma for the Monge-Ampere operator. G,Gh have tobe well defined and continuous, cf. Example 2.78: We recall

G(u) := det∇2u− f(x, u,∇u) and = uxxuyy − u2xy − f(x, u,∇u) for n = 2. (3.58)

Let Df := {u : f(x, u(x),∇(x)) is defined a.e. in Ω}. Obviously, this G(u) is welldefined for u ∈ D(G) := Df ∩ C2(Ω), but certainly is �∈ L2(Ω) for u ∈ Df ∩H2(Ω). Sofor (3.59), we choose the domain, D(G) ⊂ Df , as dense subset, here as C2(Ω) or H4

+(Ω)of H2(Ω) and the range, R(G) ⊂ L2(Ω), cf. Theorems 2.79 ff. Another possibilityis D(G) = Df ∩ {u ∈ H2(Ω) ∩W 2,∞(Ω)}. Then the exact solution, u0, satisfies thedifferential equation, and the boundary condition in the trace sense, cf. Subsections5.2.4, 5.2.5. We assume D(G) to be defined s.t.

G : D(G) ⊂ U := H2(Ω) → R(G) ⊂ V := L2(Ω) = V ′ : G(u) := Gw(·, u,∇u,∇2u),

s.t. ∀u ∈ D(G) :G(u) ∈ L2(Ω), G(u0)(x)=0∀ a.e. x ∈ Ω,⇔ (G(u0), v)L2(Ω) =0∀v∈V,

and u0|∂Ω = 0 ⇐⇒ (u0|∂Ω, vb)L2(∂Ω) = 0 ∀vb ∈ L2(∂Ω), (3.59)


with Gw : Ω× R× Rn × Rn2 → R, (see (2.304)). For this second order fully nonlinearproblem (3.59) on a polyhedral Ω, we will choose Uh ⊂ C1(Ω), with piecewise poly-nomials, Pn

d of degree d in n variables, and a triangulation, T h e.g. with simplices T ,cf. Definition 4.2. So

for a polyhedral Ω and a triangulation for Ω = ∪T∈T hT ⊂ Rn let

Uh = {uh : uh|T ∈ Pnd } ∩ C1(Ω), Uh

b = {uh ∈ Uh : uh|∂Ω = 0}. (3.60)

For this Ω, and uh0 ∈ Uh

b ⊂ H2(Ω) ∩H10 (Ω), the FE solution, uh

0 , is determined bytesting only the differential equation with respect to vh ∈ Vh, cf. Section 5.2.

uh0 ∈ (Uh

b = Vh) ∩ D(G) s.t.(G(uh

0

), vh)L2(Ω)

= 0 ∀vh ∈ Vh. (3.61)

�Proposition 3.27. For the three cases of u ∈ D(G) ⊂ Df in the previous Examplethe G(u) ∈ L2(Ω) and for uh ∈ Uh ∩ Df in (3.60) the G(uh) ∈ L2(Ω) are well defineda.e. for the Monge-Ampere equation in (3.58) and both G and Gh are continuousw.r.t. the G : H2(Ω) → L2(Ω) norms. Generalized to curved Ω, and to the order 2mwith Uh ⊂ C2m−1(Ω), this remains correct and is proved exactly the same way.

Proof. We formulate the proof only for G(uh) and n = 2. It is well known that foru ∈ H2(Ω) the uxxuyy �∈ L2(Ω). But for the piecewise polynomials uh in (3.60) weobtain uh, uh

x, uhy , u

hxx, u

hyy, u

hxy, uh

xxuhyy, (u

hxy)2, . . . ∈ L2(Ω) ∩ L∞(Ω). For uh

1 , uh2 ∈ Uh,

we apply the Cauchy-Schwarz inequality (1.48) to any T ∈ T h :

( ∫T

(uh1

∣∣Tuh

2

∣∣T)dx)2 ≤ ∫

T

(uh1

∣∣T)2dx

∫T

(uh2

∣∣T)2dx implying (3.62)

‖uh1u

h2‖4L2(T ) =

(∫T

(uh1u

h2

∣∣T)2dx

)2 =(∫

T

∣∣uh1

∣∣2T

∣∣uh2

∣∣2Tdx)2≤∫

T

(uh1

∣∣T)4dx

∫T

(uh2

∣∣T)4dx.

These integrals are defined as limits of Riemann sums with appropriately distributedN knotes xi ∈ T . The maximal distance between them vanishes and the covering withopen volume elements Mi � xi and measure μ(Mi) satisfies T = UN

i=1M i.∫T

(uh|T )4dx = limN→∝

N∑i=1

(uh(xi))4μ(Mi) (3.63)

≤ limN→∝

(N∑

i=1

(uh(xi))2μ(Mi))N

maxi=1

(uh(xi))2 = ‖uh‖2L2(T )‖uh‖2L∝(T ),

and similarly for all other up to second order derivatives of the uh, hencecontinuity. �

We prove stability for linear coercive, hence boundedly invertible operators and theircompact perturbations. These linear coercive operators are usually the principal partsof elliptic operators, sometimes, e.g. for natural boundary conditions a term such


as c(u, v)L2(Ω), c > 0 has to be added. It is important to realize that for all ellipticproblems discussed in this book, the weak principal parts always have the form

ap(·, ·) : Vb × Vb → R or V × V → R.

In a few exceptional cases, e.g. for mesh–free methods, in one of the V × V a densesubspace has to be chosen. For our generalized Petrov–Galerkin methods, always theUh

b ,Vhb , even for Uh

b �= Vhb , are approximating subspaces for the Ub = Vb, with the same

norms ‖vh‖Vh = ‖vh‖Uh . So we additionally require

for generalized Petrov–Galerkin methods: ∀ε > 0 ∃δ < ε ∈ R+ : (3.64)

∀vh ∈ Vhb ∃uh ∈ Uh

b : ‖uh − vh‖Uh ≤ δ‖vh‖Uh , always correct for Uhb = Vh

b .

Fully nonlinear elliptic problems cannot be solved by generalized Petrov–Galerkinmethods. In that case, (3.64) only plays the role for the weak linearized form of theseproblems. However, it has to be modified there for handling the violated boundaryconditions.

For nonconforming FE, the DCG, the difference, and the mesh–free methods, thefollowing proof has to be generalized by employing specific techniques. So we willformulate the corresponding stability results in those chapters.

So we restrict the discussion to conforming methods, cf. Examples 3.4, 3.8. Theseare, in our two books, the conforming FE and the spectral methods. Then theorthogonal projection Q

′h in (3.18) and the Ahuh and fh in (3.20) are stronglysimplified. For

Uhb ⊂ Ub,Vh

b ⊂ Vb : Q′h : V ′

b → V ′hb :

⟨Q

′hf − f, vh⟩V′

b×Vb= 0∀vh ∈ Vh

b , (3.65)

Ah = Q′hA|Uh

b, with

(Q

′hA|Uhb

)−1 : V ′hb → Uh

b , fh = Q′hf |Vh

b.

For Ub-coercive A ∈ L (Ub,U ′b), the stability of conforming generalized Petrov–

Galerkin methods can be verified directly. Often this A ∈ L (Ub,U ′b) = A|Ub

of anA ∈ L(U ,U ′).

Theorem 3.28. Ub-coercive operators induce stable Petrov–Galerkin methods:

1. Let A ∈ L (Ub,U ′b) be a linear Ub-coercive operator, i.e.

∃M > 0 : a(u, u) = 〈Au, u〉U ′×U ≥M ||u||2U , for all u ∈ Ub. (3.66)

Then A−1 : U ′b → Ub exists and ‖A−1‖Ub←U ′

b≤ 1/M .

2. For any conforming generalized Petrov–Galerkin method, satisfying (3.64) andwith Q

′h in (3.65), the Ah = Q′hA|Uh

bis stable and∥∥∥∥(Q′hA|Uh

b

)−1∥∥∥∥Uh

b ←Vhb

≤ 2/M for sufficiently small h. (3.67)

Proof. Relations (3.66) and (3.67) follow from Theorem 2.12 and (3.64). �

In general the existence of a bounded inverse of A is not sufficient to ensurethat Ah = Q

′hA|Uhb

is stable. However, the next result allows all types of elliptic


equation, bordered systems, hence numerical methods for bifurcation, and Navier-Stokes equations, see the following chapters.

Theorem 3.29. For conforming generalized Petrov–Galerkin methods invertibleA imply stable Ah: For the approximating spaces

(Uh

b , Vhb

)h∈H

for Ub,Vb, satisfying(3.64), let B ∈ L (Ub,V ′

b) be boundedly invertible and the Q′hB|Uh

bbe stable. Let

A := B + C, with C ∈ C (Ub,V ′b) , the set of compact operators from Ub → V ′

b. Then:

A−1 ∈ L (V ′b,Ub) ⇒ Q

′hA|Uhb

is stable.

Remark 3.30. The existence of A−1 ∈ L (V ′b,Ub) has to be discussed in a little bit

more detail. Since B−1 ∈ L (V ′b,Ub) exists, this implies B−1A = I + B−1C ∈ L(Ub,Ub)

and, as a product of a continuous and a compact operator, B−1C ∈ C(Ub,Ub) is acompact perturbation of the identity. Therefore the Riesz–Schauder theory applies, cf.Theorem 2.20. Hence for λ ∈ C \ {0} either (B−1C − λI)−1 ∈ L(Ub,Ub) or λ is oneof the at most countably many eigenvalues of B−1C. This implies that A−1 exists ifand only if λ = −1 is no eigenvalue of B−1C.

Proof. We have to show that A−1 ∈ L (V ′b,Ub) implies the stability of Q

′hA|Uhb. We

determine for an arbitrary u ∈ Ub and v := Cu the, by assumption, unique exact anddiscrete solutions, u and uh, of the equations Bu = v and Q

′hB|Uhbuh = Q

′hBuh =

Q′hv ∈ V ′h

b , cf. (3.4). With the notation Bh := Q′hB|Uh

b, T := B−1 ∈ L (V ′

b,Ub) and

Th := Bh−1Q

′h ∈ L(V ′

b,Uhb

), Theorems 3.16 and 3.21 imply that

limh→0

||u− uh||Ub= ||(T − Th)Cu||Ub

= 0 and u ∈ Ub or (3.68)

for any v ∈ V ′b and Bu = v,Bhuh = Q

′hv : limh→0

||u− uh||Ub= ||(T − Th)v||Ub

= 0.

By Lemma 3.31 this second limit implies

||(T − Th)C||Ub←Ub→ 0 for h→ 0. (3.69)

Now let uh ∈ Uhb ⊂ Ub. Because A is boundedly invertible, we can estimate

||uh||Ub≤ ||A−1||Ub←V′

b||Auh||V′

b= ||A−1||Ub←V′

b||B(I + TC)uh||V′

b

≤ ||A−1||Ub←V′b||B||V′

b←Ub||(I + TC)uh||Ub

, hence,

||(I + TC)uh||Ub≥(||A−1||Ub←V′

b||B||V′

b←Ub

)−1

||uh||Ub. (3.70)

For the following we recall that Q′hBu is defined for every u ∈ Ub. We apply the sta-

bility of Bh to wh := Bhuh to find ‖uh‖Ub≤ ‖(Bh)−1‖Uh

b ←V′hb· ‖wh‖V′

b. Furthermore

‖(I + ThC)uh‖Ub= ‖(Bh)−1Bh(I + ThC)uh‖Ub

≤ ‖(Bh)−1‖Uhb ←V′h

b‖Bh(I + ThC)uh‖V′

b

implies


‖Bh(I + ThC)uh‖V′b≥ 1‖(Bh)−1‖Uh

b ←V′hb

· ‖(I + ThC)uh‖Ub. (3.71)

We combine (3.70), (3.71) for estimating∥∥∥Q′hAuh∥∥∥V′

b

= ||Bh(I + TC)uh||V′b≥ ‖Bh(I + ThC)uh‖V′

b− ‖Q′hB(T − Th)Cuh‖V′

b

(3.71)

≥ ||(I + ThC)uh||Ub/||(Bh)−1||Uh

b ←V′hb− ‖Q′hB(T − Th)Cuh‖V′

b

≥ 1/||(Bh)−1||Uhb ←V′h

b

(||(I + TC)uh||Ub

− ||(T − Th)Cuh||Ub

)−‖Q′hB(T − Th)C‖V′

b←Ub‖uh‖Ub

(3.70)

≥ 1

||Bh−1||Uhb ←V′h

b

[1(

||A−1||Ub←V′b||B||V′

b←Ub

)−O(||(T − Th)C||V′

b←Ub

)]‖uh‖Ub

.

Because of (3.69) and the stability of (Bh)h∈H there exists a positive constant K,independent of h, such that for all h ≤ h0 the following holds:

||Q′hAuh||V′b≥ K||uh||Ub

for all uh ∈ Uhb .

This and dimUhb = dimV ′h

b <∞ shows that Q′hA|Uh

bis invertible for h ≤ h0. More-

over, we obtain ||(Q

′hA|Uhb

)−1

||Uhb ←V′h

b≤ 1/K, i.e. the stability. �

Lemma 3.31. Raasch [539]: With the notation and conditions introduced in theprevious proof, the second line of (3.68) implies (3.69).

Proof. We have to prove the implication that for any

∀v ∈ V ′b and Bu = v,Bhuh = Q

′hv

∀v ∈ V ′b : lim

h→0||u− uh||Ub

= ||(T − Th)v||Ub= 0 (3.72)

=⇒ limh→0

||(T − Th)C||Ub←Ub= 0. (3.73)

This limit in (3.72) is a consequence of the stability and consistency of Bh with B,hence the convergence of the method.

For the compact operator C : Ub → V ′b, and with the previous v := Cu, we claim

that

M := {u ∈ Ub : ‖u‖Ub= 1} � u implies ||(T − Th)Cu||Ub

≤ C ′ε = C ′ε‖u‖Ub(3.74)

implying (3.73). With C, the closed image CM is compact. Therefore by Definition1.8 ff. the following finite open coverings exist, such that


∀ε > 0∃N = N(ε) : v1, . . . , vN ∈ V ′b s.t. (3.75)

CM⊂ ⋃Ni=1

{v ∈ V ′

b : ‖v − vi‖V′b≤ ε}

and by (3.72)

=⇒ ∃h0 : ||(T − Th)vi||Ub≤ ε ∀h < h0, i = 1, . . . , N.

As a consequence of (3.73) and the equiboundedness of T, Th, hence ||(T − Th)||Ub←↩V′b

≤ C∗, we find

∀u ∈M∃i = iu ∈ {1, . . . , N} with ‖Cu− vi‖V′b≤ ε hence (3.76)

||(T − Th)Cu||Ub≤ ||(T − Th)(Cu− vi)||Ub

+ ||(T − Th)vi||Ub

≤ C∗ε + ε ≤ (1 + C∗)ε ≤ C ′ε‖u‖Ub,

hence the claim by (3.73). �

An important example for operators, which are compact perturbations of coerciveoperators, are A ∈ L (Ub,U ′

b) satisfying a Garding inequality, cf. (2.141).

Theorem 3.32. Garding inequality and compact perturbation: In the Gelfand tripleof Banach spaces Ub ⊂ W =W ′ ⊂ U ′, cf. Definition 2.17, let Ub be (continuously,densely) and compactly embedded into the Hilbert spaceW. Assume that A ∈ L (Ub,U ′

b)fulfills a Garding inequality, i.e. ∃M > 0 and m ∈ R, such that

< Au, u >U ′b×Ub

≥M ||u||2Ub−m||u||2W ∀ u ∈ Ub. (3.77)

Then A is a compact perturbation of a coercive operator.

Proof. With the identified W with W ′, the above assumptions yield Ub ⊂ W ⊂ U ′b.

Additionally the embedding IUb→U ′b

is compact, see Zeidler [677]. We obtain thesplitting

A =(A + mIUb→U ′

b

)−mIUb→U ′

b, with coercive

(A + mIUb→U ′

b

),


Finally, we discuss variational crimes of the easiest type. We allow inexact evalua-tions of the operators Q

′hA|Uhb, Q

′hG|Uhb, and Q

′hAh|Uhb, Q

′hGh|Uhb, and of Q

′hf , cf.Example 3.8. This is necessary, since the exact values often are not available or aretoo expansive. So they are approximated, e.g. by numerical integration. The influenceof this type of perturbation is analyzed in the following theorem. We only present thebasic result for conforming Petrov–Galerkin methods. More detailed results includingnonconforming FE and spectral methods will again be discussed in Chapters 4and in [120].

Theorem 3.33. Let Uhb ,Vh

b be approximating spaces for Ub,Vb in a conformingPetrov–Galerkin method, and f ∈ V ′

b. Assume Ah ∈ L(Uh

b ,V′hb

)and fh ∈ V ′h

b are


“proper” perturbations of Ah = Q′hA|Uh

band Q

′hf , respectively, i.e.

||Ah −Q′hA|Uh

b||V′h

b ←Uhb→ 0 and ||fh −Q

′hf ||Vhb

′h → 0 for h→ 0. (3.78)

1. Then, for sufficiently small h,

Ah is stable ⇔(Q

′hA|Uhb

)is stable. (3.79)

2. Furthermore, if Ah = Q′hA|Uh

bis stable, the uniquely determined solution uh

0 ofthe perturbed discrete equation

Ahuh0 = fh, uh

0 ∈ Uhb

converges to the uniquely determined solution uh0 of Ahuh

0 = fh, more precisely,for sufficiently small h, we obtain finally, with ‖(Ah)−1‖Uh

b ←V′hb≤ S,

‖uh − uh‖Uhb≤ 2S2‖Ah −Ah‖V′h

b ←Uhb‖f‖V′h

b+ S‖f − f‖V′h

b. (3.80)

Proof. The condition (3.78) immediately shows that for small enough h the relation(3.79) is correct. For proving the estimate we consider

Ahuh = f, uh = (Ah)−1f

Ahuh = (Ah + (Ah −Ah))uh = Ah(I + (Ah)−1(Ah −Ah))uh = f

uh = (I + (Ah)−1(Ah −Ah))−1(Ah)−1f

uh − uh = (I + (Ah)−1(Ah −Ah))−1(Ah)−1f − (Ah)−1f + (Ah)−1(f − f).

We assume, e.g. for sufficiently small h, that ‖(Ah)−1(Ah −Ah)‖Uhb ←Uh

b< 1/n. With

the Neumann series we obtain

(I + (Ah)−1(Ah −Ah))−1 =∞∑

j=0

(−(Ah)−1(Ah −Ah))j =⇒

(I + (Ah)−1(Ah −Ah))−1 = (I + (Ah)−1(Ah −Ah)) +O(‖(Ah)−1(Ah −Ah)‖2Uhb ←Uh

b).

This implies

uh − uh =((I + (Ah)−1(Ah −Ah)) +O(‖(Ah)−1(Ah −Ah)‖2Uh

b ←Uhb))(Ah)−1f

− (Ah)−1f + (Ah)−1(f − f)

=((Ah)−1(Ah −Ah) +O(‖(Ah)−1(Ah −Ah)‖2Uh

b ←Uhb))

(Ah)−1f + (Ah)−1(f − f).

So we obtain finally with ‖(Ah)−1‖Uhb ←V′h

b≤ S

‖uh − uh‖Uhb≤ 2S2‖Ah −Ah‖V′h

b ←Uhb‖f‖V′h

b+ S‖f − f‖V′h

b,


3.6. Stability implies invertibility 203

Vb ⊂

P hd

V hb = V ″b

h

Q′dh,

(Ad )h = Q′bh Ad⏐V ″b

h

V ″bAd

(Ad )h

Q′dh

tested by uniformly

bounded, and (3.78)

P dh

tested byU ′bh U ′

bh

U ′b Ub

.

Figure 3.3 General discretization methods for dual problems.

3.6 Stability implies invertibility

In this section we study the opposite relation: under which conditions is the invertibil-ity of an operator A a consequence of the stability of its discretization Ah. In fact weshow that stability is, for some cases, the stronger condition, but both propertiesare equivalent under rather general conditions. Again, we restrict the discussionto conforming Petrov–Galerkin methods, with Ah = Q

′hA. Extensions to the othermethods would be possible as well.

We have to simultaneously study Au0 = f , cf. (3.4), and its dual problem. So theoriginal equation and its dual form

A : Ub → V ′b, Au = f and Ad : V ′′

b → U ′b, Adv′′0 = f ′ (3.75)

induce a pair of corresponding discrete equations of the form

Ah : Uhb ⊂ Ub → V ′h

b , and (Ad)h : Vhb = V ′′h

b ⊂ V ′′b → U ′h

b , (3.76)

Ahuh0 = Q

′hAuh0 = Q

′hf and (Ad)hv′h0 = Q

′hd Adv

′h0 = Q

′hd f ′.

This Vhb = V ′′h

b ⊂ Vb is a consequence of the assumed conforming Petrov–Galerkinmethod with the bi-dual pair of approximating spaces. For Ph

d we choose an inter-polation or approximation operator, similar to the previous Ph for Uh

b . The Q′hd are

defined as special case of (3.18), as

Q′hd : U ′

b → U ′hb :

⟨Q

′hd g′ − g′, uh

⟩U ′h

b ×U ′′hb

= 0∀uh ∈ U ′′hb . (3.77)

Therefore the situation in Figure 3.2 implies a corresponding diagram for Ad ∈L (V ′′

b ,U ′b) in Figure 3.3.

For the proof of Theorem 3.37 we need the following proposition for the boundedinvertibility of a linear operator between Banach spaces. It is a consequence of Theorem2.12, however it can be proved directly via Theorem 1.19.

Proposition 3.34. Let U , V be Banach spaces and A ∈ L(U ,V ′). Then the followingconditions are equivalent:

(1) A−1 ∈ L(V ′,U);(2) (Ad)−1 ∈ L(U ′,V ′′);(3) there exist positive constants C1, C2 such that

(a) ||Au||V′ ≥ C1||u||U for all u ∈ U and(b) ||Adv′′||U ′ ≥ C2||v′′||V′′ for all v′′ ∈ V ′′.


If one of the above conditions is satisfied, we obtain

||A−1||U←V′ = ||(Ad)−1||V′′←U ′ ≤ min(1/C1, 1/C2).

Remark 3.35. The three conditions (1), (2), (3) are not symmetric: withD(A),N (A), R(A) denoting the domain, null-space and range of A, respectively,note that A ∈ L(U ,V ′) implies D(A) = U . Therefore Ad ∈ L(V ′′,U ′) is defined andA−1 ∈ L(V ′,U) exists if and only if (Ad)−1 ∈ L(U ′,V ′′) exists. Hence, we only needone of the equivalent conditions (1) or (2). However, in (3) we indeed need boothconditions, since (3)(a) still admits AU �= V ′.

Proof. The equivalence “(1) ⇔ (2)” is well known. To show “(1) ⇒ (3)” let A−1 ∈L(V ′,U) (and therefore (Ad)−1 ∈ L(U ′,V ′′)). This yields immediately condition (3)with C1 = 1/||A−1|| (and C2 = 1/||((Ad)−1|| = C1).

“(1) ⇐ (3)” 3(a) and 3(b) imply that A and Ad are injective. 3(a) yields, for aconvergent sequence {Auk}, uk ∈ U , that {uk} is a Cauchy sequence and consequentlyconverges to an u ∈ U . A is continuous by assumption, so we obtain Auk → Au andcan conclude that R(A) is closed in V ′. The closed range theorem and the injectivityof Ad yield that A is surjective. Finally, the inverse mapping theorem implies the firstcondition. �

Remark 3.36. Applying Proposition 3.34 to the bi-dual Petrov–Galerkin approxi-mation case Ah : Uh

b → V ′hb , and (Ad)h : Vh

b = V ′′hb → U ′h

b , shows that Ah is stable ifand only if (Ad)h is stable, and for both sequences the same stability constants canbe chosen. For applying Theorem 3.21 we need a solution, u0, for Au0 = f, and theproperties 1., 2., 3.

Theorem 3.37. Stability implies the existence of A−1 :

1. For A ∈ L(Ub,V ′b) let the pair of bi-dual approximating spaces (Uh

b , Vhb = V ′′h

b ),define a generalized Petrov–Galerkin method. Then the stability of Ah = Q

′hA|Uhb

implies the existence of A−1.2. If Ah = Q

′hA|Uhb

is replaced, according to Theorem 3.33, by an approximation

Ah the corresponding discrete operator (Ad)h for the dual of A has to satisfy(3.78) as well. In this sense the stability of Ah again implies the existence ofA−1 ∈ L(V ′

b,Ub).

Proof. We have to show that the existence of a bounded inverse of A follows fromthe stability of Q

′hA|Uhb. Let u0 ∈ Ub and v′′0 ∈ V ′′

b be given. Obviously, they solvef := Au0 ∈ V ′

b, g′ := Adv′′0 ∈ U ′b, respectively. The corresponding discrete solutions

uh0 , v

′′h0 , see (3.75), (3.76), uniquely exist, since Remark 3.36 and the conforming

bi-dual Petrov–Galerkin method imply ‖Q′h‖ = ‖Q′hd ‖ = 1, stability and consistency

for Ah and (Ah)d = (Ad)h, hence ‖uh0 − u0‖Ub

→ 0, ‖v′′h0 − v′′0‖V′

b→ 0 for h→ 0. One

3.7. Solving nonlinear systems 205

obtains the following estimates:∥∥uh0

∥∥Ub

=∥∥(Q′hA|Uh

b

)−1Q

′hf∥∥Ub

=∥∥(Q′hA|Uh

b

)−1Q

′hAu0

∥∥Ub

≤∥∥(Q′hA|Uh

b

)−1∥∥Uh

b ←V′hb

∥∥Q′h∥∥V′h

b ←V′b

·∥∥Au0

∥∥V′

b

≤ C∥∥Au0

∥∥V′

b

and∥∥v′′h0

∥∥V′′

b

=∥∥(Q′h

d Ad|V′′hb

)−1Q

′hd g′∥∥V′′

b

=∥∥(Q′h

d Ad|V′′hb

)−1Q

′hd Adv′′0

∥∥V′′

b

≤∥∥(Q′hA|Uh

b

)−1∥∥Uh

b ←V′hb

∥∥Adv′′0∥∥U ′

b

≤ C∥∥Adv′′0

∥∥U ′

b

.

Theorem 3.21 applied to the bi-dual Petrov–Galerkin method yields the convergenceof uh

0 and v′′h0 to the chosen solutions u0 and v′′0 , respectively. Consequently, we obtain

with the above estimates and the continuity of the norms

||Au0||V′′b≥ K||u0||Ub

and∥∥Adv′′0

∥∥U ′

b

≥ K ‖v′′0‖V′b.

Therefore Proposition 3.34 implies that A−1 ∈ L (V ′′b ,Ub) does exist. If, instead of

Ah = Q′hA|Uh

ba sequence Ah is chosen, the corresponding (Ad)h has to satisfy (3.78)

as well. �

3.7 Solving nonlinear systems: Continuation and Newton’smethod based upon the mesh independence principle (MIP)

3.7.1 Continuation methods

For the many discretization methods discussed in our two books much more specificsolution techniques, in particular for linear problems, have been studied. The jointstructure of these methods is less strong than for general discretization methods. Sowe do not aim at a synopsis of these methods. Rather we indicate two principles,allowing for all presented cases a simultaneous solution technique. We determinethe discrete solutions, u0, by combing the well-known continuation methods withthe mesh independence principle (MIP). Proving convergence results for Newton’smethod for many discrete problems avoiding the MIP often require highly technicalefforts.

In continuation methods we embed the discrete problem Gh(uh0 ) = 0 into a one-

parameter problem. We identify uh0 and x1 ∈ Rm and define

Fh : Rm+1 → Rm with Fh(x, λ) := (1− λ)Gh0 (x) + λGh

1 (x) where (3.79)

Gh1 := Gh and a Gh

0 : Rm → Rm with a known approximate solution x0 ∈ Rm.

We start we the continuation process with λ = 0 and the known (approximate) solutionx0 aiming for λ = 1: we only consider the easiest case and refer for other strategiesto, e.g. Allgower and Georg [20, 21], Decker and Keller [278, 439, 440] Rheinboldt andBurkardt [550,551], and Seydel [580,581].


This allows formulating the following algorithm:

Algorithm 3.1.Continuation method by parametrization

INPUT: (x0, λ0 = 0) ∈ Rm × R, Gh0 , Gh

1 , Gh0 (x0) = Gh(x0, 0) ≈ 0); starting point with

h > 0 (Initial step size);

OUTPUT: (xi, λi), i = 1, 2, . . . , (Stop at λie = 1 with(xie ≈ uh

0 , λie = 1));

ITERATION: FOR (i = 1; Stopping criterion)

(i) λi = min{λi−1 + h, 1}; xi = xi−1;

(ii) With Newton-like methods (MIP) improve this xi into xi ∈ Rm, such thatF h(xi, λi) ≈ 0;

(iii) Stop for λi = 1, else goto (iv);

(iv) Adapt the stepsize h > 0, goto (i).

Figure 3.4 shows the predictor and the trace of a few corrector steps.

3.7.2 MIP for nonlinear systems

As a consequence of the general discretization approach, the number of iterations in theNewton methods for the original G in (3.81) and its discrete counterpart Gh in (3.84)are strongly related. The number of iterations for a given tolerance, ‖uh

i − uh0‖2Uh ≤ ε,

for G and Gh, is nearly the same, independent of h. This is achieved by the MIP, cf.Theorem 3.40, Allgower and Bohmer [9], and Allgower et al. [17].

We only present this main form of the MIP. There are many other papers onthis subject, dealing with different aspects of discrete Newton methods: Deuflhardand Potra [285] discuss the “affine” Newton method, and Bohmer [112] combines itwith the defect correction method. In particular, Argyros [36, 37] discusses Newtonmethods, where the derivative of the operator is Holder instead of the original

h

(xi , λi )

(xi − 1, λi − 1)

Figure 3.4 Natural parametrization.

3.7. Solving nonlinear systems 207

Lipschitz-continuous, or the discretized secant iterations. In both cases the quadraticconvergence is reduced. Instead of discretizations he allows perturbed or inexactNewton methods or directly works in Banach spaces, sometimes based upon majorantprinciples.

Efficient strategies combining the MIP with sequences of increasingly finer mesheshave been studied by Allgower and Bohmer [9, 10]. They can be combined with thenested sequences of, e.g. FE spaces S1

d

(T h,1

c

)� S1

d

(T h,2

c

)� · · · yielding multiscaling

or multiresolution techniques.For formulating the MIP we recall the main condition for the existence of a unique

solution u0 for the original G(u) = 0 and the convergence for a general discretization:it is the bounded invertibility of the linearized G′(u0). For a G ∈ C1

L, this is simultane-ously the main condition for the (quadratic) convergence of the Newton–Kantorovichmethod applied to G. We do not want to formulate the sharpest possible conditionsfor the following theorems, but rather give their flavor. We compare Newton’s methodapplied to G and to Gh. We additionally assume a Lipschitz-continuous derivative, so

Let u0 be an isolated solution of G(u0) = 0, s.t. G′(u0) : U → V ′ (3.80)

is boundedly invertible and

‖G′(u)−G′(v)‖V′←↩U ≤ L‖u− v‖U ∀u, v ∈ BR(u0).

Theorem 3.38. Newton–Kantorovich method for G, Ortega and Rheinboldt [520]:Under the condition (3.80) and for small enough ‖u0 − u1‖U the method in (3.81) iswell defined and converges quadratically:

ui+1 := ui −G′(ui)−1G(ui), i = 1, . . . , and ‖u0 − ui+1‖U ≤ C‖ui − u0‖2U , (3.81)

or ≤ C ′‖ui − ui−1‖2U . These C,C ′ depend upon ‖u0 − u1‖U , L, ‖(G′(u0))−1‖.For maintaining the consistency of Gh of order p > 0 for u ∈ Us a smooth subspace

of U , during the Newton process we have to require

G : D(G) ∩ Us → V ′s, and G′(u0) : Us → V ′

s is boundedly invertible. (3.82)

Furthermore, we have to extend the concept of consistency in Definition 3.14 to thefirst derivative.

Definition 3.39. Differentiable consistency: Under the conditions of Definition 3.14a general discretization method is called differentiably consistent and of order p, if itis classically consistent and if for a Frechet differentiable nonlinear operator G,

‖(Gh)′(Phu)Phv −Q′h(G′(u)v)‖V′h

b→ 0 and (3.83)

‖(Gh)′(Phu)Phv −Q′h(G′(u)v)‖V′h

b≤ Chp(1 + ‖u‖Us

)‖v‖Usfor h→ 0.

These ‖v‖Usare usually ‖v‖W k,q(Ω) and p, k, q appropriately chosen.

Theorem 3.40. Newton method for the discrete Gh, cf. [9, 10, 17]: Under theconditions (3.80), (3.82), and for a linear discretization method M assume a stableGh, consistent and differentiably consistent of order p > 0 for u0 ∈ Us ⊂ U , a smooth


subspace, still with Lipschitz-continuous (Gh)′. Start the discrete Newton process fori = 1, . . . , with uh

1 := Phu1, u1 ∈ Us, as

uhi+1 := uh

i −(Gh(uh

i

)′)−1Gh(uh

i

). (3.84)

Then, for small enough ‖u0 − u1‖U and h, the uhi+1 in (3.84) uniquely exist and

converge quadratically to uh0 , such that∥∥uh

i+1 − uhi

∥∥Uh ≤ C

∥∥uhi − uh

i−1

∥∥2Uh and∥∥uh

i − Phui

∥∥Uh ≤ Chp‖ui‖Us

, i = 1, . . . ,

‖Gh(uh

i

)−Q

′hG(ui)‖V′h ≤ Chp‖ui‖Us, i = 1, . . . , (3.85)

‖(uh

i − uh0

)− (Ph(ui − u0))‖Uh ≤ Chp‖ui‖Us

, i = 1, . . . .

Corollary 3.41. MIP, cf. [9,17]: Under the conditions of the previous theorem thereexists h such that for all h ∈ (0, h]∥∥min{i ≥ 1 :

∥∥uhi − Phu0

∥∥Uh < ε} −min{i ≥ 1 : ‖ui − u0‖U < ε} ≤ 1. (3.86)

This result is the reason for the MIP notation and our remark concerning theessential h independence of the convergence.

Now we are able to formulate the discrete Newton algorithm, based upon the MIP.We start with a moderately good approximation for uh

0 obtained by Algorithm 3.2

Algorithm 3.2.Discrete Newton method and MIP

INPUT: Starting approximation uh1 ∈ Rm with Gh

(uh

1

)≈ 0 and

tolerance ε > 0; h > 0;

OUTPUT: Approximation uhn for uh

0 s.t.∥∥uh

n − uh0

∥∥Uh < ε.

ITERATION: FOR (i = 1; Stopping criterion)

(i) Calculate uhi+1 := uh

i −(Gh(uh

i

)′)−1

Gh(uh

i

), see (3.84);

(ii) Update i := i + 1;

(iii) Estimate∥∥uh

i − uh0

∥∥Uh (with standard Newton techniques);

(iv) Stop if∥∥uh

i − uh0

∥∥Uh < ε, else goto (i).

4

Conforming finite element methods(FEMs)

4.1 Introduction

Finite element methods (FEMs) are nowadays the most advanced methods, applicableto nearly all types of PDEs. As throughout this book, we restrict the discussion toelliptic problems, from linear, to semilinear, to quasilinear to fully nonlinear equationsand systems of orders 2 and 2m. The goal in this chapter is the proof of stability andconvergence for different types of conforming FEMs and to generalize many of theknown the results. In Chapter 3, we developed appropriate tools for proving resultsfor these operator equations and a relatively general class of discretization methods,based upon consistency and stability. Conforming FEMs are consistent for continuousproblems. We proceed in two steps to stability for nonlinear a consequence of stabilityfor the linearized operator. Let a boundedly invertible linear operator yield a stablediscretization, e.g. for an operator with coercive bilinear form. Under some technicalconditions, this stability is inherited to its compactly perturbed linear operators,if these are still boundedly invertible and consistent. These results are applicableessentially to all the elliptic problems in Chapter 2, if, for nonlinear problems, thelinearization is bounded.

As mentioned, a stable discretization of a boundedly invertible operator is a conse-quence of a coercive principal part or a Garding inequality, cf. Theorem 3.32. This iscorrect, except for (nonlinear) elliptic systems in the sense of Agmon et al. [3], withdifferent highest orders of the derivatives in different equations, cf. Definition 2.85.The most famous example is the Navier–Stokes operator. The stability and boundedinvertibility of the Stokes operator has been proved by other techniques. Then theabove arguments show the stability of the (linearized) Navier–Stokes operator formoderate Reynolds numbers. This is applicable to other examples as well. Convergencefor monotone operator is available without linearization.

Excellent books are available on FEMs. We list only some of them, best suitedto support specific points in the text. We need discretizations of general problemsand their convergence properties. Hackbusch’s [386, 387] studies of elliptic, includinghigher order PDEs, with finite difference and FEMs, and Ciarlet’s [174] Brenner andScott’s [141] books on FEMs have very strongly influenced this book and this andthe next chapters. Different numerical methods and the corresponding analysis arestudied by Grossmann et al. [374–376], for second order PDEs. FEMs, again combinedwith many interesting applications, are discussed by Babuska and Strouboulis [51]

210 4. Conforming finite element methods

and Oden and Reddy [518]. Other important applications with more or less theoryfor linear and nonlinear equations and systems are studied, e.g. by Braess [135],Brenner and Scott [141], Debnath et al. [277, 638], Leung [473], DuChateau andZachmann [305–307], Larsson and Thomee [469], Quarteroni and Vali [537], andThomas [626]. In all these books, nonlinear problems essentially are discussed viasome practically important examples. Zenısek [681] approaches nonlinear problemswith monotony methods. Haslinger et al. [393] study unilateral linear boundary value,mainly mechanical contact, problems via inequalities. The nonlinear problems in thesebooks are so-called semilinear problems. The analytic theory, special Galerkin methodsand a general discretization theory for nonlinear, here semilinear and quasilinear,problems are presented by Zeidler [675–678]. FEMs for fully nonlinear problems havenot been available before. So they are not included in previous books, c.f. Section 5.2.

In this chapter we only consider and approximately solve elliptic and monotoneproblems formulated as weak forms. Therefore we assume throughout this chapterthat our operator and approximating subspaces satisfy

A : Ub → U ′

b, G : D(G) ⊂ Ub → U ′

b, Uhb = Vh

b ∀ h

usually A and G are defined on U as well.The basic idea of FEMs is simple. We demonstrate it for a linear operator

A : U = V := H1(Ω) → V ′ := H−1(Ω), s.t. A : Ub = Vb := H10 (Ω) → V ′

b bijective.(4.1)

The index and exponent, b and h, e.g. in Ub and Uhb , indicate boundary conditions and

finite dimensional subspaces. The exact solution, u0, is determined by testing

u0 ∈ Ub ⊂ U : a(u0, v) = 〈Au0, v〉V′×V = 〈f, v〉V′×V (4.2)

∀ v ∈ Vb ⊂ V often with Ub = Vb ⊂ U = V.In FEMs the Ub,Vb in (4.2) are replaced by finite dimensional approximating subspacesUh

b ,Vhb of piecewise, e.g. linear polynomials, defined on a triangulation of Ω, cf.

Definition 4.2, with Uhb = Vh

b for conforming methods. We determine the discrete orapproximate solution, uh

0 , such that Uhb ⊂ Ub,Vh

b ⊂ Vb,

uh0 ∈ Uh

b ⊂ Ub ⊂ U : a(uh

0 , vh)

=⟨Auh

0 , v⟩V′×V = 〈f, v〉V′×V∀vh ∈ Vh

b ⊂ Vb. (4.3)

The Ub,Uhb and Vb,Vh

b , represent ansatz and test functions, respectively. If thefunctions uh ∈ Ub,Uh

b are continuous and exactly satisfy the boundary conditions,we call (4.3) a conforming FEM. For a bounded and coercive a(u, u), so ∃α > 0 ∀u ∈Ub : a(u, u) ≥ α‖u‖2U , the convergence of the uh

0 to u0 via the Cea Lemma 4.48 issurprisingly simply proved and yields for piecewise polynomial FEs of degree d− 1∥∥u0 − uh

0

∥∥U ≤ (C/α)dist(u0,Uh) = C∗hd−1‖u0‖Hd(Ω). (4.4)

We do not study super convergence results, c.f. e.g. Bank and Xu, Griewank andReddien, Jiang et al. [66, 368,417]

The conditions of continuous FEs, exactly satisfying the boundary conditions,is too restrictive for many applications. If we relax them we obtain the so-callednonconforming FEMs, cf. Chapter 5.


Our approach to FEMs differs from the usual versions in several respects. Wegeneralize known results and modify proofs in many directions. The main tool isthe systematic use of the general theory of discretization methods as presented inChapter 3. This contrasts with the standard coercivity and ellipticity arguments formainly linear problems with their bilinear forms and trilinear modifications as, e.g.for the Navier–Stokes equations, cf. Section 4.6, and to the monotony arguments forhigher nonlinearities. In fact, conforming and nonconforming FEMs, as usual withone reference element and here with fixed polynomial degree, are special cases ofgeneralized Petrov–Galerkin methods. We have to ensure stability and convergence forFEMs for elliptic problems. Stability is a consequence of the fact that for a linearA the (modified) principal part Ap induces a stable Ah

p . For conforming FEMs the(linear) discrete Ah are (classically) consistent with the original A.

The goal is an extension to nonlinear problems and to higher order conforming andnonconforming FEMs for the large class of linear, semilinear, quasilinear and fullynonlinear elliptic equations mentioned above. This is achieved, including quasilinearproblems, for FEMs based upon the classical weak form of the problems. For fullynonlinear problems we have to use nonconforming FEMs for their strong form.

Applying FEMs in the context of path-following of parameter-dependent solutionsof nonlinear problems and avoiding spurious solutions for turning and bifurcationpoints requires high order methods. They can conveniently and with a wide range ofapplications be defined as conforming and nonconforming FEMs.

We organize this chapter on conforming FEMs as follows. In Section 4.2 we startwith the standard approximation theory for FEs. We only prove results not included instandard textbooks, e.g. [141], and in easily accessible papers, e.g. [362]. As preparationfor the later nonconforming FEMs, we present FEs on curved domains Ω and smoothFEs ∈ Cr(Ω), r ≥ 1 in Subsections 4.2.7, and with Davydov, in 4.2.6. In Section 4.3 asimple example, the Helmholz equation, motivates the definition of conforming FEMs.They are extended to general linear and in Section 4.4 to quasilinear elliptic differentialequations and systems of orders 2 and 2m.

Another type of generalization represents the very active area of mixed FEMs,presented here only for the Navier–Stokes equation in Section 4.6. For the case ofmoderate Reynolds numbers, it is a compact perturbation of the well studied bound-edly invertible Stokes operator with stable discretization. Consequently a boundedlyinvertible linearized Navier–Stokes operator implies stability for the linearized andthe nonlinear operator. This stability result for the Navier–Stokes operator seems tobe the state of the art in discretizations via FEM, spectral, difference and waveletmethods anyway. Extensions to nonconforming and adaptive forms are not consideredhere. In mixed FEMs the different components of the solution, here the velocity, �u,and the pressure, p, live in different spaces and satisfy side conditions, div �u = 0.Correspondingly, for �uh, ph, different and appropriate spaces, Uh,Wh, mixed FEMshave to be chosen.

In particular, for the later studies of bifurcation, FEMs for eigenvalue problems areimportant. We discuss them in Section 4.7. The extension of the results to nonlinearscenarios is not interesting here, since the eigenvalues that decide bifurcation arealways defined by the linearized operators.


Adaptive FEMs are very well advanced these days. So the main tools and resultsare presented in Chapter 6 by W. Doerfler.

We have already mentioned that we delay the study of FEMs with crimes, ornonconforming FEMs to the next chapter. Another possibility for violated continuityand boundary conditions, the discontinuous Galerkin methods (DCGMs) follows.

We exclude the following important areas. The hp-methods and their discontinuousversions are found, e.g. in Babuska et al. [52,53,517]. Even some of the recent resultstowards superconvergence in Babuska and Strouboulis [51], and exponential accuracyas in Schwab and Melenk [485–487], would probably fit our approach. Multigridmethods, multigrid solutions of bifurcation problems, hierarchical bases and all kinds ofa posteriori and reliability estimates are studied by, e.g. Hackbusch [385], Babuska andStrouboulis [51], Brenner and Scott [142], Mittelmann and Weber [492–494,660,661],and Babuska et al. [51, 59–61, 63–66]. Special high order FEMs are studied, e.g. bySolın, Segeth and Dolezel [593].

Throughout the chapter, all convergence results require a sufficiently small dis-cretization parameter, h. In fact, we characterize, strongly simplifying, an FE dis-cretization method by a parameter h. In the following we indicate the essentialconstants depending upon essential parameters, by indices listing these constants, e.g.Ch,μ depends upon h, μ. Sometimes, we omit parameters to show the independence ofconstants, e.g. we would write C instead of Ch,μ to indicate the independence of Cfrom h, μ.

4.2 Approximation theory for finite elements

The following introduction to the approximation theoretic properties of finite elements(FEs) and many other topics is strongly influenced by the classical Ciarlet [174],and by Hackbusch, Braess, Brenner and Scott [136, 141, 386, 387]. The basis for thesuccess of FEMs are the good approximation properties and the high flexibility ofFEs. We present the basic results. After introducing subdivisions and general FEs,we consider the most interesting example of piecewise polynomials. We formulategeneral FE spaces and their approximation properties for functions. For vectors offunctions, e.g. solutions of systems of elliptic equations, they remain unchanged if eachcomponent shares the same structural properties and has the same smoothness. Wesummarize interpolation errors, inverse estimates and extensions to curved boundaries.The results in this section are strictly local, except Subsection 4.2.6 with O. Davydov.Only here do we present and prove the essentially new results for smooth FEs oncurved domains with the necessary splitting properties.

We assume throughout the whole chapter that the (open) domain Ω satisfies

Ω ⊂ Rn bounded domain, piecewise smooth Lipschitz-continuous ∂Ω. (4.5)

4.2.1 Subdivisions and finite elements

The power of FEs is based upon approximating functions, uh : Ω ⊂ R, by splitting thedomain into subsets Ti ⊂ Ω. For simplified computations we assume each of these Ti

4.2. Approximation theory for finite elements 213

to be affinely (or isoparametrically, see Section 4.2.7), mapped onto the same referencedomain, K. In the simplest case an approximating function uh, restricted to a triangleT = Ti, is uh ∈ P1, hence a linear polynomial, defined by its function values in thethree vertices of T . This is generalized as, see [174]:

Definition 4.1. We assume that

1. the element or reference domain K ⊆ Rn is an open domain with piecewisesmooth boundary;

2. the space of shape functions P is a finite-dimensional space of functions definedon K; and

3. the set, N = {N1, N2, . . . , Nd′}, of nodal variables are linear, linearly indepen-dent, usually interpolation functionals defined on P. They represent a basis forP ′, hence, P ′ = span N .

We need the same dimension dimP = dimP ′ = d′. Then (K,P,N ) is called a ref-erence element or a finite element. There are many possible combinations of P,N .If P = span {φ1, φ2, . . . , φd′} is a basis for P dual to N , hence Ni(φj) = δij, thenφ1, . . . , φd′ is called the nodal basis for P. A set of nodal variables, N , is calledunisolvent for P, or K,P,N a unisolvent FE, compare 3., if

∀(βi)d′

i=1 ∈ Rd′ ∃1 φ ∈ P : Ni(φ) = βi, i = 1, . . . , d′.

This choice of a reference element (K,P,N ) has the important consequence thatmuch of the following analysis and computations can be performed in the referencedomain K.

In the previous simplest case, motivating Definition 4.1, K is a (unit) triangle withvertices P1, P2, P3, the P = P1 and N are the point evaluations in the P1, P2, P3.It is implicitly assumed that the nodal variables, Ni, lie in the dual space of somelarger function space, U , allowing Ni(φ)∀φ ∈ U . The Ni usually denote evaluation offunctions or derivatives in given interior and boundary points of K. Obviously N isunisolvent for P if and only if dimP = dim span N = d′ and φ ∈ P, Ni(φ) = 0, i =1, . . . , d′ implies φ ≡ 0.

For a given u ∈ U , a nodal basis of shape functions P = span{φ1, . . . , φd′} and aunisolvent N = {N1, N2, . . . , Nd′}, applicable to functions in a space U : K → R, alocal interpolation operator is uniquely defined, as

IK := I : U → P, Iu :=d′∑

i=1

Ni(u)φi, Ni(u) = Ni(Iu), i = 1, . . . , d′. (4.6)

We use the standard notation of the Dirac delta function δ(Pi) and δ(Pi) ∈ N ifNi(u) = u(Pi) = δ(Pi)u. We mainly consider these or more general interpolations.

Usually the K is chosen as a triangle, tetrahedron, an n-dimensional simplex,rectangle, parallelogram or an n-dimensional affine cuboid. Usually the following Ti areaffinely (or isoparametrically), see below, mapped onto this same K. Then properties2.–4. in Definition 4.2 make sense. We introduce subdivisions for general finite elementsand will combine it below with the above local interpolation operator on K.


Figure 4.1 Admissible triangulation for Ω with curved boundary.

Definition 4.2. Subdivision: A set T h = {T1, . . . , TM} of open subsets Ti ⊂ Ω ⊂ Rn

is called a subdivision and an admissible subdivision for Ω if it satisfies 1. and 1.–4.,respectively.

1. Ti ∩ Tj = ∅ for i �= j and Ω =M∪i=1

T i;

2. each T ∈ T h has at least n + 1 and at most n ≥ n + 1 vertices, e.g. trianglesand quadrangles with straight or curved edges;

3. T i ∩ T j for i �= j is either empty or has one vertex or a common edge or one,in general (n− 1) -dimensional surface spanned by joint vertices of Ti and Tj,see Figure 4.1;

4. for paralellotopes or cuboids, hence Qi = Πnj=1[aj , bj ], generally Ti, 3. can be

relaxed, e.g. for rectangles, Ti ∩ Tj may be one half of the larger edge of thelarger Tj, see Figure 4.15.

T h is called a triangulation when the T ∈ T h are triangles, tetrahedrons or n-dimensional simplices, possibly with curved edges, cf. Figure 4.1.

4.2.2 Polynomial finite elements, triangular and rectangular K

Chosing piecewise polynomials as P are certainly the most important FEs. We startchoosing K as triangles and quadrangles for n = 2 and their generalizations to n > 2.To avoid difficulties near the boundary, we start with

a polyhedral domain Ω, for n = 2 called a polygon. (4.7)

Curved boundaries Ω are treated by interpolation, approximation or isoparametricFEs in Subsections 4.2.7 and 4.2.6. We choose the reference domain K as a unitrectangular n-triangle or a unit n-cube, hence diam K = 1 or diam K =

√n. We

give some examples of FE triples (K,P,N ). The most important choice for P arepolynomials with degree d:

P = Pnd :=

⎧⎨⎩∑|i|≤d

αixi : x = (x1, . . . , xn) ∈ Rn

⎫⎬⎭ ,Pd := P2d ,Pd := P1

d , (4.8)

with i = (i1, . . . , in) ≥ 0, |i| = i1 + · · ·+ in, xi = xi11 · · ·xin

n , αi ∈ R.


Sometimes, we allow P ⊂ Pnd . The dimensions of these Pn

d are

dimP1d = dimPd = (d + 1),dimP2

d = dimPd =(d + 1)(d + 2)

2, (4.9)

and in general

dimPnd =

(n + d

n

)=(n + d

d

).

A convenient criterion for checking the unisolvence property for n-triangles combinesthe edges L ∈ {L1, L2, . . . , Ln+1} of the triangle K with Proposition 4.3, see [141]. Leta nondegenerate hyperplane L be defined as

L :={x ∈ Rn : L(x) =

∑|i|=1

αixi − β = 0

}withαi ∈ R,

∑|αi| > 0. (4.10)

Proposition 4.3. Let a polynomial P ∈ Pnd vanish on L. Then there exists a Q ∈

Pnd−1, such that P (x) = L(x)Q(x). Otherwise apply computer algebraic tools.

Triangular finite elements

In Figures 4.2–4.11 we show combinations of K with Pd for d = 1, . . . , 5, and appro-priate N , unisolvent for Pd. The different cases are indicated by points zi in K or onthe boundary of K according to

zi is marked by • if Ni(φ) = φ(zi), Ni ∈ N , for φ ∈ Pd,

and by # if Ni(φ) = φ(zi) and Ngi (φ) = grad φ(zi), Ni, N

gi ∈ N .

If only function values, •, are required, we have a Langrange finite element. If functionvalues and first (#) or, even additionally, second derivatives, indicated by two circlesare required, e.g. the Argyris element, see below, we have Hermite finite elements.Proposition 4.3 is visualized in Figures 4.2–4.5.

In some cases, e.g. for the Argyris P5-element, see Figure 4.9, in addition tothe values of the function and first and second derivative, normal derivatives are

z 1 z 2

z 3

Figure 4.2 P1: Linear Lagrange FEs.


z 1 z 2

z 3

z 4z 5

z 6

Figure 4.3 P2: Quadratic Lagrange FEs.

z 10

z 1 z 2 z 4

z 5z 7

z 8z 9

Figure 4.4 P3: Cubic Lagrange FEs.

z 15

z 13z 14

z 10z 12

z 9

z 5z 3z 1

Figure 4.5 P4: Quartic Lagrange FEs.


z 1 z 2

z 3

Figure 4.6 P3: Cubic Hermite FEs.

z 1 z 2

z 3

z 6

z 9

Figure 4.7 P4: Quartic Hermite FEs.

Figure 4.8 Inequivalent cubic Hermite FEs: Incompatible normal derivatives.


z 1 z 2

z 3

Figure 4.9 Quintic Argyris FEs.

z 3 z 2

z 1

Figure 4.10 P1: Nonconforming linear Lagrange FEs (Crouzeix–Raviart).

prescribed. So in the sense of Definition 4.6 below, the Argyris elements are notaffine equivalent, but only nearly affine equivalent [135, 174]. On the other hand, seeSubsection 4.2.6, there are C1 FEs as modified Argyris elements, important for thelater FEMs. The Bell element corresponds to an 18-dimensional subset of P5.

For unisolvence we need the same number of values of the function (and/or itsderivative) as dimP, e.g. dimPd = (d + 1)(d + 2)/2. All figures show unisolvent,hence dimPd = (n + 1)(n + 2)/2 = dimN , and conforming finite elements. Figures4.10 and 4.11 show nonconforming elements, see Section 5.5. The case P1 for the linearconforming and nonconforming Lagrange elements shows that the N is certainly notuniquely determined by P.

Remark 4.4. All the following figures show a high symmetry with respect to theboundary (and interior) points. For general nondegenerate subdivisions, see Definition4.12 below, we have to impose Ni ∈ N for boundary points symmetric and of thesame kind on all edges, corresponding to those in neighboring subtriangles. For highly


z 2z 1

z 3

z 5

z 4

z 6

Figure 4.11 P2: Nonconforming quadratic Lagrange FEs.

symmetric subdivisions, e.g. rectangular triangles in a rectangle, relaxations of thissymmetry of the boundary points are possible.

Remark 4.5. Simple inductive computation of the FE-interpolant: Let the knots bechosen in a triangle K on d + 1 parallel lines with altogether s = 1 + 2 + · · · (d + 1)points z1, z2, . . . , zs, see [135]. We determine

φ ∈ Pd s.t. φ(zi) = fi, i = 1, . . . , s =(d + 1)(d + 2)

2= dim Pd. (4.11)

Using the affine equivalence, see Definition 4.6, we can choose the parallel lines asy = constant. The φ for the d + 1 = 1 line is obvious. So we apply the following (4.12)inductively. We assume the first d lines for φ ∈ P2

d−1 in (4.11). Let the last line, withthe first d + 1 knots, z1, · · · ., zd+1, be located at y = 0.

Determine p0 ∈ P1d : p0(zi) = fi, i = 1, . . . , d + 1, and, by induction, (4.12)

determine the unique q ∈ P2d−1 : q(zi) =

1yi

(fi − p0(zi)), i = d + 2, . . . , s.

Then p ∈ P2d = Pd, with p(x, y) := p0(x) + y q(x, y) solves (4.11).

A similar approach is possible for higher dimensions and some rectangular elements.

Bilinear, biquadratic, . . . finite elements

For rectangular K we often use instead of Pd the tensor products of bilinear,biquadratic, . . . polynomials, see Figures 4.12–4.15 and [141,572]

Q2d = P1

d ⊗ P1d =

⎧⎨⎩u =

⎛⎝ ∑0≤i≤d

ci xi

⎞⎠⎛⎝ ∑0≤j≤d

dj yj

⎞⎠ , i, dj , x, y ∈ R

⎫⎬⎭ ⊂ P22d

Pnd ⊂ Qn

d =n∏

i=1

P1d ⊂ Pn

nd, Qd = Q2d, dimQn

d =(dimP1

d

)n= (d + 1)n. (4.13)

z 1 z 2

z 3 z 4

Figure 4.12 Q1: Bilinear Lagrange FEs.

z 1 z 2 z 3

z 6

z 7 z 9

Figure 4.13 Q2: Biquadratic Lagrange FEs.

z1 z2 z3 z4

z8

z13 z16

z12

Figure 4.14 Q3: Bicubic Lagrange FEs.


quadraticinterpolation

Figure 4.15 Quadratic subdivision (mesh-refinement), violating Definition 4.2, 3., see 4..

4.2.3 Interpolation in finite element spaces, an example

We want to interpolate on general subdivisions of Ω. This requires the extension fromthe (K,P,N ) situation to the collection of the K = T ∈ T h, denoted as (K, P, N ).For efficient computations they have to be closely related to the original (K,P,N ).Therefore we introduce

Definition 4.6. Affine equivalent FEs: Let (K,P,N ) and (K, P, N ) be two finiteelements, let F : K → K, F (x) = Ax + b, A ∈ Rn×n nonsingular, b ∈ Rn, be an affine(or isoparametric) map such that, see Figure 4.16,

1. F (K) = K,2. P ◦ F = P or P = P ◦ F−1 and3. N (f) = N (f ◦ F ) = N (f) for f : K → R and f := f ◦ F.

Then (K,P,N )�F (K, P, N ) and (K, P, N ) is called (affine) equivalent to (K,P,N ).

Obviously affine equivalence is an equivalence relation. If P = span{φ1, φ2, . . . , φd′},cf. Definition 4.1, is a dual basis for N with Ni(φj) = δij , then P = span{φ1 = φ1 ◦F−1, . . . , φd′ = φd′ ◦ F−1} is a dual basis for N and Niv = Ni(v ◦ F ).

The elements in Figures 4.8 and 4.17 are inequivalent.Now we can handle the local interpolation in (K,P,N ) and its affine equivalent

elements, e.g. the (T,PT ,NT ). If (4.7) is satisfied, we assume for simplicity the same(K,P,N ) and appropriate FT and PT ◦ FT = P. A generalization to different K isstraightforward, but we need additional technical tools, e.g. combining triangular


K KF

z1

z 1 z 2

z 3

z2

z3

Figure 4.16 Affine transformation.

Figure 4.17 Inequivalent quadratic Lagrange FEs: Affine mappings impossible.

and rectangular elements. For introducing the space of FEs, we need K,P,N , thesubdivision T h and the affine (or isoparametric) mappings

F = FT : K → T, Fx = Ax + b, A = AT ∈ Rn×n, b = bT ∈ Rn∀ T ∈ T h. (4.14)

Definition 4.7. Let T h be a subdivision for Ω and let FT : K → T be chosen as in(4.14). Then

Uh := {uh ∈ L∞(Ω) : uh|T ∈ PT := P ◦ (FT )−1} (4.15)

is denoted as (approximating) space of FEs, uh as FEs. Since the T ∈ T h are open,(4.15) does not imply transition properties for uh|T and uh|T1 if T ∩ T1 �= ∅. Fornonconforming FEs, uh, see below, and an edge e ⊂ T ∩ T1, usually uh|T |e �= uh|T1|e .Therefore, in many cases additional properties are imposed such as

Uh ∩ L2(Ω), Uh ∩ C0(Ω), Uh ∩H10 (Ω).

For the following local and global interpolation operators we combine K, P ={φ1, . . . , φd′}, and N = {N1, . . . , Nd′}, with the bijection FT , see Definitions 4.2 ff.Choose U � v : Ω → R such that ∀T ∈ T h the NT

i (v) := Ni(v ◦ FT ) are defined.


Definition 4.8. Choose the FE space Uh : Ω → R with the underlying (K,P,N ), Pa nodal basis for N , the subdivision T h and a function v : Ω → R:

1. Let (T,PT ,NT ) and the fixed (K,P,N ) be affine equivalent ∀T ∈ T h such thatFTx = ATx + bT satisfies FT (K) = T. cf. Figure 4.18.

2. Define the interpolation operator Ih = IhT h : U → Uh with NT

i (v) :=Ni(v ◦ FT )

(= NT

i (Ihv ◦ FT ))

and φTi := φi ◦ F−1

T , as, see (4.6),

Ihv|T = IhT hv|T = IT v =

d′∑i=1

NTi (v)φT

i : T → R ∀ T ∈ T h. (4.16)

3. The φTi = φi ◦ F−1

T , i = 1, . . . d′, ∀ T ∈ T h are called an interpolationbasis for PT for a fixed T ∈ T h (for ∀T ∈ T h not a basis for Uh,see 4.

4. Assume P ∈ T ∩ T1 to be a joint vertex or a point on an edge e or an (n− 1)-dimensional boundary surface of T with, e.g. δ(P ) ∈ NT . Then ∀ Tj ∈ T h

with P ∈ Tj the δ(P ) ∈ NTjand identical function values in this point P have

to be used for all these Tj � P defining, e.g. an N(v ◦ FTj) = δ(P )v. (This

implies continuity of the interpolation function in P from T to Tj, but noton e and in Ω and δ(P ) yields exactly one basis element for Uh.) Analogousrestrictions are to be imposed for derivatives. Exceptions are the DCGMs inChapter 7.

5. Let l be the highest derivative required in N . A reference element (K,P,N ),K ⊂ Ω is said to be a Cr element if r is the largest nonnegative integer suchthat

Uh = Ih(Cl(Ω)) ⊆ Cr(Ω) ∩W r+1∞ (Ω).

6. Let Pnd−1 = P ⊂W d

∞(K). Then K or the FEs are called polynomial FEs of (total)degree d− 1 or order d and denoted as Pn

d−1(T h). For Pnd−1 ⊆ P ⊆ Pn

d+τ , P ⊂W d

∞(K), with the smallest possible τ ≥ −1, e.g. the bilinear FEs, we call themFEs of at least degree d− 1.

Proposition 4.9. Under the conditions of Definition 4.8 we get, cf. (4.15),

IKv = v ∀ v ∈ P and Ihvh = vh ∀ vh ∈ Uh implying (4.17)

(IK)2 = IK on P and (Ih)2 = Ih on Uh, hence, IK , Ih are projectors.

Remark 4.10. There are different reasons to use the reference elements (K,P,N ).Most theoretical results can be proved by combining the specific situation (K,P,N )with the affine equivalence.

For the numerical praxis another point is even more important: The Ni(v) andφi, i = 1, . . . , d′ can be computed once. Then the NT

i (v) := Ni(v ◦ FT ) and φTi = φi ◦

F−1T are obtained by simple transformation formulas via the FT . This applies to the

terms needed in the FEMs below, e.g. elements of the mass and stiffness matrices aswell, cf. (4.121).


K T = KF = FT

z 1

z 1 z 2

z 3

z 2

z 3

Figure 4.18 Affine transformations here applied to linear FEs.

Example 4.11.

We want to demonstrate this procedure for a triangulation applied to two cases of lin-ear and cubic two-dimensional polynomials defined by the values indicated in Figures4.19 and 4.20, see [576], p. 68 ff. We start with the transformation from the referencetriangle K with the vertices z1 := (0, 0), z2 := (1, 0), z3 := (0, 1), and the coordinates(ξ , η) ∈ K onto the triangle T . We denote edges of T as zi := (xi, yi), i = 1, 2, 3 andits coordinates as (x , y) ∈ T , thus defining FT = ((FT )1, (FT )2)T as

x = (FT )1 (ξ , η) := x1 + (x2 − x1) ξ + (x3 − x1) η

y = (FT )2 (ξ , η) := y1 + (y2 − y1) ξ + (y3 − y1) η. (4.18)

Since FT is linear its derivative is constant and

F ′T =

((x2 − x1) (x3 − x1)(y2 − y1) (y3 − y1)

).

z 1 z 2

z 3

Figure 4.19 P1: Again linear Lagrange FEs.


z 1 z 2

z 3

Figure 4.20 P3: Again cubic Hermite FEs.

This FT is bijective if and only if the Jacobian, J, of F ′T is

J = det (F ′T ) = det

∣∣∣∣∣ (x2 − x1) (x3 − x1)(y2 − y1) (y3 − y1)

∣∣∣∣∣= (x2 − x1)(y3 − y1)− (y2 − y1)(x3 − x1) �= 0. (4.19)

Then FT is invertible. (FT )−1 has the form

ξ = (FT )−11 (x , y) := a1 + a2x + a3 y

η = (FT )−12 (x , y) := b1 + b2x + b3 y. (4.20)

By equating the coefficients in FT ◦ (FT )−1 = I we determine the a1, . . . , b3 as

a1 = −(x1(y3 − y1)− y1(x3 − x1))/J, a2 = (y3 − y1)/J,

a3 = −(x3 − x1)/J, b1 = (x1(y2 − y1)− y1(x2 − x1))/J,

b2 = −(y2 − y1)/J, b3 = (x2 − x1)/J. (4.21)

We use the notation

u : (ξ , η) ∈ K → R and v : (x , y) ∈ T → R.

For the later computations of the FE terms in mass and stiffness matrices, see Example4.51 below, we need the corresponding transformations of the partials via the chainrule:

For v := u ◦ F−1T we find vx = uξξx + uηηx

vy = uξξy + uηηy.


For computing the partials of the ξ, η we differentiate (4.18) with respect to x andfind

1 = (x2 − x1) ξx + (x3 − x1) ηx

0 = (y2 − y1) ξx + (y3 − y1) ηx.

Analogously, we differentiate with respect to y. Both systems yield the solutions

ξx = (y3 − y1)/J, ξy = − (x3 − x1)/J

ηx = −(y2 − y1)/J, ηy = (x2 − x1)/J. (4.22)

To give a feeling for the interplay of the transformations in Ni(v ◦ FT ) ·(φi ◦ F−1

T

)we

will study below a special case of (4.18), the scaling

F (ξ, η) := FT (ξ, η) := (x = hξ, y = hη) with

ξx = 1/h, ξy = 0, ηx = 0, ηy = 1/h (4.23)

implying

vx = uξ/h, vy = uη/h. (4.24)

We apply these results to two special cases:

Linear Lagrange FEs: We choose P = PT = P1 and v and u have the form

v = v(x, y) = c1 + c2 x + c3 y,

u = u(ξ, η) = α1 + α2 ξ + α3 η.

For the z1 = (0, 0), z2 = (1, 0), z3 = (0, 1), and N = {δ(z1), δ(z2), δ(z3)} we wantto determine a nodal basis P = span {φ1, φ2, φ3}. More generally, to determine aφ interpolating in the zi or the φi in (4.6), we compute in a first step the αj forgiven ui := u(zi) in Figure 4.19. We introduce the vectors α := (α1, α2, α3)T and u :=(u1, u2, u3)T and obtain the system

α1 = u1

α1 +α2 = u2

α1 +α3 = u3

implyingu1 = α1

−u1 +u2 = α2

−u1 +u3 = α3.(4.25)

This (4.25) has the form

u = Bα with B :=

⎛⎝ 1 0 01 1 01 0 1

⎞⎠, or α = Au with A :=

⎛⎝ 1 0 0−1 1 0−1 0 1

⎞⎠, (4.26)

where AB = I.We employ the unit vectors ei, i = 1, 2, 3, e.g. e1 := (1, 0, 0)T , for determining the

nodal basis φi. We obtain the coefficients αi =(αi

1, αi2, α

i3

)T for the φi as

αi = Aei, e.g. α1 = (1,−1,−1)T , φ1(ξ, η) = 1− ξ − η.


For determining the φi(F−1T ) we insert in φi(ξ, η) for the ξ, η the terms in (4.20)

with the a1, . . . , b3 determined in (4.21). Finally IKu =∑3

j=1 φju(Pj) and IT v =∑3j=1 φj

(F−1

T

)v(Pj).

In cases of simple T,P,N we can often determine the φj and φj

(F−1

T

)and the Ihv|T

directly, e.g. for our linear Lagrange FEs and the scaling in (4.23). With T = hK,we get obviously φ1 = 1− ξ − η, φ2 = ξ, φ3 = η from (4.26) and with F−1

T in (4.23)we find, e.g. φT

1 = φ1 ◦ F−1T = (h− x− y)/h. The zi = Pi and zi = Pi show NT

i v =Ni(v ◦ FT ) = v(FT (Pi)) = v(Pi). Then (4.16) yields

Ihv|T = [v(P1)(h− x− y) + v(P2)x + v(Pi)y]/h

(Ihv|T )′ = ((v(P2)− v(P1)), (v(P3)− v(P1)))/h

with

‖Ihv|T ‖∞ ≤ max{|v(Pi)| : i = 1, 2, 3} ≤ ‖v|T ‖∞,

‖(Ihv|T )′‖∞ ≤ max{|(v(P2)− v(P1))|, |(v(P3)− v(P1))|}/h ≤ ‖v′|T ‖∞.

Cubic Hermite FEs: The ansatz for v and u is now

v = v(x, y) = c1 + c2 x + c3 y + c4 x2 + c5 xy + c6 y2

+ c7 x3 + c8 x2y + c9 xy2 + c10 y3,

u = u(ξ, η) = α1 + α2 ξ + α3 η + α4 ξ2 + α5 ξ η + α6 η2

+α7 ξ3 + α8 ξ2 η + α9 ξ η2 + α10 η3. (4.27)

In addition we need the partials of u

uξ = α2 +2α4ξ +α5η +3α7ξ2 +2α8ξη +α9η

2

uη = α3 +α5ξ +2α6η +α8ξ2 +2α9ξη +3α10η

2.

With zi = Pi, zi = Pi, z4 := (z1 + z2 + z3)/3, we determine the FEs in (4.6), see Figure4.20, by the following function values and derivatives:

ui := u(zi), i = 1, . . . , 4, pi := uξ(zi), qi := uη(zi), i = 1, 2, 3. (4.28)

We determine any φ in (4.6), by evaluating in the first step, the u and its derivativesin (4.27), then compute, in the next step, the α := (α1, . . . , α10)T from the system

u := (u1, p1, q1, u2, p2, q2, u3, p3, q3, u4)T = Bα,

with a matrix B obtained as follows. Every row of B, e.g. numbers 1 and 5, yield thevalues of the corresponding components of u. For numbers 1 and 5 these are the u1


and p2. Then (4.28) tells us u1 = u(z1 = (0, 0)), p2 = uξ(z2 = (1, 0)). The evaluationaccording to (4.27) and the following uξ then yields rows 1 and 5 of B, and so on.

B :=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 | 0 0 0 | 0 0 0 00 1 0 | 0 0 0 | 0 0 0 00 0 1 | 0 0 0 | 0 0 0 0− − − | − − − | − − − −1 1 0 | 1 0 0 | 1 0 0 00 1 0 | 2 0 0 | 3 0 0 00 0 1 | 0 1 0 | 0 1 0 0− − − | − − − | − − − −1 0 1 | 0 0 1 | 0 0 0 10 1 0 | 0 1 0 | 0 0 1 00 0 1 | 0 0 2 | 0 0 0 3− − − | − − − | − − − −1 1/3 1/3 | 1/9 1/9 1/9 | 1/27 1/27 1/27 1/27

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

For computing the φi we recall Ni(φj) = δi,j , hence, we determine αi from Bαi = ei,with the unit vectors ei, i = 1, . . . , 10, or we invert B as A := B−1. Then the coefficientvectors α for a FE and the αi for the φi are obtained from the systems α = Au andαi = Aei, e.g. for i = 8 as column number 8 of A, as

α = Au with αi = Aei, e.g., φ8 = −ξη + ξ2η + 2ξη2 and

A :=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 | 0 0 0 | 0 0 0 00 1 0 | 0 0 0 | 0 0 0 00 0 1 | 0 0 0 | 0 0 0 0− − − | − − − | − − − −−3 −2 0 | 3 −1 0 | 0 0 0 0−13 −3 −3 | −7 2 −1 | −7 −1 2 27−3 0 −2 | 0 0 0 | 3 0 −1 0− − − | − − − | − − − −2 1 0 | −2 1 0 | 0 0 0 0

13 3 2 | 7 −2 2 | 7 1 −2 −2713 2 3 | 7 −2 1 | 7 2 −2 −272 0 1 | 0 0 0 | −2 0 1 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

Then we proceed as for the Lagrange case, however we do not directly computethe φT

i .In addition we do need the transformation of the partials in the zi to the zi. We

explicitly formulate this for the above special case of scaling in (4.23). With (4.24)and, e.g. N8(v ◦ FT ) ·

(φ8 ◦ F−1

T

), we obtain(

φ8 ◦ F−1T

)(x, y) = −xy/h2 + (x2y + 2xy2)/h3

N8(v ◦ FT ) = (v ◦ FT )ξ(0, 1) = v′(FT (0, 1)) · (FT )ξ(0, 1)

= ((vx, vy)(0, h))(h, 0)T = hvx(0, h)


hence

N8(v ◦ FT ) ·(φ8 ◦ F−1

T

)(x, y) = vx(0, h)(−xy/h + (x2y + 2xy2)/h2)

for 0 ≤ x, y, x + y ≤ h.

This shows, in particular, that the contribution of N8(v ◦ FT ) ·(φ8 ◦ F−1

T

)(x, y) in T

is O(h).30 Similarly the other terms Ni(v ◦ FT ) ·(φi ◦ F−1

T

)can be computed. �

We do not intend formulating the sharpest results under the weakest possibleconditions for the different cases. Rather we formulate Condition 4.16 below, whichallows the following implications: The interpolation operator Ih is a bounded operator,a good approximation and the FEs allow inverse estimates. We introduce

Definition 4.12. Let {T h}, 0 < h ≤ 1, be a family of subdivisions for Ω such that

max{diam T : T ∈ T h} ≤ h diam Ω.

Then h is denoted as the maximal step size. The family is said to be quasiuniform ifthere exists χ > 0 such that

min{diam BT : BT ⊂ T ∈ T h} ≥ χ h (4.29)

for all h ∈ (0, 1]; here BT is the largest ball such that T is star-shaped with respect toBT , cf. Definition 4.14. The family is said to be nondegenerate if there exists χ > 0such that

diam BT ≥ χ diam T ∀ T ∈ T h, hT := diamT, h ∈ (0, 1]. (4.30)

Remark 4.13.

1. The sequence of {T h} is nondegenerate if and only if the chunkiness parameterγ = γT , see (4.32), is uniformly bounded for all T ∈ T h and for all h ∈ (0, 1],hence γT = 1/χ. Let all T ∈ T h be triangles. Then T h is nondegenerate if andonly if for every T ∈ T h the interior angles α ≥ α0 > 0. This is used as thedefinition for quasiuniform T h e.g. in [135, 387]. A quasiuniform family isnondegenerate, but not conversely.

2. If we start with an arbitrary triangulation in two dimensions and repeatedlysubdivide by connecting midpoints of edges, we obtain a nondegenerate familyof subdivisions. For a well-defined alternative, see [58].

3. For quasiuniform T h each T is uniformly star-shaped.

4.2.4 Interpolation errors and inverse estimates

We have already formulated the general interpolating uh = Ihu on Ω in (4.16). Itreproduces function values and/or derivatives of the original u according to N . Wewant to study the question of how well these Ihu approximate the original u. We collect

30 f(h) = O(hp) and f(h) = o(hp) iff |f(h)|/hp ≤ C and |f(h)|/hp → 0 for h → 0.


the appropriate assumptions in Condition 4.16. Mainly for the study of variationalcrimes we need estimates of higher Sobolev norms of FEs by lower Sobolev norms, theso-called inverse estimates.

We need the standard and broken Sobolev norms, cf. (4.36), and some lemmas. Weuse the notation for multi-indices, α, and partials, ∂α, see (2.73),

‖v‖W d,p(Ω) :=( d∑

j=0

(|v|W j,p(Ω))p)1/p

, |v|W j,p(Ω) :=( ∑

|α|=j

‖∂αv‖pLp(Ω)

)1/p

. (4.31)

For w ∈Wm,p(Ω) the classical derivatives and hence their point evaluation andTaylor polynomials Tm

y w(x) =∑

|α|<m ∂αw(y)(x− y)α/α! are not defined. So we usethe ∂α w ∈ L2(Ω), |α| < m, directly and introduce the averaged Taylor polynomialsQmw for w ∈Wm,p(Ω), see [141], in

Definition 4.14. Let G ⊂ Rn, e.g. G = Ω, have finite diam G and let B := Bρ(x0)be a ball of radius ρ. We assume B ⊂ G such that for all x ∈ G the closed convex hullof {x} ∪ B ⊂ G. Then G is called star-shaped with respect to B. With

ρmax := sup{ρ : G star-shaped with respect to B}, and γ = diam G/ρmax, (4.32)

this γ is called the chunkiness parameter of G. Let G be star-shaped with respect to Band let φ be a cut-off function for B, that is (i) supp φ = B and (ii)

∫Rn φ(x)dx = 1.

Then the following averaged Taylor polynomials Qdw of degree d− 1, and the T dy w(x)

are well defined, ∀x ∈ B and ∀y ∈ B \M , M a set of Lebesgue measure 0, as

Qdw(x) :=∫

B

T dy w(x)φ(y)dy with T d

y w(x) =∑|α|<d

∂αw(y)(x− y)α/α!. (4.33)

Obviously, triangles are star-shaped. Note that only ∂αu with |α| < d are employedin (4.33). A standard example for a cut-off function on B = Bρ(x0) is

φ(x) = c exp (−(1− (|x− x0|/ρ)2)−1) for x ∈ B and φ(x) ≡ 0 outside B.

We only consider star-shaped G (or T ∈ T h, see below) with bounded chunkinessparameter, γ, see (4.32). We obtain, see [141],

Lemma 4.15. Bramble–Hilbert lemma: Let G be star-shaped with respect to B ofradius ρ such that ρ > (1/2)ρmax, G has the chunkiness parameter γ, and Qdw be theaveraged Taylor polynomial. Then for all w ∈W d,p(G), � = 0, 1, . . . , d, 1 ≤ p ≤ ∞,there exist constants Cd,n,γ such that

|w −Qdw|W �,p(G) ≤ ‖w −Qdw‖W �,p(G) ≤ Cd,n,γ(diam G)d−�|w|W d,p(G). (4.34)

Since the uh ∈ Uh might not be continuous across common edges e ⊂ T ∩ T 1, andhence uh �∈ H1(Ω) [174], pp. 207–208, the usual Sobolev norms ‖ · ‖W d,p(Ω) mightnot be defined. However, for all practically important cases the ‖uh|T ‖W d,p(T ) and


‖u|T ‖Cd(T ). So the broken Sobolev spaces and norms ‖uh‖W d,p(T h) make sense:

UT h := W k,p(T h) := {uh : Ω → R : ∀ T ∈ T uh|T ∈W k,p(T )}, with (4.35)

‖uh‖hk,q := ‖uh‖W k,q(T h) =

( ∑T∈T h

∑|α|≤k

‖∂αuh|T ‖qLq(T )

)1/q

,

for 1 ≤ q <∞, (4.36)

‖uh‖W k∞(T h) := ess sup{|∂αuh(x)| : ∀α |α| ≤ k∀x ∈ T ∀ T ∈ T h}. (4.37)

These norms satisfy Holder inequalities as the standard norms:

‖uhvh‖W k,1(T h) ≤ ‖uh‖W k,p(T h)‖vh‖W k,q(T h) for 1 ≤ p, q ≤ ∞, 1/p + 1/q = 1 (4.38)

The corresponding seminorms with∑

|α|=k in (4.36) are

|uh|W k,q(T h) =( ∑

T∈T h

∑|α|=k

‖∂αuh|T ‖qLq(T )

)1/q

=( ∑

T∈T h

|uh|qW k,q(T )

)1/q

. (4.39)

These norms coincide with the original norm ‖uh‖W k,q(T h) for uh ∈W d,p(Ω). Analo-gous definitions and equalities may have to be used for scalar products and bilinearforms as well. The decision about which of the following choices of d, p in the spaces

UhW d,p(Ω) := {uh ∈ Uh : ‖uh‖W d,p(T h) <∞} for h→ 0 (4.40)

is appropriate, depends upon the interplay of ansatz- and test-functions and the(non)linear elliptic problem. They are related by the corresponding bilinear or otherforms. Certainly, for each fixed h we want the ‖uh‖W d,p(T h) <∞. We start with apolyhedral domain Ω ⊂ Rn, for curved ∂Ω see Subsections 4.2.7, 4.2.6.

Condition 4.16. For the reference FE (K,P,N ) and the subdivision T h assume:

1. K is star-shaped with respect to some ball.2. Choose the smallest τ ≥ −1 such that Pn

d−1 ⊆ P ⊆ Pnd+τ , P ⊂W d

∞(K), usuallyτ = −1. For τ ≥ 0, let P � Pn

d+τ , hence, there exist v ∈ P ∩ Pnd+τ .

3. N ⊂ (Cl(K))′, hence the elements of N are evaluations of u and derivativesof u up to the order ≤ l in (different) points P for P ∈ K and additionally let1 ≤ p ≤ ∞ and{

d− l − n/p > 0 for p > 1 ord− l − n ≥ 0 for p = 1.

4. Alternatively to 3. we assume N ⊂ (W d,p(K))′.5. {T h}, 0 < h ≤ 1 , is a nondegenerate family of admissible subdivisions of a

bounded polyhedral domain Ω ⊂ Rn with the parameter χ, see (4.30).6. For all T ∈ T h, 0 < h ≤ 1, the (T,PT ,NT ) is affine equivalent to (K,P,N ).7. The combination of 1. and 5. shows that T is uniformly star-shaped ∀ T ∈ T h

and that the chunkiness parameter γ of all T ∈ T h is bounded, see e.g. [141],Theorem 4.4.20.


The unusual condition τ ≥ −1 such that Pnd−1 ⊆ P � Pn

d+τ allows us to study, e.gthe families of tensor product FEs as in (4.13), the serendipity FEs, see [141], pp. 88ff. and pp. 112 ff. For all these families and for Davydov’s C1 splines, see Subsection4.2.6 and [263], specific error estimates are possible. For linear and quadratic tensorproduct FEs we get τ = 1 and τ = 3 for n = 2 or τ = 4 and τ = 17 for n = 3. Thefollowing results are proved in [141], Theorems 4.4.4, 4.4.20, 4.6.11 and 4.6.14, compare[135, 387] as well. Note that the conditions, relating d, l, n, p in [141], Theorems 4.4.4and 4.4.20 are imposed to guarantee Condition 4.16, 3. and to apply the Sobolevembedding lemma. They are not necessary for 4. Furthermore, these results extend toa combination of v ∈W d,p(Ω) and FEs of higher order.

Theorem 4.17. Interpolation errors: Let the {T h}, 0 < h ≤ 1, and the referenceelement, (K,P,N ), satisfy Condition 4.16 for some l, d, p, 1 ≤ p ≤ ∞, cf. (4.36),(4.37). Let Ih be the (local) interpolation operator in (4.16) defined by (K,P,N ).

1. Then there exists a constant C > 0, such that ∀ 0 ≤ s ≤ d, the following localand global interpolation error estimates hold ∀v ∈W d,p(Ω):

‖v − Ihv‖W s,p(T ) ≤ C (diam T )d−s|v|W d,p(T ),

‖Ihv‖W s,p(T ) ≤ ‖v‖W s,p(T ) + C(diam T )d−s|v|W d,p(T ). (4.41)

‖v − Ihv‖W s,p(T h) ≤ C hd−s|v|W d,p(Ω),

‖Ihv‖W s,p(T h) ≤ ‖v‖W s,p(Ω) + C hd−s|v|W d,p(Ω) ≤ C‖v‖W d,p(Ω). (4.42)

C depends on (K,P,N ), n, d, p and the number χ in (4.30).2. For 0 ≤ s ≤ l < d, see 3. in Condition 4.16, we have

‖v − Ihv‖W s,∞(T h) ≤ C hd−s−n/p|v|W d,p(Ω) ∀ v ∈W d,p(Ω). (4.43)

3. For the case of tensor product FEs the estimates for the interpolation errors in(4.41)–(4.43) can be modified by replacing the seminorms |v|W d,p(T ), |v|W d,p(Ω)

by |v|tW d,p(T ), |v|tW d,p(Ω) :=(∑n

i=1

∥∥∂dv/∂xdi

∥∥p

Lp(Ω)

)1/p.

Proposition 4.18. Our FE spaces Uh are approximating spaces for U = W s,p(Ω)and d > s ≥ 0, in the sense of Definition 3.5, hence,

limh→0

dist (u,Uh) = infuh∈Uh

||u− uh||Uh = 0 ∀ u ∈ U ⇒ limh→0

||Ihu||Uh = ||u||U . (4.44)

Proof. For fixed u ∈W s,p(Ω) and arbitrary ε > 0 we choose a v ∈W d,p(Ω) such that‖v − u‖W s,p(T h) < ε/2. Now by ‖v − Ihv‖W s,p(T h) ≤ C hd−s|v|W d,p(Ω) we can chooseh0 such that ‖v − Ihv‖W s,p(T h) < ε/2 for h < h0 , so finally ‖u− Ihv‖W s,p(T h) < ε forh < h0, Hence, our FE spaces for d > s are approximating spaces. For the last claimwe combine (4.42) with ‖u− Ihu‖ < ‖u− v‖+ ‖v − Ihv‖+ ‖Ihv − Ihu‖. �

Inverse estimates, presented in the sequel, relate various norms ‖vh‖W s,p(T h) forfinite element spaces for s ≥ d. This is necessary to handle variational crimes. We again


present local and global estimates. Note that for the next two results, see [141], pp. 110ff., we need T h and the affine (isoparametric) maps FT : K → T . However, no referenceto N ,NT nor the unisolvence is made. Since P always satisfies P ⊆W j,p(T ) ∩W l

q(T ),1 ≤ q ≤ p, 0 ≤ l ≤ j, we get

Theorem 4.19. Inverse estimates: Let {T h}, 0 < h ≤ 1, (K,P,N ),PT = P satisfyCondition 4.16, let j ≤ d + τ , 1 ≤ q ≤ p, 0 ≤ l ≤ j, and

Uh = {vh : Ω → R, vh|T ∈ PT ∀ T ∈ T h}. (4.45)

Then there exists C = C(j, p, q, χ), with χ in (4.30), such that

‖vh‖W j,p(T ) ≤ C (diam T )l−j+n/p−n/q‖vh‖W lq(T ), ∀vh ∈ Uh (4.46)

and

‖vh‖W j,p(T h) ≤ C hl−j+min(0,n/p−n/q)‖vh‖W lq(T h), (4.47)

where {T h}, 0 < h ≤ 1, is quasiuniform for (4.47), and for all vh ∈ Uh. These inequal-ities only make sense for j ≤ d + τ , otherwise ‖vh‖W j,p(T h) = ‖vh‖W d+τ,p(T h).

4.2.5 Inverse estimates on nonquasiuniform triangulations

This condition of quasiuniformity for (4.47) can be relaxed by applying the results forboundary and finite element methods in Graham, Hackbusch and Sauter [360–364],and with Dahmen, Faermann and Grasedyck in [244, 359]. We present some of theirwide range of inverse-type inequalities, here applications to finite element functionson general classes of meshes, including degenerate meshes obtained by an isotropicrefinement. These are obtained for Sobolev norms of positive, zero and negativeorder. In contrast to classical inverse estimates, negative powers of the minimum meshdiameter are avoided.

In the whole subsection the functions u = uh are always FEs.For n = 2 or 3, let Ω ⊂ Rn denote a bounded domain. Suppose that Ω is decom-

posed into a mesh, T = T h, of tetrahedra/bricks (n = 3) or curvilinear triangles/quadrilaterals (n = 2). Then classical inverse estimates give, cf. (4.47) for the leftinequality, for ∼, � compare the index,

‖u‖Hs(T h) � h−smin‖u‖L2(T h) � h−2s

min‖u‖H−s(T h), (4.48)

for a suitable range of positive s and for all functions u ∈ Hs(T h) which are piecewisepolynomials of degree ≤ d− 1 with d− 1 ≥ 0 with respect to this mesh. The quantityhmin is the minimum diameter of all the elements of the mesh and (4.48) holds underthe assumption of shape regularity, i.e. ρτ � hτ for each τ ∈ T h where hτ is thediameter of τ and ρτ = ρ is the diameter of the largest inscribed sphere, cf. Definition4.14. We will use such estimates regularly in the finite element analysis, particularlyfor nonconforming FEMs, cf. Chapter 5. When the mesh is quasiuniform, cf. Definition4.12, so h � hmin, with h the maximum diameter of all the elements, they can be usedto obtain convergence rates in powers of h for various quantities in various norms.


However, practical meshes are often nonquasiuniform and then the negative powersof hmin in (4.48) may give rise to overly pessimistic convergence rates. In [361], lesspessimistic replacements for (4.48) have been derived, a particular case being

‖u‖Hs(T h) � ‖h−su‖L2(T h) � ‖h−2su‖H−s(T h), (4.49)

where h : Ω −→ R is now a continuous piecewise linear mesh function whose value oneach element τ reflects the diameter of that element, i.e. hτ � h|τ � hτ .

Estimates (4.49) have several applications, e.g. to the analysis of the Mortar elementmethod [361]. In fact [361] contains more general versions of (4.49), e.g. in the Sobolevspace W s,p(T h) and in related Besov spaces. While the left-hand inequality in (4.49)is well-known, at least in the Sobolev space case, the right-hand inequality requiresrather delicate analysis.

In this subsection we obtain more general versions of (4.49) which do not requirethe mesh sequence to be shape-regular. A typical estimate is

‖u‖Hs(T h) � ‖ρ−su‖L2(T h) � ‖ρ−2su‖H−s(T h), (4.50)

where the mesh function ρ : Ω −→ R is now a continuous piecewise linear functionwhose value on each element τ reflects the diameter of the largest inscribed sphere,introduced in Definition 4.12. Estimates (4.50) hold under the rather weak assump-tions that (i) the quantities hτ and ρτ are locally quasiuniform (i.e. hτ/hτ ′ � 1and ρτ/ρτ ′ � 1 for all neighboring elements τ, τ ′) and (ii) the number of elementswhich touch any element remains bounded as the mesh is refined (see Assumption4.26). These assumptions admit degenerate meshes, containing long thin “stretched”elements, which are typically used for approximating edge singularities or boundarylayers in solutions of PDEs. Our estimates (4.50) hold true when all the elements τof a mesh are obtained by suitable maps from a single unit element, as for the abovefinite element spaces. For the purpose of a readable introduction we have here writtenour estimates (4.50) in a very compact form. In fact the range of s for which theright-hand inequality in (4.50) holds may be greater then that for which the left-handinequality holds and we shall give precise ranges in the last subsubsection.

Meshes and finite elements

Throughout, Ω will denote a bounded n-dimensional subset of Rn, for n = 2 or 3.We define the Sobolev spaces Hs(Ω), and Hs(T h), s ≥ 0, as in Subsection 1.4.3.Throughout, we let [−k, k] denote the range of Sobolev indices for which we are goingto prove the inverse estimates (where k is a positive integer), and we assume thatHs(Ω) is defined for all s ∈ [−k, k], and that H−s(Ω) is the dual of Hs(Ω), for s > 0.We assume that Ω is decomposed as above into a mesh T = T h of relatively openpairwise-disjoint finite elements τ ⊂ T h with the property T h = ∪{τ : τ ∈ T }.

Definition 4.20. Mesh parameters: For each τ ∈ T , |τ | denotes its n-dimensionalmeasure, hr denotes its diameter and ρτ is the diameter of the largest sphere centeredat a point in τ whose intersection with T h lies entirely inside τ . For any other simplexor cube t ∈ Rn (not necessarily an element of T ) we define ht and ρt in the same way.


In order to impose a simple geometric character on the mesh τ , we assume that eachτ ∈ T is diffeomorphic to a simple unit element. More precisely, let σ3 in R3 denote theunit simplex with vertices (0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1), and let κ3 denote the unitcube with vertices {(x, y, z) : x, y, z ∈ {0, 1}}. In R2, define σ2 to be the unit simplexwith vertices (0, 0), (1, 0) and (0, 1) and define κ2 to be the unit square with vertices{(x, y) : x, y ∈ {0, 1}}.

Then we assume, as above, that for each τ ∈ T , there exists a unit element τ = σn

or κn and a bijective map χτ : τ → τ , with χτ and χ−1τ both smooth, here ∈ C∞. Since

χτ is smooth, each element τ ∈ T is either a curvilinear tetrahedron/brick (n = 3) ora curvilinear triangle/rectangle (n = 2). The mesh T is allowed to contain both typesof elements. Each element has vertices and edges. In the R3 case the element also hasfaces, comprising the images of the faces of the unit element. For a suitable indexset Q, we let {xp : p ∈ Q} denote the set of all vertices T . We assume the mesh isconforming, cf. Definition 4.2, i.e. for each τ, τ ′ ∈ T with τ �= τ ′, τ ∩ τ ′ is allowed to beeither empty, a node, an edge or (when n = 3) a face of both τ and τ ′. The requirementthat χτ is smooth ensures that edges of T h (n = 2) and edges ∂T h (n = 3) are confinedto edges of elements τ ∈ T . Let Jτ denote the n× n Jacobian of χτ . Then

gτ :={det JT

τ Jτ

}1/2

is the Gram determinant of the map χτ , which appears in the change of variableformula:

∫τf(x)dx =

∫τf(χτ (x))gτ (x)dx. To ensure that the map χτ is sufficiently

regular we shall make the following assumptions on Jτ :

Assumption 4.21. Mapping properties:

D−1|τ |2 ≤ det{Jτ (x)TJτ (x)} ≤ D|τ |2, (4.51)

Eρ2r ≤ λmin{Jτ (x)TJτ (x)}

uniformly in x ∈ τ , for all τ ∈ T , with positive constants D,E independent of τ .

(Throughout this section, for a symmetric matrix A, λmin(A) and λmax(A) denoterespectively the minimum and maximum eigenvalues of A.) Assumption 4.21 issatisfied in a number of standard cases.

Example 4.22.

Suppose τ ⊂ Ω, where τ is for n = 2 and n = 3 a polygon and a polyeder, respectively.Then χτ can be chosen as an affine map. Then the Jacobian Jτ is identical to an n× nconstant matrix and it is well known that detJτ = |τ |/|τ | and that ‖J−1

τ ‖2 ≤ hτρ−1τ ,

from which the estimates (4.51) follow. �

Example 4.23.

In many applications quadrilateral or hexahedral elements are important. Consider,for example, quadrilateral elements τ obtained by mapping from the unit elementκ2 = (0, 1)2. If the map is affine, then the estimates for (4.51) obtained in Example4.22 carry over verbatim. However only parallelograms can be obtained by applying


affine maps to κ2. More general quadrilaterals can be obtained using bilinear mapsand it turns out that, under quite moderate assumptions, (4.51) still hold. �

This is a consequence of the following considerations for quadrilateral elements τ .A parallelogram π (considered as an open subset of R2) will be called an inscribedparallelogram for τ if π ⊂ τ and if two adjacent edges of π are identical with twoadjacent edges of τ . It is easy to show that any convex quadrilateral τ has at least oneinscribed parallelogram.

Proposition 4.24. cf. [364]: Let τ be a convex planar quadrilateral obtained byapplying a bilinear mapping to κ2. Let π be any inscribed parallelogram for τ . Thenthe estimates (4.51) hold uniformly in x ∈ τ with D,E depending continuously on theratio r := |π|/|τ | ∈ (0, 1].

Remark 4.25. If follows that if the ratio |π|/|τ | is bounded below by some constantγ > 0 say, for all elements τ as the mesh is refined, then (4.51) hold (with D and Edependent on γ). There are obvious degenerate elements which satisfy this, for exampleany parallelograms (no matter how small the smallest interior angles are) satisfy it.Similarly “moderately” distorted parallelograms also satisfy it.

Assumption 4.21 describes the quality of the maps which take the unit element τ toeach τ . We also need assumptions on how the size and shape of neighboring elementsin our mesh may vary. Here we impose only very weak local conditions which requirethe meshes to be neither quasiuniform nor shape-regular. Throughout the rest of thissubsubsection we make the following assumption.

Assumption 4.26. Mesh properties: For some K,L ∈ R+ and M ∈ N, we assumethat, for all τ, τ ′ ∈ T with τ ∩ τ ′ �= ∅,

hτ ≤ Khτ ′ , ρτ ≤ Lρτ ′ , (4.52)

maxp∈Q

#{τ ∈ T : xp ∈ τ} ≤M. (4.53)

Note that condition (4.52) requires that hτ and πτ do not vary too rapidly betweenneighboring elements. This allows elements with large aspect ratio, provided theirimmediate neighbors have a comparable aspect ratio.

Example 4.27.

Shape-regular meshes are easily shown to satisfy Assumption 4.26 with moderateK,L,M . Also, meshes which are anisotropically graded towards an edge typically liein this class. Near an edge, but away from the corners, the solution typically is badlybehaved only in the direction orthogonal to the edge and efficient approximationsrequire meshes which are anisotropically graded.

For example, for the square screen [0, 1]× [0, 1], a typical tensor product anisotropicmesh would be: xi,j = (ti, tj), where ti = (i/n)g/2 and t2n−i = 1− (i/n)g/2 for i =0, . . . , n, for some grading exponent g ≥ 1. In this case the elements become very longand thin near smooth parts of edges. In the hp version of the finite element method


similar meshes but with more extreme grading may be used and these also satisfyAssumption 4.26. �

We denote the class of meshes which satisfy Assumptions 4.21 and 4.26 asMD,E,K,L,M.From now on, if A(T ) and B(T ) are two mesh-dependent quantities, then theinequality A(T ) � B(T ) means that A(T ) ≤ CB(T ) with the constant C independentof T ∈ MD,E,K,L,M , but not necessarily independent of D,E,K,L,M , similarly forA(T ) ∼ B(T ) and B(T ) � A(T ).

Now we generalize the above definition of finite element spaces on the mesh T .

Definition 4.28. Finite element spaces: For d− 1 ≥ 0 and τ ∈ {σn, κn}, we define

Pd−1(τ) ={

polynomials of total degree ≤ d− 1 on τ if τ = σn,polynomials of coordinate degree ≤ d− 1 on τ if τ = κn,

with the σn, κn on top of p. 234. Then we define

Sd−10 (T ) = {u ∈ L∞(T h) : u ◦ χτ ∈ Pd−1(τ), τ ∈ T for d− 1 ≥ 0.

Sd−11 (T ) = {u ∈ C0(T h) : u ◦ χτ ∈ Pd−1(τ), τ ∈ T for d− 1 ≥ 1.

Inverse estimates

In this subsubsection we summarize the inverse estimates from [364], which weremotivated above (see (4.50)). To define the scaling function ρ, recall the parametersρτ introduced in Definition 4.20. From these we construct a continuous mesh functionρ ∈ S1

1 on T h as follows.

Definition 4.29. Mesh function: For each p ∈ Q, set ρp = max{ρτ : xp ∈ τ}. Themesh function ρ is the unique function in S1

1 (T ) such that ρ(xp) = ρp, for each p ∈ Q.

Clearly ρ is a positive, continuous function on Ω and, by Assumption 4.26, it followsthat ρ(x) ∼ ρτ for x ∈ τ , and all τ ∈ T . The main results of this section are Theorems4.30, 4.32 and 4.33. The first two of these provide inverse estimates in positive Sobolevnorms for functions u ∈ Sd−1

i (T ) with continuity index i = 1, 0, respectively. The thirdtheorem provides inverse estimates in negative norms.

Theorem 4.30. Inverse estimates: Let 0 ≤ s ≤ 1 and −∞ < α < α <∞. Then

‖ραu‖Hs(T h) � ‖ρα−su‖L2(T h),

uniformly in α ∈ [α, α], u ∈ Sd−11 (T ).

Remark 4.31. Since Sd−11 (T ) ⊂ Hs(T h) for all s < 3/2, it may be expected that

the range of Sobolev indices for which Theorem 4.30 holds may be extended. Such anextension has been obtained in [361] for shape-regular meshes at the expense of workingin Besov norms. We have avoided such extensions here in order to simplify the presentsubsubsection.

Theorem 4.32. Inverse estimates: Let 0 ≤ s < 1/2 and −∞ < α < α <∞. Then

‖ραu‖Hs(T h) � ‖ρα−su‖L2(T h),


uniformly in α ∈ [α, α], u ∈ Sd−10 (T ).

Theorem 4.33. Inverse estimates, negative norms: Let i ∈ {0, 1}, d− 1 ≥ i, 0 ≤ s ≤ 1and −∞ < α < α <∞. Then the inequality

‖ρs+αu‖L2(T h) � ‖ραu‖H−s(T h), (4.54)

holds uniformly in u ∈ Sd−1i (T ) and α ∈ [α, α]. If χτ is affine for all τ then (4.54)

holds for all 0 ≤ s ≤ k, where k is as described in the first paragraph of the lastsubsubsection.

Remark 4.34. Note that the test function w constructed in the proof of Theorem4.33 in Graham et al. [364] vanishes at the boundaries of elements. Hence w belongsto the closure of the space C∞

0 (Ω) with respect to the Hs(Ω) norm (this space is usuallydenoted Hs

0(Ω)). Thus the result of Theorem 4.33 also holds if H−s(Ω) was defined asthe dual of Hs

0(Ω), although we have not so defined it here.

4.2.6 Smooth FEs on polyhedral domains, with O. Davydov

This subsection is an adapted version of Davydov [264]. In the context of this book,we need smooth FEs for two kinds of problems. Conforming FEMs for weak 2m-thorder equations require FEs (piecewise polynomial splines) in Cm−1(Ω). Clearly suchpiecewise polynomial functions belong to Hm(Ω). The new FEM for fully nonlinearstrong 2m-th order elliptic problems can only be used on (curved) domains with C2m

boundary, and require FEs in C2m−1(Ω) on such domains in the case m > 1. Form = 1 we still need C1(Ω) splines, but can admit convex domains, which allows convexpolyhedral domains in particular. Moreover, applications to fully nonlinear equationsin Section 5.2.5 require certain stable splitting of the spline space. Consequently, weconsider smooth FEs in three different settings:

(1) Approximation theory of Cr(Ω) splines of local degree d, for r ≥ 1 on poly-hedral domains in Rn with d ≥ r2n + 1. For this setting we prove inverseestimates (Theorem 4.35) and error bounds for the quasi-interpolation operators(Theorem 4.36).

(2) C1(Ω) splines on polygonal domains in R2. A modification of the Argyris FEpresented below is shown to admit a stable splitting, see Theorem 4.39.

(3) C2m−1(Ω) splines, for m ≥ 1, on curved C2m domains in Rn as needed inFEMs for fully nonlinear problems. Although the question of constructing suchsplines remains widely open, we formulate Conjecture 4.41 as a basis for thesemethods.

In this and the next chapter we, in particular, prove the full convergence theoryfor elliptic equations and systems of order 2m with FEs in Hm(Ω). Smooth FEs intwo variables have been studied, e.g. in the classical book by Ciarlet [174]. For n > 2,see Zenısek [679] and Le Mehaute [484]. For most FEMs, smooth FEs or splines aresuccessfully avoided nowadays by using the standard FEMs in C(Ω) and in H1(Ω).Then equations or systems of order 2m are transformed into a system of m second order


equations, analogously for systems, sometimes losing information. The correspondingvariational crimes, see e.g. Sections 5.5, allow further generalizations. This is not yetpossible for fully nonlinear elliptic problems.

In the literature, several different approaches for smooth FEs are discussed. EitherFEs on a polyhedral or on approximate curved domains approximate the functions onthe original polyhedral or curved domain. C1(Ω) FEs on polyhedral or C2 domains,and Cm−1(Ω) FEs for m ≥ 1, are studied in many papers.C1 FEs on curved domains are studied and applied to the Karman equations by

Bernadou, e.g. [85,86] for the “curved” Argyris and Bell FEs, hence with degree d = 5.The results in [85, 86, 680, 681] and the compilations only apply to R2 and providepartial answers in the above setting (3). A totally different approach is based on theweighted extended B-splines, see e.g. Hollig, Reif and Wipper [400] They allow theincorporation of curved boundaries into splines on uniform grids with a new technique.The results about stable splitting mentioned in (2) above are new (see Davydov [264]).

On polyhedral domains many results are available for bivariate polynomial splines,see e.g. the recent book [467]. However, certain important questions have not beenaddressed, in particular the error bounds for multivariate smooth piecewise polyno-mials on general triangulations in Rn.

The error bounds below are formulated for a triangulation T h of a polyhedraldomain Ω. Thus, we deal with a family of triangulations {T h} and spaces {Sh}parametrized by h. It is assumed that the other parameters, e.g. of a basis for Sh

characterizing its stability and locality, or n, d, p, ωT h , remain independent of h. Inthis sense, one obtains respective asymptotic estimates and rates of convergence.

Splines on polyhedral domains in Rn

The triangulation T h for polyhedral domains has been introduced in Definition 4.2.T h has to satisfy the standard conditions, see Definition 4.2. In Definition 4.12 weintroduced the maximal and local step sizes h and hT = diam T. For a quasiuniformand nondegenerate T h, see (4.30), we require

∃χ > 0 :minT∈T h diam BT

h≥ χ and

1ωT h

:= minT∈T h

diamBT

hT≥ χ, (4.55)

respectively, and ωT h is called the shape regularity constant of T h.For any d ≥ 0, we consider the spaces of all (including discontinuous) piecewise

polynomials Sd(T h) and the space Srd(T h) of splines of smoothness r ≥ 0, defined by

s ∈ Sd(T h) ⇔ s|T ∈ Pnd ∀T ∈ T h and

Srd(T h) = Sd(T h) ∩ Cr(Ωh), 0 ≤ r < d, (4.56)

where Pnd denotes the space of n-variate polynomials of total degree ≤ d. In particular,

S0d(T h) is the classical space of continuous FEs.The spaces Sr

d(T h) are subspaces of the W r+1,p(T h) with norms and subnorms in(4.35)–(4.39) and corresponding scalar products for p = 2.

As a consequence of the differentiability of the Srd(T h) we have to change the

assumptions here and in Section 5.2 compared with the earlier sections. The dif-ferentiability imposes a semilocal condition, cf. (4.62), not directly comparable with


the preceding Pd−1 ⊂ P ⊂ Pd+τ , cf. Definition 4.8, 6. Condition (4.62) allows us forexample to include reduced FEs, e.g. the Bell element into consideration.

Note that for nested sequences of triangulations T h1 � T h

2 � · · · , nested splinespaces Sr

d

(T h

1

)⊂ Sr

d

(T h

2

)⊂ · · · are available, allowing multiresolution techniques, see

Davydov and Petrushev [265] and Davydov and Stevenson [268].

Inverse estimates

We start by establishing the inverse estimates needed in the proof of the interpolationerror bound and of independent interest in the FEM.

The multivariate Markov inequality for a simplex T ⊂ Rn, see Coatmelec [181],

‖Dαs‖L∞(T ) ≤ cnd2

diamBT‖s‖L∞(T ), s ∈ Pn

d , |α| = 1,

implies the following inverse estimates.

Theorem 4.35. Inverse estimates: For any 1 ≤ p ≤ ∞, 0 ≤ k < μ ≤ d, we have

|s|W μ,p(T ) ≤K

hμ−kT

|s|W k,p(T ), s ∈ Sd(T h) for simplices T ∈ T h, (4.57)

where the constant K depends only on n, d, p, ωT h , see (4.55). For a quasiuniform T h,hT = diam T can be replaced by h.

Proof. Let |α| = μ and g = s|T ∈ Pnd . Since Dαg = DβDγp for some β, γ with |β| =

μ− k, |γ| = k, and since Dγg ∈ Pnd−k, a repeated application of the Markov inequality

implies

‖Dαg‖L∞(T ) ≤a1

diamBμ−kT

‖Dγg‖L∞(T ) ≤a2

hμ−kT

‖Dγg‖L∞(T ),

where a1 depends only on n, d, and a2 = a1ωμ−kT h . This already proves (4.57) in the

case p = ∞. For 1 ≤ p <∞, a simple scaling argument shows that

a−13 vol1/p (T )‖q‖L∞(T ) ≤ ‖q‖Lp(T ) ≤ a4vol1/p (T )‖q‖L∞(T ), q ∈ Pn

d , (4.58)

where vol (T ) is the n-dimensional volume of the simplex T , and the constants a3, a4

depend only on n, d, p. Therefore, we get

‖Dαg‖Lp(T ) ≤a2a3a4

hμ−kT

‖Dγg‖Lp(T ),

and (4.57) follows. The statement for the quasiuniform case is obvious. �

Error bounds

For smooth splines on a polyhedral domain Ω in Rn we now prove error bounds ofquasi-interpolation. For FE subspaces these bounds can be found e.g. in [141], and forbivariate spline spaces in [467], Theorem 5.18. Anisotropic triangulations have beenconsidered in [265].

We introduce the star of a vertex v of T h, denoted by star(v), as the closure ofthe union of all n-simplices T ∈ T h with a vertex at v. We set star1(v) := star(v),


and define starγ(v), γ ≥ 2, recursively as the union of the stars of all vertices of T h

contained in starγ−1(v).Assuming that Sh is a subspace of Sd(T h), we choose a basis for Sh and its dual

(Sh)′, such that

Sh = span [s1, . . . , sN ], (Sh)′ = span [λ1, . . . , λN ], λisj = δij , i, j = 1, . . . , N. (4.59)

The basis [s1, . . . , sN ] is said to be stable and local if for each k = 1, . . . , N , a setEk ⊂ Ω exists, such that

supp sk ⊂ Ek ⊂ starγ(vk) for an appropriate vertex vk of T h, (4.60)

‖sk‖L∞(Ω) ≤ C1 and |λks| ≤ C2‖s‖L∞(Ek), ∀s ∈ Sh, (4.61)

for some C1, C2 and γ. Indeed, (4.60) implies that the basis functions sk have localsupport, and it follows from (4.60), (4.61) that they are stable in L∞ in the sense thatfor any real α1, . . . , αN ,

K1 max1≤k≤N

|αk| ≤∥∥∥ N∑

k=1

αksk∥∥∥

L∞(Ω)≤ K2 max

1≤k≤N|αk|,

with K1,K2 depending only on n, d, ωT h , γ, C1, C2.It can also be shown [263], Lemma 6.2, that, after renorming, the new basis

s1, . . . , sN with sk := vol−1/p(Ek) sk, is stable in Lp, 1 ≤ p <∞, that is

K1

(N∑

k=1

|αk|p)1/p

≤∥∥∥∥∥

N∑k=1

αksk

∥∥∥∥∥Lp(Ω)

≤ K2

(N∑

k=1

|αk|p)1/p

.

In addition, we assume that Sh contains the space of polynomials of certain degree,such that

Pn�−1 ⊂ Sh ⊂ Sd(T h) for some 0 ≤ �− 1 ≤ d. (4.62)

The following theorem provides error bounds for certain quasi-interpolation operatorIh : L1(Ω) → Sh. For Sr

d−1(T h) in (4.56), we have Pnd−1 ⊂ Sr

d−1(T h). Therefore therole of the earlier d in the context of interpolation errors is taken over by �− 1 here.Note that in applications to FEMs usually only the existence of an operator Ih withdesired approximation properties is important. Hence it is acceptable to use the Hahn–Banach Theorem 1.14 as a tool in the definition of Ih.

Theorem 4.36. Quasi-interpolation error on polyhedral domains Ω in Rn:

1. Under conditions (4.59)–(4.62) there exists a linear operator Ih : L1(Ω) →Sh, such that for any T ∈ T h, 1 ≤ p ≤ ∞, 0 ≤ |α| ≤ �− 1, u ∈ L1(Ω) with|u|W �,p(Ωγ

T ) <∞,

‖Dα(u− Ihu)‖Lp(T ) ≤ Kh�−|α|T |u|W �,p(Ωγ

T ), (4.63)


where ΩγT :=

⋃v∈T star2γ−1(v), K = K(n, d, ωT h , γ, C1C2, L∂Ω), and L∂Ω is the

Lipschitz constant of the boundary ∂Ω of Ω. Moreover, if u ∈W �,p(Ω), then forall 1 ≤ p ≤ ∞, 0 ≤ |α| ≤ �− 1,

‖Dα(u− Ihu)‖Lp(T h) ≤ Kh�−|α||u|W �,p(Ω). (4.64)

2. Rewriting (4.64) for Sobolev norms, we obtain for all 0 ≤ k ≤ �− 1,

‖u− Ihu‖W k,p(T h) ≤ Kh�−k‖u‖W �,p(Ω) and (4.65)

limh→0

‖Ihu‖W k,p(T h) = ‖u‖W k,p(Ω).

3. Moreover, it follows that for all 0 ≤ k ≤ �− 1,

‖u− Ihu‖W k,p(∂Ω) ≤ Kh�−k−1/p‖u‖W �,p(Ω), and (4.66)

limh→0

‖Ihu‖W k,p(∂Ω) = ‖u‖W k,p(∂Ω),

where p > 1 if k = �− 1.

Proof.

(1) Let us define the operator Ih. In view of (4.61), each functional λk, k = 1, . . . , N ,is well defined on Sh|Ek

. By the Hahn–Banach Theorem 1.14, we extend λk fromSh|Ek

to Sd(T h)|Ek, such that

|λks| ≤ C2‖s‖L∞(Ek), for all s ∈ Sd(T h)|Ek,

and define Ih : Sd(T h) → Sh by

Ihs =N∑

k=1

λk(s|Ek)sk, s ∈ Sd(T h). (4.67)

Let T be a simplex in T h, and let T ⊂ Ek. In view of (4.60), diam (Ek) ≤ c1hT

and vol (Ek) ≤ c2vol (T ), with some constants c1, c2 depending only on n, γ andωT h .31 Hence, for any s ∈ Sd(T h) we have in view of (4.58),

c−13 vol1/p (T )‖s‖L∞(Ek) ≤ ‖s‖Lp(Ek) ≤ c4vol1/p (T )‖s‖L∞(Ek),

where c3, c4 depend only on n, d, p, γ, ωT h . Since the basis splines sk are alsopiecewise polynomials, we have by (4.61),

‖sk‖Lp(T ) ≤ c5vol1/p (T )‖sk‖L∞(T ) ≤ c5vol1/p (T )C1,

31 This is easy to show considering that the number of simplices in Ek is bounded by a constantdepending only on n, γ and ωT h , and that any two simplices T ′, T ′′ with a common facet satisfyhT ′′ ≤ c1hT ′ and vol (T ′′) ≤ c2vol (T ′), for some c1, c2 depending only on n and ωT h .


where c5 depends only on n, d and p. Hence, assuming s ∈ Sd(T h) and applying(4.60) and (4.61), we obtain for any T ∈ T h,

‖Ihs‖Lp(T ) =

∥∥∥∥∥∥∥N∑

k=1T⊂Ek

λk(s|Ek)sk

∥∥∥∥∥∥∥Lp(T )

≤N∑

k=1T⊂Ek

|λk(s|Ek)|‖sk‖Lp(T )

≤ c5C1C2

N∑k=1

T⊂Ek

‖s‖L∞(Ek)vol1/p (T )

≤ c3c5C1C2

N∑k=1

T⊂Ek

‖s‖Lp(Ek) ≤ c6c3c5C1C2‖s‖Lp(ΩγT ),

where the last inequality follows fromN⋃

k=1T⊂Ek

Ek ⊂ ΩγT , T ∈ T h,

and

#{k : T ⊂ Ek} ≤ dimSd(T h)|ΩγT

=(n+d

n

)#{T ′ ∈ T h : T ′ ⊂ Ωγ

T

}≤(n+d

n

)c6,

where c6 depends only on γ and ωT h . Thus, we have shown that

‖Ihs‖Lp(T ) ≤ c‖s‖Lp(ΩγT ), s ∈ Sd(T h), T ∈ T h, (4.68)

with c depending only on n, p, d, ωT h , C, and γ.(2) In order to extend Ih = Ih from Sd(T h) to L1(Ω), we consider, for any u ∈

L1(Ω) and any T ∈ T h, the averaged Taylor polynomial pT (u) of degree �− 1with respect to the inscribed ball of T (see Definition 4.14). By Lemma 4.15,

|u− pT (u)|W k,p(T ) ≤ Bh�−kT |u|W �,p(T ), 0 ≤ k ≤ �− 1, (4.69)

where B depends only on �, n, ωT h . We define s(u) ∈ Sd(T h) by

s(u)|T = pT (u), T ∈ T h,

and set

Ihu := Ihs(u), u ∈ L1(Ω).

Clearly, Ih is a projector onto Sh, and, in particular, in view of (4.62),

Ihp = p, for any p ∈ Pn�−1. (4.70)

(3) Let us prove (4.63). Suppose that T ∈ T h, 0 ≤ |α| ≤ �− 1, and |u|W �,p(ΩγT ) <

∞. By using the Stein extension theorem (see e.g. [141], Theorem 1.4.5), we


extend u|ΩγT

to a function u defined on the convex hull U of ΩγT such that

|u|W �,p(U) ≤ c7|u|W �,p(ΩγT ),

where c7 depends only on n, p, �, ωT hc

and, possibly, on the Lipschitz constantL∂Ω if Ω is nonconvex, and the boundary of Ωγ

T contains a part of ∂Ω, compare[467], Theorem 1.9. Now, let pT ∈ Pn

�−1 be the average Taylor polynomial foru with respect to a ball in U of the greatest diameter. Again by the Bramble–Hilbert Lemma 4.15, we have32

|u− pT (u)|W k,p(ΩγT ) ≤ ch�−k

T |u|W �,p(ΩγT ), 0 ≤ k ≤ �− 1, (4.71)

which implies, in particular,

‖Dβ(u− pT (u))‖Lp(ΩγT ) ≤ c8h

�−|β|T |u|W �,p(Ωγ

T ), 0 ≤ |β| ≤ �− 1, (4.72)

with c8 depending only on n, p, �, ωT h , L∂Ω. Therefore,

‖Dα(u− Ihu)‖Lp(T ) ≤ ‖Dα(u− pT (u))‖Lp(T ) + ‖Dα(pT (u)− Ihu)‖Lp(T )

≤ c8h�−|α|T |u|W �,p(Ωγ

T ) + ‖Dα(pT (u)− Ihu)‖Lp(T ).

By (4.70), (4.57), (4.68), and (4.72),

‖Dα(pT (u)− Ihu)‖Lp(T ) = ‖DαIh(pT (u)− s(u))‖Lp(T )

≤ A

h|α|T

‖Ih(pT (u)− s(u))‖Lp(T )

≤ c

h|α|T

‖pT (u)− s(u)‖Lp(ΩγT )

≤ c

h|α|T

‖u− pT (u)‖Lp(ΩγT ) +

c

h|α|T

‖u− s(u)‖Lp(ΩγT )

≤ ch�−|α|T |u|W �,p(Ωγ

T ) +c

h|α|T

‖u− s(u)‖Lp(ΩγT ).

Now, since hT ′ ≤ c9hT , for all T ′ ∈ T h such that T ′ ⊂ ΩγT , where c9 depends

only on n, γ and ωT h , we have by (4.69)

‖u− s(u)‖L∞(ΩγT ) = max

T ′∈T h

T ′⊂ΩγT

‖u− pT ′(u)‖L∞(T ′)

≤ B maxT ′∈T h

T ′⊂ΩγT

h�T ′ |u|W �,∞(T ′) ≤ Bc9h

�T |u|W �,∞(Ωγ

T )

32 For a convex Ω it is possible to show (4.71) by arguments similar to the proof of Lemma 2.5in [436], without any resort to the Stein extension theorem.


in the case p = ∞, and

‖u− s(u)‖p

Lp(ΩγT ) =

∑T ′∈T h

T ′⊂ΩγT

‖u− pT ′(u)‖pLp(T ′)

≤ Bp∑

T ′∈T h

T ′⊂ΩγT

h�pT ′ |u|pW �,p(T ′) ≤ Bpcp

9h�pT |u|

p

W �,p(ΩγT )

in the case 1 ≤ p <∞, which completes the proof of (4.63).(4) To show (4.64), we first consider the case p =∞. For some T ∗ ∈ T h, we have

by (4.63),

‖Dα(u− Ihu)‖T h

L∞(Ω) = ‖Dα(u− Ihu)‖L∞(T∗) ≤ Kh�−|α|T∗ |u|W �,∞(Ωγ

T∗)

≤ K|T h|�−|α||u|W �,∞(Ω).

Assume now that 1 ≤ p <∞. Then by (4.63),(‖Dα(u− Ihu)‖T h

Lp(Ω)

)p

=∑

T∈T h

‖Dα(u− Ihu)‖pLp(T )

≤ Kp∑

T∈T h

h(�−|α|)pT |u|p

W �,p(ΩγT )

≤ Kp|T h|(�−|α|)p∑

T∈T h

∑T ′∈T h

T ′⊂ΩγT

|u|pW �,p(T ′).

Now ∑T∈T h

∑T ′∈T h

T ′⊂ΩγT

|u|pW �,p(T ′) =

∑T∈T h

#{T ′ ∈ T h : T ⊂ Ωγ

T ′}|u|p

W �,p(T ),

and since T ⊂ ΩγT ′ ⇔ T ′ ⊂ Ωγ

T , we have

#{T ′ ∈ T h : T ⊂ Ωγ

T ′}

= #{T ′ ∈ T h : T ′ ⊂ Ωγ

T

}≤ c6,

and, hence,(‖Dα(u− Ihu)‖T h

Lp(Ω)

)p

≤ c6Kp|T h|(�−|α|)p

∑T∈T h

|u|pW �,p(T )

= c6Kp|T h|(�−|α|)p|u|p

W �,p(Ω),

which completes the proof of (4.64).


(5) The last statement of the theorem is a direct consequence of (4.64) and Theorem1.38. Indeed, by (1.68),

‖u− Ihu‖W k,p(∂Ω) ≤(‖u− Ihu‖W k,p(Ω)

)1−1/p(‖u− Ihu‖W k+1,p(T h)

)1/p

≤ Kh�−k−1/p‖u‖W �,p(Ω),

and (4.66) follows. �

From the different types of extension operators mentioned above we refer toTreves [632]. He proves Theorem 26.A.3 for functions in u ∈ Ck(Ω) and extends themcontinuously to u ∈W k,p(Ω), 1 ≤ p <∞, in Corollary 26.A.1, yielding (4.73), (4.74).This technique can obviously be extended the following situation to show (4.75). Forthe FEMs for fully nonlinear elliptic problems and Ω ∈ C2 we will need approximatepolyhedral Ωh or curved Ωh

c with dist(Ω,Ωh

c

)≤ Ch.

Theorem 4.37. Modified Sobolev–Stein extension operator: Let Ω ∈ Ck, Ωhc , Ω0 ⊃

(Ω ∪ Ωhc ), and k ∈ N be given. Then there exist a linear bounded extension Ec :

Hk(Ω) → Hk(Ω ∪ Ωh

c

)→ Hk

0 (Ω0) and a constant, C = C(k,Ω,Ωh

c ,Ω0

), such that

Ecu|Ω = u on Ω and ‖Ecu‖Hk(Ω∪Ωhc ) ≤ C‖u‖Hk(Ω) ∀u ∈ Hk(Ω). (4.73)

The traces of the partials of u and Ecu ∈ Hk(Ω ∪ Ωh

c

)to the boundary ∂Ω coincide

and for dist(Ω,Ωh

c

)≤ Ch we obtain

Hk−|β|−1/2(∂Ω) � ∂β(u− Ecu)|∂Ω = 0 ∀|β| ≤ k,∀u ∈ Hk(Ω) (4.74)

=⇒ ‖∂β(Ecu(x)

)− Ec

(∂βu(x)

)‖L2(Ω∪Ωh

c ) ≤ Ch‖u‖H|β|(Ω) = 0∀u ∈ Hk0 (Ω). (4.75)

Based upon this theorem we immediately obtain

Theorem 4.38. Interpolation errors: Under the conditions (4.59)–(4.62), those ofTheorems 4.36 and 4.37, and for Ω piecewise in Cp, 2 ≤ p ≤ 5, a positive constantK = K

(n, d, ωT h

c, γ, C := C1C2, L∂Ωh

)exists, such that for u ∈ H�(Ω), and 0 ≤ s ≤

�− 1, cf. (4.62), we obtain for the operator

Ph = IhEc : ‖Ecu− Phu‖Hs(T hc ) ≤ C h�−s|u|H�(Ω), ‖Phu‖Hs(T h

c ) ≤ C‖u‖H�(Ω)

‖Ecu− Phu‖Hs(∂Ωhc ) ≤ C h�−s−1/2|u|H�(Ω), ‖Phu‖Hs(∂Ωh

c ) ≤ C‖u‖H�(Ω). (4.76)

The Uh : = {Phu : u ∈ H�(Ω)} approximate the Hs(Ω), � ≥ s, and Hs(∂Ω), � ≥ s +1/2, e.g. in the sense that

dist(Ecu,Uh) := infuh∈Uh

||Ecu− uh||Hs(T hc ) → 0 ∀u ∈ Hs(Ω) for h→ 0. (4.77)

Spaces of smooth piecewise polynomials with stable local bases

Bases with properties (4.60), (4.61), with γ = 1 and C1, C2 depending only on d, areavailable for Sr

d(T h) and certain subspaces of it on arbitrary T h if d ≥ r2n + 1, see[263] and references therein. In the case of two variables, [267] provides a construction


of bases for Srd(T h) satisfying (4.60)–(4.62), with γ = 3 and C1, C2 depending only

on d, as soon as d ≥ 3r + 2. There are many more results on stable local spline basesin the recent literature, especially in the contexts of high order macro-element andLagrange interpolation methods, see e.g. [467,516] and references therein.

Some classical spaces of smooth finite elements, for example those based on theArgyris element, see Figure 4.9, and [141] Example 3.2.10, belong to the class ofso-called superspline subspaces of Sr

d(T h) [573], and their associated bases satisfy therequirements of Theorem 4.36.

In particular, for the Argyris FE we have in the notation of Theorem 4.36,

Sh ={s ∈ S1

5 (T h) : s is C2 smooth at any vertex v of T h}. (4.78)

Clearly, (4.62) is satisfied with n = 2 and � = 6. The functionals λk : Sh → R arefunction evaluations, weighted first and second derivatives at the vertices,

s(v), hT∂s

∂x1(v), hT

∂s

∂x2(v), h2

T

∂2s

∂x21

(v), h2T

∂2s

∂x22

(v), h2T

∂2s

∂x1∂x2(v), (4.79)

and weighted first order normal derivatives at the midpoints of the edges of T h,

hT∂s

∂n

((v1 + v2)/2

), (4.80)

where hT is the diameter of a triangle in T h containing the corresponding evaluationpoint v or (v1 + v2)/2 from (4.79), (4.80). The sets Ek = supp sk are either star (v) forthe functionals of type (4.79), or the unions of two triangles sharing the edge [v1, v2]in case (4.80). Hence, (4.60) is satisfied with γ = 1. Furthermore, by estimating thenorms of local Hermite interpolation operators, it can be shown that ‖sk‖L∞(Ω) ≤ C1,where C1 is an absolute constant, see [263, Lemma 3.3]. Similarly, in view of the inverseestimates (4.57), |λks| ≤ C2‖s‖L∞(Ek), for any s ∈ Sh, where C2 is again an absoluteconstant, see [263, p. 292]. Thus, (4.61) is correct with C = C1C2 being an absoluteconstant.

Stable splitting Sh = Sh0 ⊕ Sh

b

In the finite element method an important role is played by the spaces of finite elementsvanishing on (parts of) the boundary. We set

Sh0 = {s ∈ Sh : s|∂Ω = 0}.

For the sake of applications to the numerical solution of fully nonlinear problems, weconstruct a splitting of Sh into a direct sum

Sh = Sh0 ⊕ Sh

b

such that there exists a basis {s1, . . . , sN} for Sh satisfying the hypotheses(4.60), (4.61) of Theorem 4.36 (i.e. a stable local basis), where {s1, . . . , sN0} and{sN0+1, . . . , sN} are bases for Sh

0 and Shb , respectively.

We provide a construction for n = 2 and Sh ⊂ S15 (Ω) based on a modified Argyris

finite element. We replace (4.78) by

Sh ={s ∈ S1

5 (T h) : s is C2 smooth at any interior vertex v of T h}. (4.81)


Thus, in contrast to the Argyris element, the functions in Sh are not necessarily C2

at the boundary vertices.We now describe a set of functionals {λ1, . . . , λN} ⊂ (Sh)′, such that the desired

basis {s1, . . . , sN} for Sh will be uniquely defined by duality

λisj = δi,j .

The set {λ1, . . . , λN} includes

(a) the functionals (4.79) for all interior vertices v of T h;(b) the functionals (4.80) for all edges of T h; and(c) the following functionals for each boundary vertex v of T h:

s(v), hT∂s

∂e0(v), hT

∂s

∂e⊥0(v), h2

T

∂2s

∂e20(v), . . . , h2

T

∂2s

∂e2n(v), h2

T

∂2s

∂e0∂e1(v),

where e0, . . . , en are all edges of T h emanating from v, in counterclockwiseorder, with e0 and en being the boundary edges.

Here the symbol ∂/∂e denotes the usual directional derivative in the direction of edgee, and ∂/∂e⊥ in the orthogonal direction. The above second order edge derivatives

∂2s

∂e20(v), . . . ,

∂2s

∂e2n(v),

∂2s

∂e0∂e1(v)

are well defined despite s being only C1 at the boundary vertices, see [266]. Note thatthese second order derivatives are independent of each other even if some edges arecollinear. This choice of the degrees of freedom at boundary vertices in (c) is motivatedby the construction of the Morgan–Scott basis [497] and is shown to be stable in [266].

Following the argument in [266], one can see that the basis {s1, . . . , sN} for Sh,defined by duality, satisfies (4.60), (4.61) with γ = 1 and bounded C1, C2. Hence it isa stable local basis. Moreover, (4.62) is obviously true for Sh with n = 2 and � = 6.Therefore, Theorem 4.36 applies to this basis.

To determine the subsets of {s1, . . . , sN} which generate Sh0 and Sh

b , respectively,we now split the functionals in (c) into two groups (c1) and (c2) as follows.

(c1) The first group includes

h2T

∂2s

∂e21(v), . . . , h2

T

∂2s

∂e2n−1

(v), h2T

∂2s

∂e0∂e1(v),

for all boundary vertices, and, in addition, hT∂s

∂e⊥0(v) for those boundary

vertices, where e0 and en are collinear.(c2) The second group includes

s(v), hT∂s

∂e0(v), h2

T

∂2s

∂e20(v), h2

T

∂2s

∂e2n(v), (4.82)


for all boundary vertices, and, in addition, hT∂s

∂e⊥0(v) for those boundary

vertices, where e0 and en are not collinear.

Now let {λ1, . . . , λN0} list all functionals λi in (a), (b), and (c1), and let {λN0+1, . . . ,λN} be those in (c2). It is easy to see that

Sh0 = {s ∈ Sh : λN0+1s = · · · = λNs = 0}. (4.83)

Therefore Sh0 = span {s1, . . . , sN0}, and Sh

b := span {sN0+1, . . . , sN} is its complementin Sh as required.

Clearly, both {s1, . . . , sN0} and {sN0+1, . . . , sN} are stable local bases as subsets ofthe stable local basis {s1, . . . , sN}.

Note that hT∂s

∂e⊥0(v) belongs to (c1) or (c2) depending on whether e0 and en are

exactly collinear or not. In particular it is in (c2) if e0 and en are near-collinear, butnot collinear, a situation which may appear quite often when a polygonal domainis an approximation of a smooth domain. As e0 and en become exactly collinear,

hT∂s

∂e⊥0(v) is moved into (c1). Thus, the dimensions of Sh

0 and Shb jump if a vertex

shared by collinear boundary edges is slightly perturbed. This ‘dimension instability’ isnot a new phenomenon in the theory of multivariate splines. Indeed, it is related to thewell-known fact that the dimension formulas for the spline spaces may depend on somegeometric information about the placement of the vertices, see [467]. This behavior iscompatible with the availability of stable bases, see for example the discussion in [267],Remark 13.1.

Since λi are function evaluations or derivatives of at most second order, they may beapplied to any sufficiently smooth functions, thus leading to the interpolation operatorIh : C2(Ω) → Sh, defined by

Ih(v) =N∑

i=1

λi(u)si, u ∈ C2(Ω). (4.84)

The following is an obvious property of this operator:

v|∂Ω = 0 =⇒ Ih(v) ∈ Sh0 . (4.85)

It is easy to see that the operator Ih can be used in the proof of Theorem 4.36 inplace of the quasi-interpolation operator assuming that u is sufficiently smooth.

Finally, consider the operator Ihb : C2(Ω) → Sh

b , defined by

Ihb (u) =

N∑i=N0+1

λi(u)si. (4.86)

Clearly,

Ihb (u)|∂Ω = Ih(u)|∂Ω. (4.87)


Hence, by Trace Theorem 1.38 we obtain for any 1 ≤ p ≤ ∞,∥∥u− Ihb (u)

∥∥Lp(∂Ω)

≤ C‖u− Ih(u)‖1−1/pLp(Ω)‖u− Ih(u)‖1/p

W 1,p(Ω),

where C is a constant depending only on p and the Lipschitz constant L∂Ω of ∂Ω. Thisallows us to find a bound for

∥∥u− Ihb (u)

∥∥Lp(∂Ω)

, see (4.90) below, using the estimates

available for ‖u− Ih(u)‖Lp(Ω) and ‖Dα(v − Ih(u))‖Lp(Ω), |α| = 1.We summarize the results for the stable splitting in the following theorem.

Theorem 4.39. Stable splitting: For the space Sh defined in (4.81), there exists abasis {s1, . . . , sN} satisfying (4.60)–(4.62), such that {s1, . . . , sN0} is a basis for Sh

0 ={s ∈ Sh : s|∂Ω = 0}. The interpolation operators Ih : C2(Ω) → Sh and Ih

b : C2(Ω) →Sh

b defined by (4.84) and (4.86), respectively, have the following properties, cf. (4.63),(4.64).

1. If u|∂Ω = 0, then Ih(u) ∈ Sh0 .

2. Let 1 ≤ p ≤ ∞, � ≤ 6, be such that W �,p(Ω) ⊂ C2(Ω). Then for any 0 ≤ |α| ≤�− 1,

‖Dα(u− Ih(u))‖Lp(T ) ≤ Kh�−|α|T |u|W �,p(Ω1

T ), for any T ∈ T h, (4.88)

and, as a consequence

‖Dα(u− Ih(u))‖Lp(T h) ≤ K ′h�−|α||u|W �,p(Ω). (4.89)

3. Moreover, ∥∥u− Ihb (u)

∥∥Lp(∂Ω)

≤ K ′′h�−1/p|u|W �,p(Ω). (4.90)

The constants K,K ′,K ′′ depend only on p, ωT h , and the Lipschitz constant L∂Ω.

Conjecture 4.40. The results in Theorems 4.35 and 4.36 are proved in Rn and forSh = Sr

d(T h) and n, r if d ≥ r2n + 1. We conjecture that the techniques of [263] canbe used to generalize Theorem 4.39 as well to Sr

d(T h) and appropriate subspaces of it.

Conjecture 4.41. We extend the previous Conjecture 4.40 to suitable nonpolyhedral(curved) boundaries ∂Ωh

c ≈ ∂Ω with dist(∂Ωh

c , ∂Ω)

= O(hq), q > 2, and the corre-sponding triangulations T h

c . We expect that Theorems 4.35–4.39 can be generalized inthis case to appropriate subspaces of Sh = Sr

d

(T h

c

)for any r if d ≥ r2n + 1, at least

for n = 2.

4.2.7 Curved boundaries

In our earlier discussion we have excluded curved boundaries. We study them inthis and the next subsection. We will formulate corresponding FEMs in Subsections5.5.6, 5.5.10. The results here are valid partially for Rn, n ≥ 2, or Rn, n ≤ 3. So weconsider

Ω ⊂ Rn, n ≥ 2, with piecewise smooth ∂Ω ∈ Ctp, t ≥ 1, P = Pn

d−1, d ≥ 1. (4.91)


Figure 4.21 Admissible triangulation for a curved boundary.

To simplify the presentation, we restrict the construction to two-variate FEs inthis subsection. Extensions to Rn, n ≥ 3, are theoretically pretty obvious, in contrastto the practical realization. In a first step we define an approximating Ωh and thecorresponding T h:

Choose points Pj ∈ ∂Ω with distance ≤ h for neighboring points.Include all “non-smooth” points on ∂Ω ∈ Ct

p into these Pj .Replace ∂Ω and Ω by ∂Ωh and Ωh : ∂Ωh is obtainedby connecting the neighboring Pj ∈ ∂Ω by straight lines, thusdefining the new edges e ⊂ ∂Ωh. The polygonal Ωh is then theinterior of ∂Ωh. Choose a nondegenerate triangulation T h for Ωh.

(4.92)

Then automatically the T ∈ T h are star-shaped. Figure 4.21 shows the original ∂Ωwith an interior T h. We have to be prepared that the necessarily modified FEs willsatisfy the boundary conditions only approximately. The discussion here is restrictedto triangulations, possibly curved at the boundary, and to

K triangle,∀ e ⊂ K only function evaluations and no derivatives ∈ N . (4.93)

Note that vertices Pj ∈ e, see (4.98), are included.Here, we present two different possibilities to handle Ωh and T h. For the “boundary

FEs” (T,PT ,NT ) with |T ∩ ∂Ω| ≥ 2 we change NT or PT , thus modifying Definition4.6. In the first case, see Figure 4.22, we replace, along the boundary,

NT by interpolation along the curved∂Ω instead of the straight ∂Ωh. (4.94)

Secondly, we use the original FT for T with respect to Ωh, see (4.92), Figures 4.22,4.23, but,

replace FT : K → T affine, by Fh|T ◦ FT : K → Tc, Fh|T ∈ Pn

d−1, (4.95)

for the T ∈ T h at the boundary and with the original polynomials P = Pnd−1. This

yields the so-called isoparametric FEs. We describe in Definition 4.42 and the followingpages, how these Fh

c , Tc are constructed. Obviously the basis functions φi ◦ (Fh|T ◦FT )−1 �∈ P are more complicated nonpolynomial functions. For a third choice seeSubsection 4.2.6.


Pe

Pi

k = 2Tc

Figure 4.22 Polynomial interpolation on ∂Ω.

Pe

∂Tc

∂T

∂Ω

Pc

x (s )

xa(s )

Pi

0 £ y £ ya(s )

Figure 4.23 Isoparametric interpolation on ∂Ω.

Polynomial approximation and interpolation in points of ∂ΩFor the new triangulation, T h, we modify the T and NT at the boundary with |T ∩∂Ω| ≥ 2, see (4.94): We denote the vertex points of T as Tv and define

Tv ∩ ∂Ωh = {Pi, Pe}, Pi �= Pe, Tv := {vertices of T}. (4.96)

According to (4.94), we modify these triangles T into Tc by replacing the straightline Pi, Pe by that part of the boundary ∂Ω between Pi and Pe, see Figure 4.22.


The modified triangulation we denote as T hc . So we still obtain, in contrast to the

isoparametric FEs below,

Ω =⋃

Tc∈T hc

Tc. (4.97)

For modifying NT we proceed in two steps. Choose a smooth parametrization x = x(s)of ∂Ω between Pi and Pe with respect to the arc length s, 0 ≤ s ≤ he, such thats ∈ C2d−1[0, he]. Now we choose d Gauss–Lobatto points, cf. (5.307)–(5.311), withρ = 2 and the yρ

j = y2j there. Let ξ0, . . . , ξd−1, ξj := he

(1 + y2

j

)/2, ξ0 = 0, ξd−1 = he

in [0, he] (or Gauss points in Subsubsections 5.5.5 and 5.5.6, 5.5.7). Determine, on thecurved edge e = ∂T , cf. Figure 4.22,

∂Ω � P ej := Pj := x(ξj) ≈ P s

j := Pi + ξj(Pe − Pi)/‖(Pe − Pi)‖R2 ,

j = 0, . . . , d− 1, Pi = P0, Pe = Pd−1. (4.98)

In principle, we could use any d interpolation points on e. The choice of the Gauss-typepoints is motivated by the applications to FEMs in Section 5.5.

Now, we replace the d function evaluations in the P sj along the straight edge PiPe for

the original T , by the d function evaluations in the P ej = Pj , j = 0, . . . , d− 1. Thus

we replace NT by a modified N cT corresponding to one curved edge, see Figure 4.22.

For Gauss and Gauss–Radau points we obtain the corresponding 0 < ξ0 ≤ ξd−1 < he

and ξ0 = 0, ξd−1 < he, respectively. Then discontiuous Uh are generated. We discussthese cases in Subsections 5.5.6, 5.5.7. For small enough h, the edge PiPe will be atmost O(h2) away from the curved part of ∂Ω between Pi and Pe, see [141], 8 ex.3.Hence PT ,N c

T will be unisolvent simultaneously with PT ,NT for small enough h. Theseperturbation arguments show that the results in Subsection 4.2.4 remain essentiallyvalid.

For R3 the above Gauss-type points could be replaced by the cubature points ofthe best types of cubature formulas in tetrahedra in R3. Their order of convergence issmaller than for the R2 case. This is inherited by the FEMs. We do not present thosemodifications here, since we consider discontinuous Galerkin methods in Chapter 7,avoiding these difficulties. For details we refer to Cools [208–210].

Isoparametric nonpolynomial approximation

The Gauss type points decrease, as far as possible, interpolation and, for applicationsin FEMs, the quadrature errors along the relevant boundary part of ∂Ω. There isanother method, well established in engineering applications, allowing much morefreedom, but with the same order of convergence as before. In addition this method isapplicable in FEMs to Rn, n ≥ 2. Particularly efficient is the isoparametric polynomialapproach in (4.95). As in (4.96) we assume one curved and two straight edges Pi, Pe

and P0 Pi, P0 Pe for Tc. Well known results of Ciarlet and Raviart [176], Ciarlet [174]and Lenoir [471], construct a bijective, piecewise polynomial mapping Fh : Ωh → Ωh

c ≈Ω ⊂ R2, see [471] for Rn, n ≥ 2, such that Fh = id away from the boundary. We presenta very rough idea of this construction with the goal of showing that the isoparametric


FEMs fit into our general discretization framework. Figure 4.23 motivates the nextdefinition. In 3. below we need dist(Fh(∂Ωh), ∂Ω).33

Definition 4.42. We define the triangulation T hc , the bijective isoparametric

mapping Fh : Ωh → Ωhc ≈ Ω, and the boundary ∂Ωh

c := Fh(∂Ωh): Choose Ωh, ∂Ωh

and a nondegenerate triangulation T h for Ωh as in (4.92), (4.96). Then, correlatingT h with T h

c , we define Fh : Ωh → Ωhc := Fh(Ωh) ⊂ Rn, n ≥ 2, T h

c and Tc ∈ T hc as:

1. Let (Fh)|T = id|T ∀T ∈ T h with |Tv ∩ ∂Ω| ≤ 1, see (4.96), and let Fh|e = id|e,∀e ⊂ T with |e ∩ ∂Ω| ≤ 1.

2. For the components of F h, the (Fh)i, i = 1, 2, let (Fh)i|T ∈ Pnd−1 ∀T ∈ T h with

|Tv ∩ ∂Ω| ≥ 2, see Figure 4.23, and T hc � Tc := FhT ∀T ∈ T h.

3. dist (Fh(∂Ωh), ∂Ω) = O(hd), dist (Fh(Ωh),Ω) = O(hd).4. ‖(Fh)′‖W d,∞(Tc)←↩W d,∞(T ), ‖((Fh)′)−1‖··· ≤ C, h independent, ∀Tc ∈ T h

c .5. Define the Tc as Tc = Fh(T ) ∀ T ∈ T h with T = Tc ∀ T ∈ T h for |T ∩ ∂Ω| ≤

1 and Tc �= T for |T ∩ ∂Ω| ≥ 2, unless ∂Ω between Pi, Pe is a straight line.6. Define T h

c :={Th

c

}for the Th

c in (v), and Ωhc := ∪

Tc∈T hc

Tc.

Note that the conditions 1.–6. are strictly local, including 3.We only describe this construction, see above, for a triangle T ∈ T h

c with 2 =|Tv ∩ ∂Ω| = {Pi, Pe}, see Figure 4.23. For all other cases, (Fh)|T = id|T by 1.Our chosen T ∈ T h is defined by the vertices Pi, Pe, Pc. The solid line ∂Ω marksthe original part of the boundary between Pi and Pe. Then Fh|T : T → Tc, by 2.The isoparametric triangle Tc has the straight edges Pc, Pi, Pc, Pe and the curvededge ∂Tc connecting Pi, Pe. By 2. the components of Fh are (Fh)i|T ∈ Pn

d−1

with Fh|e = id|e, for the two straight edges Pc, Pi Pc, Pe. By 3. we require dist(Fh(∂Ωh), ∂Ω) = O(hd). For the two points marked as xa(s), x(s) on the ray fromPc the distance in Figure 4.23 is O(hd). We did not mark any intersection point of∂Ω and ∂Tc, since only the distance O(hd) and 4. are required. Finally 5., 6. yieldTc ∈ T h

c .Definition 4.42 introduces an isoparametric subdivision, T h

c . We proceed by intro-ducing isoparametric FEs, Uh, and, later on, an isoparametric interpolation operator,Ih. We combine the affine mapping F = FT : K → T ∈ T h with the polynomialsFh|T : T → Tc, see Figure 4.24. The construction of this Fh is a main point in thissubsection. So we get nonconforming FEs violating the boundary conditions. This willbe treated in Subsection 5.5.10, see Theorem 5.85.

We start with the original T h defined for Ωh in (4.92), the Fh, T hc and Ωh

c inDefinition 4.42, and the points P e

i on curved Tc with edges e, approximating ∂Ω, ford′ cf. Definition 4.1:

33 For compact A, B, the distance between A and B is defined as dist (A, B) = max {maxx∈A :{min {|x − y| : y ∈ B}}, maxy∈B : {min {|x − y| : x ∈ A}}}. It represents the maximal distance fromany point of one set to the nearest point of the other set.


affineK

FT

T Tc

F h

Figure 4.24 Isoparametric mapping FT : K → T, F h ◦ FT : K → Tc.

Uh :={uh : Ωh

c = Fh(Ωh) → R : uh|Tc=

d′∑i=1

αi · φi ◦ (Fh|T ◦ FT )−1,

with (non)polynomials φi ◦ (Fh|T ◦ FT )−1, αi ∈ R ∀ Tc = Fh(T ) ∈ T hc

},

‖uh‖W k,q(T hc ) as in (4.36) with Ω, T h replaced byFh(Ωh), T h

c (4.99)

Uhb :=

{uh ∈ Uh : ∀e ⊂ Tc uh (P e

i ) = 0, i = 0, . . . , d′ − 1, interpolating in e ≈ ∂Ωhc

},

where these P e0 = Pi, P e

d′−1 = Pe, but P ei , i = 1, . . . , d′ − 2 are d′ − 2 points on e,

unequal to those in (4.98).Let e ⊂ K denote the edge mapped by the original FT : K → T onto Pi Pe. Then

∂ΩhT := Fh(FT (e)), marked as ∂Tc in Figure 4.23, only approximates that part of ∂Ω

between Pi and Pe up to O(hd). For the following interpolation process this might havethe consequence that some of the interpolation points Pj ∈ ∂Ωh

T , j = 1, . . . , d− 2, arePj �∈ Ω ∪ ∂Ω and so u(Pj) might not be defined. In this case, Theorem 4.37 yieldsan appropriate extension ue of u, allowing the evaluation of ue(Pj), see [141]. Moreappropriate is a kind of piecewise interpolation, Fc, of the above Fh.

The complicated construction of Fc and the following properties of Ec are presentedfor Rn, n ≥ 2, in [471]. We only indicate the R2 results and summarize the essentialproperties in (4.100) and (4.101). They allow formulating error and inverse estimatesand the proofs for the FEMs below in Rn, n ≥ 2, cf. Theorem 4.43 and 5.85 ff. Thisauxiliary Fc exactly maps Ωh → Ω.

Fc = ((Fc)i)2i=1 : Ωh → Ω ⊂ Rn s.t. ∀T ∈ T h : Fc|T : T → Tc ≈ Tc, and (4.100)

(Fc|T )−1 : Tc → T are Cd diffeomorphism s.t with the operator norm ||| · |||∞,

|||Ds(Fc|T − idT )|||∞, |||Ds((Fc|T )−1 − idTc)|||∞ ≤ Chd−s∀s ≤ d

|J(D(Fc|T )− 1|∞, |J(D(Fc|T )−1)− 1|∞ ≤ Chd

and ∂Fc(Ωh) = Fc(∂Ωh) = ∂Ω and Fh(∂Ωh) = ∂Ω +O(hd),

and, replacing Fh by Fc, this Fc satisfies Definition 4.42 1.; here |||Ds(Fc|T − idT )|||∞indicates the sup-norm of the sth derivative and J(D(Fc|T ) the absolute value of the


Jacobian. Thus Fc has the property that in Figure 4.23 the part of ∂Ω from Pi to Pe

is the image Fc(∂T ) where ∂T indicates the straight segment Pi, Pe.We introduce Ec, transforming functions defined on the original Ω to those defined

on the approximate Ωhc = Fh(Ωh) ≈ Ω and its inverse (Ec)−1:

1. Ec : Ω → Ωhc = Fh(Ωh) as Ec := Fh ◦ F−1

c : t ∈ Ω → x ∈ Fh(Ωh) ≈ Ω,

2. (Ec)−1 : Fh(Ωh) → Ω with t− Ec(t) = O(hd);x− (Ec)−1(x) = O(hd) and

3. (Ec)′ − IdΩ = O(hd−1), ((Ec)−1)′ − IdF h(Ωh) = O(hd−1), as in (4.100),

4. Ec|T = IdT and Ec|e = Ide ∀ T, e in Definition 4.42 1. (4.101)

With this Ec, the uh,Uh and Ωhc = Fh(Ωh) in (4.99) we start with

uh : Fh(Ωh) → R, uh ∈ Uh, and we define the new uh as

uh : Ω → R, uh(t) := (uh ◦ Ec)(t), and Uhb =

{uh : uh ∈ Uh

b

}. (4.102)

Note that Uhb ⊂ U , but trivial Dirichlet boundary conditions are satisfied only approx-

imately on the exact ∂Ω for u ∈ Uhb . Vice versa

for f : Ω → R the f : Fh(Ωh) → R is defined as

f(x) := f((Ec)−1(x)) = (f ◦ (Ec)−1)(x), (4.103)

With Fh in Definition 4.42 the isoparametric interpolation operator, Ih, is

Ihu := (Ihu) ◦ Ec ∈ Uh, with Ih : U := {u : Ω → R} → Uh :={uh : Ωh

c → R},

Ihu|T = Ihu|T as in (4.16) ∀ inner T = Tc ∈ T hc with |Tc ∩ ∂Ωh| ≤ 1, else (4.104)

Ihu|Tc=

d′∑i=1

Ni(u ◦ Fh|Tc◦ FTc

) · (φi ◦ (Fh|Tc◦ FTc

)−1) =d′∑

i=1

NTci (u)φTc

i .

We have replaced the affine FT by isoparametric generalizations with components(Fh|Tc

◦ FTc)i ∈ Pn

d−1, i = 1, 2, for Tc with two boundary vertices. Obviously, Ih isa bounded linear operator as the original Ih, if the norms are now defined withrespect to the T h

c above. Furthermore, the original φi ◦ (FT )−1 and Ni(u ◦ FT ) haveto be replaced by the more complicated (non)polynomial φi ◦ (Fh|Tc

◦ FTc)−1 but

still linear Ni(u ◦ Fh|Tc◦ FTc

). Since, in mapping a polyhedral to a smooth domain,a C1 mapping is not appropriate, we only choose C0 reference elements, cf. [141],p.118.

Theorem 4.43. Interpolation errors: For Ω as in (4.91) define Ωh as in (4.92). Let,for 0 < h ≤ 1, the T h for Ωh satisfy Condition 4.16 for some l, d, p. We combine Fh

and T he from Definition 4.42, with Fh, piecewise of degree d− 1. For a C0 reference

element (K,P,N ), define Uh,Uhb , Ih, Ihu as in (4.99), (4.102), (4.104). Then there

4.3. FEMs for linear problems 257

exists a positive constant C = C((K,P,N ), n, d, p, χ), for χ cf. (4.30), such that for0 ≤ s ≤ d, and ∀u ∈W d,p(Ω),

‖u− Ihu‖W s,p(T ) ≤ C (diam T )d−s|u|W d,p(T ), (4.105)

‖Ihu‖W s,p(T ) ≤ (1 +O((diam T )d−s))‖u‖W d,p(T ) with

maxT∈T h

‖u− Ihu‖W s,∞(T ) ≤ C hd−s−n/p|u|W d,∞(Ω) ∀ u ∈W d,∞(Ω), and

‖u− Ihu‖W s,p(T h) ≤ C hd−s|u|W d,p(Ω),

‖Ihu‖W s,p(T h) ≤ (1 + C hd−s)‖u‖W d,p(Ω).

Since P always satisfies P ⊆W j,p(T ) ∩W lq(T ), 1 ≤ q ≤ p, 0 ≤ l ≤ j, we get

Theorem 4.44. Inverse estimates: Under the conditions of Theorem 4.43 and forT ∈ T h

c of a quasiuniform triangulation T h let

Uh ={uh : uh is measurable and uh ∈ Uh according to (4.99) ∀Tc ∈ T h

c

}.

Then there exists a constant C = C(l, p, q, χ) such that

‖uh‖W j,p(T hc ) ≤ Chl−j−(n/q−n/p)‖uh‖W l

q(T hc ) ∀uh ∈ Uh, ∀j ≥ l. (4.106)

4.3 FEMs for linear problems

The usual approach in finite element methods considers the weak operator, A,G andits corresponding weak bilinear or nonlinear form, a(·, ·) as in this subsection and inSection 4.4. We will modify this approach for fully nonlinear problems in Section 5.2.In contrast to the following Chapters 5 and 7, we consider here conforming FEMs,i.e. the exact and approximate solutions and test functions are elements of the samespaces, so u0, u

h0 ∈ Ub, v, vh ∈ Vb, here with Ub = Vb = H1

0 (Ω) and Ub = Vb = Hm0 (Ω)

for elliptic differential equations of order 2 in this and order 2m in Subsection 4.3.2.A straightforward modification allows us to include these equations and systems oforder 2m. Extensions to quasilinear problems are studied in Section 4.4.

For conforming FEMs and our problems here, we have Ub = Vb and Uhb = Vh

b , sofrom here on we use the notation Vb and Vh

b , the standard notation.More involved discussions would be necessary to study the different combinations

of natural and induced boundary conditions, see e.g. [387] and Chapter 2.We start with the main idea for FEMs, presented for a simple example in Subsection

4.3.1. In particular, we show how the definition of FEs via the triple K,P,N can beused efficiently to determine the elements of the mass and stiffness matrices for a FEM.Next, general elliptic operators of order 2 and 2m and their corresponding coerciveand elliptic bilinear forms and the induced boundary operators are recapitulated as abasis for FEMs in Subsection 4.3.2. Finally, the convergence for conforming FEMs for


these linear elliptic boundary value problems is formulated in Theorem 4.57, with theAubin–Nitsche improvement in Theorem 4.60.

4.3.1 Finite element methods: a simple example, essential tools

The main idea for FEs is very simple and is demonstrated for a simple example. Wechoose the Helmholz operator and Dirichlet boundary conditions:

Example 4.45. Homogeneous Dirichlet boundary value problemFor u ∈ H1

0 (Ω) := {u ∈ H1(Ω) : u|∂Ω = 0}, f ∈ H−10 (Ω), cf. Proposition 2.34, we have

defined the bilinear form a(·, ·) and the corresponding operator A as

a(·, ·) : H10 (Ω)×H1

0 (Ω) → R and A : H10 (Ω) → H−1

0 (Ω) are defined by

a(u, v) :=∫

Ω

(∇u, ∇ v)n + cuvdx =: 〈Au, v〉H−1(Ω)×H1(Ω) ∀v ∈ H10 (Ω),

with the Euclidean product (∇u(x),∇v(x))n = (∇u(x))T∇v(x) in Rn. For f ∈ L2(Ω)or f ∈ H−1

0 (Ω), the exact (weak) solution u0 is then defined by

u0 ∈ Ub = Vb := H10 (Ω) : a(u0, v) =

∫Ω

(∇u0, ∇ v)n + c u0 v dx (4.107)

= 〈Au0, v〉H−1(Ω)×H1(Ω) = (f, v) or 〈f, v〉∀ v ∈ Vb. �

Basic idea for conforming finite elements methods

For conforming FEs we replace Vb in (4.107) by a sequence of finite dimensionalspaces

{Vh

b

}h∈H

. For nonconforming FEMs, we sometimes even choose two sequences{Uh

b

}h∈H

and{Vh

b

}h∈H

. The indices h satisfy 0 < h ∈ H, with infh∈H h = 0.

Definition 4.46. Conforming FEs are characterized, for second order problems, byVh

b as subsets of solution and test spaces, exactly satisfying the boundary conditions,here the trivial Dirichlet conditions, and

uh, vh ∈ Vhb ⊂ Vb = H1

0 (Ω), ∀T ∈ T h : uh|T , vh|T ∈ P,Pd−1 ⊂ P ⊂ Pd+τ , d > 1,

τ ≥ −1, thus a(uh, vh) : Vhb × Vh

b → R, and 〈f, vh〉 : Vhb → R for f ∈ V ′

b = H−10 (Ω)

are well defined and Vhb ⊂ H1

0 (Ω) ∩ C(Ω). Conforming FEMs are defined with con-forming FEs.

For problems of order 2m, we require Vhb ⊂ Hm

0 (Ω).

We determine the approximate and the exact solutions uh0 ∈ Vh

b and u0 from

uh0 ∈ Vh

b : a(uh

0 , vh)

=∫

Ω

(∇uh

0 , ∇ vh)n

+ c uh0 v

h dx =⟨Auh

0 , vh⟩V′

b×Vb(4.108)

= 〈f, vh〉V′b×Vb

∀ vh ∈ Vhb ⊂ Vb = H1

0 (Ω) with Ah ∈ L(Vhb ,V

′hb )

and u0 ∈ Vb by omitting all h. (4.109)


Remark 4.47. We want to discuss an often used notation. For practically all spaces,V, discussed here, we find V �= Vb, and V ′ �= V ′

b, cf. Proposition 2.34. The same holdsfor the discrete spaces. Nevertheless, for test functions v ∈ Vb �= V and functionalsf ∈ V ′ �= V ′

b we find f |Vb∈ V ′

b. We summarize

V �= Vb ⇒ V ′ �= V ′b, e.g., H−1(Ω) = (H1(Ω))′ �=

(H1

0 (Ω))′

= H−10 (Ω) (4.110)

f ∈ V ′ : 〈f, v〉V′×V = 〈f, v〉V′×Vb∀ v ∈ Vb ⇒ f |H1

0 (Ω) = f |Vb∈ V ′

b = H−10 (Ω)

and similarly Vh′ �= Vh′

b ,Vh′ �= Vh′

b , fh ∈ Vh′ �= Vh′

b ⇒ fh|Vhb∈ Vh′

b .

Usually we will use the notation 〈fh, vh〉Vh′b ×Vh

b, similarly 〈f, v〉V′

b×Vb, but now and

then, 〈fh, vh〉Vh′×Vh ∀vh ∈ Vhb as well. By (4.110) there is not much difference for

conforming FEs, Vhb ⊂ Vb. For nonconforming FEs, Vh

b �⊂ Vb, cf. Chapter 5, this willchange.

For general cases we will prove the convergence via the results of Chapter 3.Nevertheless, honoring Cea, we include the surprisingly simple proof for convergence‖u0 − uh

0‖H1(Ω) → 0 for h→ 0 for his lemma. It applies not only to (4.109), but to all“conforming” FEMs for “coercive” bilinear forms.

Lemma 4.48. Cea lemma: Let Vhb ⊂ Vb ⊂ V, with dim Vh

b <∞ and assume a(bounded) Vb-coercive bilinear form, hence, cf. Proposition 1.25,

|a(u, v)| ≤ C‖u‖V‖v‖V , |a(u, u)| ≥ α‖u‖2V ≥ α′‖u‖2Vb,∀u, v ∈ Vb (4.111)

with constants C,α, α′ > 0. Then, with the projector Q′h in (4.117) the discrete Ah :=

(Q′hA|Vh

b)−1 ∈ L(Vh′

b ,Vhb ) is bounded and boundedly invertible with ‖Ah‖Vh′

b ←Vhb< C,

‖(Ah)−1‖Vhb ←Vh′

b< 1/α. Then unique exact solutions u0 ∈ Vb and discrete solutions

uh0 ∈ Vh

b of (4.107) and (4.109) uniquely exist, and satisfy the error estimate:∥∥u0 − uh0

∥∥V ≤

C

αmin

uh∈Vhb

‖uh − u0‖V . (4.112)

Proof. By (4.111) the a(., .) : Vb × Vb → R, and a(., .) : Vhb × Vh

b → R, satisfy theinf–sup conditions in Theorem 2.12. Thus ‖A‖Vb←V′

b< C, ‖A−1‖Vb←V′

b< 1/α and

‖Ah‖Vhb ←Vh′

b< C, ‖(Ah)−1‖Vh′

b ←Vhb< 1/α are bounded and boundedly invertible. So

the solutions u0 ∈ Vb and uh0 ∈ Vh

b of (4.107) and (4.109) uniquely exist and imply

a(u0 − uh

0 , vh)

= 0 ∀ vh ∈ Vhb ,

and furthermore

α∥∥u0 − uh

0

∥∥2V ≤ a

(u0 − uh

0 , u0 − uh0

)by coercivity

= a(u0 − uh

0 , u0 − uh)

+ a(u0 − uh

0 , uh − uh

0

)= a(u0 − uh

0 , u0 − uh)

with uh − uh0 ∈ Vh

≤ C∥∥u0 − uh

0

∥∥V ‖u0 − uh‖V by continuity.


This implies ∥∥u0 − uh0

∥∥V ≤

C

α

∥∥u0 − uh∥∥V ∀ uh ∈ Vh

b .

Since dimVhb <∞ we find, with inf = min,∥∥u0 − uh

0

∥∥V ≤

C

αmin

uh∈Vhb

‖uh − u0‖.

This estimate is exactly the same as that obtained below via Chapter 3. �

We re-interpret (4.109) by defining and applying a projector Q′h. For later more

general applications we formulate the following Proposition for broken Sobolev spacesVb = W 1,p

0 (T h),V ′b = W−1,q

0 (T h), 1/q + 1/p = 1, Vhb ⊂W 1,p

0 (T h), Vh′

b :=(Vh

b

)′, and a

generalized bounded fh ∈ V ′b in the form, so without boundary terms, cf. Proposition

2.34 and Remark 4.47,

〈f, vh〉 := 〈fh, vh〉V′

b×Vb:=∑

T∈T h

∫T

(n∑

i=0

fi∂ivh

)dx ∀vh ∈ Vb ∪ Vh

b . (4.113)

Proposition 4.49. For this f = fh ∈ V ′b = W−�,q

0 (T h) and the restriction fh = fh|Vhb

the norms are asymptotically equal ∀ 0 ≤ �, 1 ≤ q ≤ ∞, 1/p + 1/q = 1,

‖fh‖Vh′b

= ‖〈fh, ·〉Vh′b ×Vh

b‖ ≤ ‖f‖V′

b= ‖fh‖Vh′

b(1 + o(1)), and (4.114)

‖fh‖Vh′ = ‖〈fh, ·〉Vh′×Vh‖ ≤ ‖f‖V′ = ‖fh‖Vh′ (1 + o(1)).

For smoother f , e.g. f ∈ W−�,q0,k (T h), k > 0, cf. (4.137), satisfying homoge-

neous boundary conditions, we can estimate the ‖f‖Lq(T ), ‖f‖Lq(T h) in (4.116) by‖f‖W−�,q

0,k (T ), ‖f‖W−�,q0,k (T h) and still obtain the estimate (4.114).

Particularly interesting are the 〈f, v〉W−1,q0 (Ω)×W 1,p

0 (Ω) =∫Ω(f−1,∇ v)n + f0vdx.

For nonconforming FEMs in Sections 5.2, 5.5 we need the piecewise form as in(4.116). In the quadrature context, cf. Section 5.4, (4.114) applies to “smoother”f ∈ W−�,q

k (T h), k > 0, cf. Theorem 5.28 and (5.186).

Proof. We start with the V = Lp(T h) result: It is well known that

|(f, v)L2(T h)| ≤ ‖f‖Lq(T h)‖v‖Lp(T h), ‖f‖Lq(T h) =(∫

T h

(|f |q)dx)1/q

, (4.115)

valid for T h and T . Combined with vh ∈ Vh, dense in V we get ‖f‖Vh′ ≤ ‖f‖Lq(T h).This is combined with Proposition 4.18. For any ε > 0 there is a v1 �= 0 such that

|‖f‖Lq(T h) − (f, v1)L2(T h)/‖v1‖Lp(T h)| ≤ ε/2.


Proposition 4.18 implies the existence of a vh such that for small enough h

‖v1 − vh‖Lp(T h) < ε/(2‖f‖Lq(T h)) ⇒

‖f‖Lq(T h) < |(f, vh)L2(T h)|/‖vh‖Lp(T h) + ε ≤ ‖f‖Vh′ + ε.

Now (4.115) shows the asymptotic equality, hence (4.114) for the Lq(T h) case.For the W−�,q

0 (T h) case we formulate the first step, W−1,q0 (T h). We apply the

discrete Holder inequality (1.44) to the f ∈W−1,q0 (T h), cf. Proposition 2.34,

〈f, vh〉W−1,q0 (T h)×W 1,p

0 (T h) (4.116)

=∑

T∈T h

∫T

(f−1, ∇ vh)n + f0vdx

≤∑

T∈T h

n∑i=1

‖f i−1‖Lq(T )‖∂i vh‖Lp(T ) + ‖f0‖Lq(T )‖vh‖Lp(T )

≤

⎛⎝ ∑T∈T h

‖f−1‖qLq(T ) + ‖f0‖q

Lq(T )

⎞⎠1/q⎛⎝ ∑T∈T h

‖∇ vh‖pLp(T ) + ‖vh‖p

Lp(T )

⎞⎠1/p

= ‖f‖W−1,q0 (T h)‖vh‖W 1,p

0 (T h) ∀ vh ∈ W 1,p0 (Ω) ∪W 1,p

0 (T h).

yielding (4.114). �

For our special case V = H1(Ω),V ′ = H−1(Ω),Vb = H10 (Ω), Vh

b ⊂ H10 (Ω), Vh′

b , wedefine for the f ∈ V ′

b in (4.113), but valid for f ∈ V ′b = W−m,q

0 (Ω) as well,

Q′h ∈ L

(V ′

b,Vh′

b

)as 〈Q′hf − f, vh〉V′

b×Vb= 0 ∀ vh ∈ Vh

b ⇐⇒ Q′hf − f ⊥ Vh

b ,

‖Q′hf‖Vh′b

= supvh∈Vh

b ,‖vh‖V=1

〈Q′hf, vh〉V′b×Vb

= supvh∈Vh

b ,‖vh‖V=1

〈f, vh〉V′b×Vb

= ‖f‖Vh′b≤ sup

v∈Vb,‖v‖V=1

〈f, v〉V′b×Vb

= ‖f‖V′b

=⇒ ‖Q′h‖L(V′b,Vh′

b ) ≤ 1, (4.117)

and, with dense Vhb ⊂ Vb, limh→0 ‖Q

′hf‖Vh′b

= ‖f‖V′b.

Proposition 4.50. These projectors Q′h in (4.117) satisfy the conditions in Defini-

tion 3.12 for convergence of general discretization methods.

Consequently (4.109) can be re-interpreted as

uh0 ∈ Vh

b :⟨(Auh

0 − f), vh⟩V′

b×Vb= 0∀ vh ∈ Vh

b ⇐⇒ Q′hAuh

0 = Q′hf. (4.118)

Since Vhb ⊂ Vb, the (4.107), (4.109) imply

a(u0 − uh

0 , vh)

= (f, vh)− (f, vh) = 0 ∀ vh ∈ Vhb . (4.119)


K K = TF

z 1 z 2

z 3z3

z1

z2

Figure 4.25 Affine transformations applied to a triangle.

In Subsection 4.3.3 and Section 5.5 we will call a(u0 − uh0 , v

h) the (variational)consistency error for u0. So it vanishes for conforming FEMs.

Reducing the computational costs

By choosing a basis {φi}di=1 for P and using the affine equivalence in Definition 4.6,

it is possible to strongly reduce the cost for computing the∫T

(∇v,∇w)ndxdy,

∫T

vwdxdy,

∫T

vdxdy, mainly for v = vh, w = wh ∈ Vh (4.120)

by computing the integrals over the reference domain K only once, cf. Proposition5.41 and Figures 4.25 and 4.26. An analogous procedure applies to the general caseand, modified, to nonconforming FEs as well. The values for i, j = 1, . . . d′,∫

T

(∇uh

i , ∇uhj

)ndxdy and

∫T

uhi , u

hj dxdy, with d′ = dimVh

b , (4.121)

are denoted as elements of the stiffness and mass matrices, respectively

z 2

z 4

z 1

z 3

z1

z

z2

z3

z4

K F

Figure 4.26 Affine transformations applied to a rhombus.


Example 4.51. We demonstrate this procedure for Example 4.11 and a subdivisioninto triangles or rhombuses. Obviously, it can be extended to many other cases as well.K is either the unit triangle or the square. We memorize the transformation FT , itsJacobian and its partials from (4.14), (4.19), (4.22).

x = (FT )1 (ξ , η) := x1 + (x2 − x1) ξ + (x3 − x1) η

y = (FT )2 (ξ , η) := y1 + (y2 − y1) ξ + (y3 − y1) η. (4.122)

For the substitution in the integrals we need the Jacobian

J = det (F ′T ) = (x2 − x1)(y3 − y1)− (y2 − y1)(x3 − x1) �= 0 (4.123)

and the partials (note that J, ξx, ξy, ηx, ηy are constant for each T )

vx = uξξx + uηηx, ξx = (y3 − y1)/J, ξy = − (x3 − x1)/J

vy = uξξy + uηηy, ηx = −(y2 − y1)/J, ηy = (x2 − x1)/J. (4.124)

With the notation

v = u ◦ F−1T , u = v ◦ FT , w = z ◦ F−1

T , z = w ◦ FT ,

we obtain for the integrals in (4.120) the following relations cf. Figure 4.25∫T

(∇ v )2dxdy =∫

T

(v2

x + v2y

)dxdy

=∫

K

[(uξξx + uηηx)2 + (uξξy + uηηy)2]Jdξdη

= a

∫K

u2ξdξdη + 2b

∫K

uξuηdξdη + c

∫K

u2ηdξdη (4.125)

with

a := [(x3 − x1)2 + (y3 − y1)2]/J,

b := −[(x3 − x1)(x2 − x1) + (y3 − y1)(y2 − y1)]/J,

c := [(x2 − x1)2 + (y2 − y1)2]/J.

Similarly, we obtain∫T

(∇ v, ∇w )ndxdy =∫

T

(vxwx + vywy)dxdy

=∫

K

[(uξξx + uηηx)(zξξx + zηηx) (4.126)

+ (uξξy + uηηy)(zξξy + zηηy)]Jdξdη

= a

∫K

uξzξdξdη + b

∫K

(uξzη + zξuη)dξdη + c

∫K

uηzηdξdη


and ∫T

vwdxdy = J∫

K

uzdξdη,

∫T

vdxdy, J∫

K

udξdη. (4.127)

Whenever we use polynomial FEs, we are thus left with the evaluation of the integrals∫Kξpηqdξdη.∫

K

ξpηqdξdη =: Itriangpq = p!q!/(p + q + 2)! for the unit triangle K (4.128)

∫K

ξpηqdξdη =: Isquarepq = 1/(p + 1)(q + 1) for the unit square K.

Finally, a basis for the polynomials in P has to be inserted into (4.125), (4.126), (4.127)to determine the system (4.109). This allows us in an efficient way to compute theelements of the stiffness and mass matrices for our system. �

4.3.2 Finite element methods for general linear equations and systemsof orders 2 and 2m

These methods represent a straightforward generalization of the preceding (4.107),(4.117), (4.118), (4.119). So we do not repeat the original forms of the equations andsystems, cf. Sections 2.3, 2.4, 2.6, but rather formulate the FE equations for thesecases. For the convenience of the reader we recapitulate the notation: For the multi-indices α, we defined, see (2.73),

∂iu =∂u

∂xi,∇u = (∂1u, . . . , ∂nu) and ∂αu = ∂α1

1 . . . ∂αnn u =

∂|α|

∂xα11 . . . ∂xn

αnu,

∇0u = u,∇ku = (∂αu)|α|=k,∇u = ∇1u = (∂αu)|α|=1, and vectors and reals

ϑ = (ϑ1, . . . , ϑn) ∈ Rn, ϑα = (ϑ1)α1 · · · (ϑn)αn , ϑi,Θ0 ∈ R, i ≥ 0, choose the

nk, Nk, s.t. Θk = (ϑα)|α|=k ∈ Rnk , (ϑα)|α|≤k ∈ RNk ,Θ = Θ1, (4.129)

with ∂iu(x), ∂αu(x) ∈ R, ∇u(x) ∈ Rn, ∇ku(x) ∈ Rnk .

The FEM for a linear second order elliptic equation, see (2.160), determines uh0 such

that

uh0 ∈ Uh

b : a(uh

0 , vh)

=⟨Auh

0 , vh⟩V′

b×Vb=∫

Ω

n∑i,j=0

aij∂iuh

0∂jvhdx = 〈f, vh〉V′

b×Vb

∀ vh ∈ Vhb , with aij ∈ L∞(Ω), Vh

b ⊂ Vb = H10 (Ω). (4.130)

The linear form f and our elliptic bilinear form are bounded, so

|〈f, v〉V′b×Vb

| ≤ C ′‖v‖V , |a(u, v)| ≤ C‖u‖U‖v‖V ∀u, v ∈ Vb.


The principal part, ap(u, v), satisfies for all ϑ ∈ Rn

λ|ϑ|2n ≤ λ(x)|ϑ|2n ≤ (−1)mn∑

i,j=1

aij(x)ϑiϑj ≤ Λ|ϑ|2n ∀ x ∈ Ω with λ > 0. (4.131)

By (2.22) the principal part, ap(u, v), is Vb = H10 (Ω)-coercive, see (2.79), hence,

ap(u, u) =∫

Ω

n∑i,j=1

aij∂iu∂judx ≥ λ∗‖u‖2U , λ∗ > 0,∀u ∈ Ub ⊃ Uh

b . (4.132)

Conforming FEMs for elliptic equations of order 2m require Vhb ⊂ Vb = Hm

0 (Ω), hencesmooth FEs, cf. (2.160) the references in Subsections 4.2.6, and 4.3.2. For linear A,determine uh

0 ∈ Uhb ⊂ Hm

0 (Ω)

a(uh

0 , vh)

=⟨Auh

0 , vh⟩V′

b×Vb=∫

Ω

∑|α|,|β|≤m

aαβ∂αuh

0∂βvhdx = 〈f, vh〉V′

b×Vb(4.133)

∀vh ∈ Vhb ⊂ Vb with aαβ ∈ L∞(Ω), and for 1 < m = |α| = |β| : aαβ ∈ C(Ω).

This conformity condition Uhb ⊂ Hm

0 (Ω) is somehow cumbersome, since Cm−1 FEs arenot too common. In fact, we have only given some references and the correspondinginterpolation error and inverse estimates in Subsection 4.2.6.

We had assumed an elliptic equation, hence a(u, v) is bounded and the principalpart, ap(u, v), satisfies with λ > 0 and ∀ϑ ∈ Rn, see (2.79),

λ|ϑ|2mn ≤ λ(x)|ϑ|2m

n ≤ (−1)m∑

|α|=|β|=m

aαβ(x)ϑαϑβ ≤ Λ|ϑ|2mn ∀ x ∈ Ω. (4.134)

By (2.160), and Theorem 2.43, the principal part, ap(u, v), is Vb-coercive, hence

ap(u, u) =∫

Ω

∑|α|=|β|=m

aαβ∂αu0∂

βudx ≥ λ∗‖u‖2V , λ∗ > 0,∀u ∈ Ub ⊃ Uhb . (4.135)

For systems we have to extend and modify the above notation: We have to applythe partials ∂l, l = 1, . . . , n, to q components uj , vi, i, j = 1, . . . , q and modify (4.129)from reals ϑl, ∂lu(x) ∈ R, to vectors �ϑl, �ϑα, ∂l�u(x), ∂α�u(x) ∈ Rq, and from n-vectorsϑ = (ϑ1, . . . , ϑn),∇u(x) ∈ Rn to n× q-matrices �ϑ,∇�u(x) ∈ Rn×q, see (2.335), (2.389).In addition to the notation in (4.129) we need:34

�u = (u1, . . . , uq), ∂α�u = (∂αu1, . . . , ∂αuq), �ϑl =

(ϑl

1, . . . , ϑlq

), �ϑα =

(ϑα

1 , . . . , ϑαq

),

�ϑ = (�ϑ1, . . . , �ϑn), with |�ϑ| = |�ϑ|nq ∈ R, �u(x), ∂α�u(x), �ϑl, �ϑα ∈ Rq, (4.136)

∇�u(x) = (∂1�u, . . . , ∂n�u)(x), �Θ = (�ϑ1, . . . , �ϑn) ∈ Rn×q, ∂α�u = �u for |α| = 0.

34 The usual notation for θ and �u motivates the certainly not optimal notation �ϑl, �Θ, which turnsout to be appropriate below.


The FEM for a linear second order elliptic system requires ansatz and test-functions�uh =

(uh

1 , · · · , uhq

)�vh ∈ Vh

b (Rq) ⊂ Vb = H10 (Ω,Rq), with uh

j , vhj ∈ Vh

b . With the Euclid-ean product (fk, ∂

k�vh)q in Rq, see (2.340), (2.341), we obtain the FE solution�uh

0 ∈ Vhb (Rq), and the exact solution �u0 ∈ Vb(Rq) by omitting all the h, as

〈�f,�vh〉 =∫

Ω

n∑k=0

(fk, ∂k�vh)qdx = a

(�uh

0 , �vh)

=∫

Ω

n∑k,l=0

(Akl∂

l�uh0 , ∂

k�vh)qdx

∀ �vh ∈ Vhb ⊂ H1

0 (Ω,Rq), with Akl ∈ L∞(Ω,Rq×q), fk ∈ L2(Ω,Rq×q), (4.137)

and ‖�f‖H−10,j (Ω) := ‖�f‖H−1

0,j (Ω,Rq×q) := maxk=0,...,n

{‖fk‖Hj(Ω,Rq×q)},

with bounded 〈�f,�vh〉, a(�uh, �vh). The Vb-coercivity of the principal part is guaranteedfor uniformly elliptic systems, satisfying the uniform Legendre condition, see (2.343),(2.345), hence

∃ λ,Λ, s.t. 0 < λ < Λ <∞ : λ|�ϑ|2nq ≤n∑

k,l=1

(Akl(x)�ϑl)�ϑk ≤ Λ|�ϑ|2∀x ∈ Ω ⇒ (4.138)

ap(�u, �u) > λ∗‖�u‖2V and a(�u, �u) ≥ λ∗∗‖�u‖2V − Cc‖�u‖2W ∀�u ∈ Vb,W = L2(Ω,Rq).

(4.139)

For the FEM for a linear elliptic system of order 2m we choose �uh, �vh = (v1, . . . , vq) ∈Vh

b ⊂ Vb = Hm0 (Ω,Rq), and determine �uh

0 ∈ Uhb such that, see (2.392),

�uh0 ∈ Vh

b ⊂ Vb = Hm0 (Ω,Rq) : 〈�f,�vh〉 =

∫Ω

∑|α|≤m

(fα, ∂α�vh)qdx = a

(�uh

0 , �vh)

= (4.140)

∫Ω

∑|α|,|β|≤m

(Aαβ∂

β�uh0 , ∂

α�vh)qdx∀ �vh ∈ Vh

b , Aαβ(x) ∈ L∞(Ω,Rq×q), fα ∈ L2(Ω,Rq).

We assume the strong Legendre–Hadamard condition

∃λ > 0∀ϑ ∈ Rn, η ∈ Cq, x ∈ Ω :∑

|α|=|β|=m

(Aαβ(x)ϑβη, ϑαη)q ≥ λ|ϑ|2m|η|2. (4.141)

By Theorem 2.104 the principal part is Ub- and Uhb -coercive

ap(�uh, �uh) =∫

Ω

∑|α|=|β|=m

(Aαβ∂β�uh, ∂α�uh)qdx ≥ λ∗‖�uh‖2V (4.142)

if additionally Aαβ ∈ C(Ω) ∀|α| = |β| = m.

4.3.3 General convergence theory for conforming FEMs

For all the previous cases (4.109), (4.130), (4.133), (4.137), (4.140), the conformingFEMs have a very similar structure: we have replaced Vb by a sequence of finite


dimensional spaces{Vh

b

}h∈H

. These Vhb ⊂ Vb do satisfy the boundary conditions and

the appropriate regularity conditions, e.g. Vhb ⊂ Vb = Hm

0 (Ω) ⊂ V = Hm(Ω), e.g. Vhb ⊂

Cm−1(Ω). Thus a(uh, vh) : Vhb × Vh

b → R is well defined. We consider the exact and theFE solutions u0 ∈ Vb = Hm

0 (Ω), and uh0 ∈ Vh

b ⊂ Hm0 (Ω) of equations and �u0 ∈ Vb ⊂

Hm0 (Ω,Rq), �uh

0 ∈ Vhb ⊂ Hm

0 (Ω,Rq) of systems. We denote the approximate solutionuh

0 ∈ Vhb , using the

same symbol for u0 ∈ Vb = Hm0 (Ω,Rq), uh

0 ∈ Vhb ⊂ Vb = Hm

0 (Ω,Rq), q ≥ 1,

so uh0 ∈ Vh

b ⊂ Vb : a(uh

0 , vh)

=⟨Ahuh

0 , vh⟩V′

b×Vb= 〈f, vh〉V′

b×Vb∀ vh ∈ Vh

b . (4.143)

Here Ahuh is induced by the bilinear form a(uh, vh), analogous for quasilinear prob-lems.

All these methods are special cases of the general discretization methods inChapter 3. For the proof of convergence we have to verify the conditions in Subec-tions 3.3–3.4 for the different spaces and operators constituting these methods. Wesummarize the conditions and definitions in Summary 4.52. The wavelet and spectralmethods in Chapter 9 and in [120] are conforming variational methods as well. So thefollowing summary is applicable to these cases and to non conforming FEMs as well.Their relations are sketched in the diagram in Figure 4.27.

Summary 4.52. We formulate it in a more general form to include FEMs withvariational crimes, by using the necessary spaces U ,Ub,V,Vb,Uh

b ,Vhb and extensions

Ah, fh, Gh, cf. Chapter 5. For conforming methods, we have Ah = A, f = fh, Gh = G.

1. The method has to be applicable to the problem Au0 = f or Gu0 = 0, seeDefinition 3.12. This requires, throughout with ∀h ∈ H,h < h0, and fixed h0,a sequence of approximating spaces Uh

b ,Vhb for the Ub,Vb, projectors Ph, Q

′h,and the discrete problems Ah or Gh, with inf{0 < h ∈ H} = 0, such that

{Uh,Vh, Ph, Q′h, Gh}h∈H , dimUh

b = dimVhb <∞, (4.144)

thus limh→0 = limh∈H,h→0 makes sense. Definition 3.12 does not require that,but for our methods the Uh

b ,Vhb are approximating spaces, such that

∀u ∈ U : limh→0

dist (u,Uh) = 0, e.g. limh→0

||u− Phu||Uh = 0, (4.145)

similarly for V, with Ph = Ih for conforming methods, cf. Proposition 4.18.

Ub V ′b

Vbh ′

Vb

P h Φh Q′h

Ubh Vb

hAh

A tested by

tested by

Figure 4.27 Our methods: spaces, operators, often Ub = Vb,Uhb = Vh

b .


2. The Ph ∈ L(Ub,Uh

b

), Q

′h ∈ L(V ′

b,Vh′

b

), are linear approximation and projec-

tion operators, cf. (4.117), Proposition 4.50 and Definitions 3.6, 3.12, such that

limh→0

‖Phu‖Uh = ‖u‖U by (4.145) and limh→0

‖Q′hf‖Vh′b

= ‖f‖V′b. (4.146)

3. Our methods define a mapping Φh : D(Φh) ⊂ (U → V ′) → (Uh → Vh′), trans-

forming Au0 = f or Gu0 = 0 into Ahuh0 = fh or Ghuh

0 = 0. For nonconform-ing FEs, we have to extend the A,G : U → V ′ onto the corresponding spaces,UT h ,V ′

T h , piecewise defined on the subdivision, T h, cf. (4.35), (4.36), and(5.251) ff., (7.57) ff.

A,G : U → V ′ into Ah, Gh : U ∪ UT h → V ′ ∪ V ′T h with (4.147)

Ah, Gh|U = A,G, and Ah = A,Gh = G, for conforming methods.

In fact, the A,G,Ah, Gh do not, but the unique solvability does need the bound-ary conditions. For our methods, Φh is defined as

Ah := ΦhA := Qh′Ah|Uh

b, and fh := Φhf := Qh′

fh|Uhb, hence (4.148)

Φh(A · −f) = Qh′(Ah · −fh)|Uh

bor Gh := ΦhG := Qh′

Gh|Uhb.

4. According to Definition 3.14, we choose u, uh, related as Ahuh = Q′hAhu

h =Q

′hAu and Ghuh = Q′hGhu

h = Q′hGu. Then

Q′hAhu−Ahuh = Q

′h(Ahu−Ahuh), Q

′hGhu−Ghuh = Q′h(Ghu−Ghu

h)

(4.149)

are the so-called (variational) consistency errors.5. The (classical) consistency error or local discretization error in u is

AhPhu−Q′hAu = AhPhu−Q

′hAhu or GhPhu−Q′hGu. (4.150)

As a consequence of the above extension A,G to Ah, Gh we always have Au =Ahu,Gu = Ghu, cf. Theorem 4.54, (4.157).

6. The Gh are called stable in uh, and Ah stable, if for fixed h0, r, S ∈ R+, and∀h ∈ H

uhi ∈ Br(uh), i = 1, 2,⇒

∥∥uh1 − uh

2

∥∥Uh ≤ S

∥∥Gh(uh

1

)−Gh

(uh

2

)∥∥Vh′ . (4.151)

For Ah or Ah − fh this reduces to (independent of uh, but for ∀r > 0, h < h0)

(Ah)−1 ∈ L(Vh′

b ,Uhb

)exists and ||(Ah)−1||Uh

b ←Vh′b≤ S.

The stability of Gh is essentially implied by the stability of its derivative(G′(u0))h, cf. Theorem 3.23.

7. If the exact and discrete solutions, u0 and uh0 exist, the uh

0 are called convergentand convergent of order p, if

limh→0

∥∥uh0 − u0

∥∥U = 0 and

∥∥uh0 − u0

∥∥U ≤ Chp. (4.152)


For nonconforming FEMs∥∥uh

0 − u0

∥∥U have to be replaced by

∥∥uh0 − u0

∥∥Uh .

8. For quadrature approximations the Q′h, Ah, Gh are replaced by Q

′h, Ah, Gh.9. For the nonconforming FEMs in Chapters 5 and 7, (4.150) remains correct.

10. For isoparametric FEMs Q′h are replaced by Q

′hc , applied to functionals defined

on a curved approximation Ωhc ≈ Ω.

11. For fully nonlinear problems in strong form, F := (G,BD), cf. (5.41),{Uh,Vh

Π = Vh × Vhb , P

h, QhΠ, Q

hc,Π, F

h}

h∈H, with 0 < h ∈ H, (4.153)

replaces (4.144) and modifies all the following concepts, cf. Section 5.2. Here Gand BD indicate the differential and the Dirichlet boundary operators, respec-tively, approximated by Gh and Bh

D and QhΠ projects both components (G,BD)

to Fh =(Gh, Bh

D

).

12. All our methods are so-called linear methods in the sense

Φh(B + C) = ΦhB + ΦhC, and Φh(G + f) = ΦhG + Φhf. (4.154)

We reformulate Theorem 3.21 in Chapter 3 for FEMs. In fact, we only have to updatethe previous Gh,Uh

b ,Vh′

b , Ph, Q′h, Gh, to the following definitions. The obvious updates

for the fully nonlinear case, according to (4.153), are not explicitly formulated in thenext theorem.

Theorem 4.53. Unique existence and convergence of discrete solutions: Let theoriginal problems Gu = 0 have the exact solution u0 ∈ U . Let the FE discretizationmethod be applicable to G and yield Gh = Q

′hGh|Uh : Uh → Vh′in (5.62), satisfying

1. Gh : Uh → Vh′is defined and continuous in Br(Phu0), r > 0, h-independent;

2. Gh is consistent with G in Phu0;3. Gh is stable for Phu0.

Then the discrete problems Ghuh = 0 possess unique solutions uh0 ∈ Uh near u0 for all

sufficiently small h ∈ H and uh0 converge to u0. If Gh is consistent and consistent of

order p, then uh0 converge and converge of order p, respectively:∥∥uh

0 − u0

∥∥U ≤ S‖Q′hGh(Phu0)‖Vh′

b→ 0 and

∥∥uh0 − u0

∥∥U ≤ O(hp). (4.155)

Hence, good convergence, essentially determined by the consistency, is usually onlysatisfied if u0 is smoother than u0 ∈ U .

The exact formulation for the different nonconforming FE and other methods isdelayed to Chapters 5, 7 and 8. In particular, Theorem 4.53 will be specified for theother cases.

We have restricted the discussion in this chapter to conforming FEMs: the varia-tional consistency errors vanish by (4.148) for the exact solution u0. This is a simpleconsequence of Vh

b ⊂ Vb, e.g. (4.143) implies⟨Au0 −Ahuh

0 , vh⟩Vh′×V = a

(u0 − uh

0 , vh)

= (f, vh)− (f, vh) = 0 ∀vh ∈ Vhb . (4.156)

Theorem 4.54. Variational and classical consistency errors for conforming FEMs:


1. For conforming FEMs applied to linear and nonlinear problems, Au− f = 0 andGu = 0, the variational consistency errors in u vanish, cf. (4.143), (4.149).

2. For A ∈ L(U ,V ′), and a nonlinear G : D(G) ⊂ U → V ′ with Lipschitz-continuousG ∈ CL(D(G)) theclassical consistency or local discretization error is

‖(ΦhA)(Ihu)−Q′hAu‖V′

b= ‖Q′hA(Ihu− u)‖V′

b≤ C‖A‖V′

b←↩Ub‖Ihu− u‖U .

‖(ΦhG)(Ihu)−Q′hGu‖V′

b= ‖Q′h(GIhu−Gu)‖V′

b≤ CLV′

b←↩Ub‖Ihu− u‖U .

(4.157)

3. By Proposition 4.18, the classical consistency error tends to 0, according, e.g. to‖Q′hA(Ihu− u)‖V′

b≤ C‖A‖V′

b←↩Ub‖Ihu− u‖U → 0 for u ∈ U .

Proof. For the classical consistency error we use Vhb ⊂ Vb and (ΦhA)uh = Ahuh =

Q′hAuh. We find ‖AhIhu−Q

′hAu‖Vh′b≤ C‖A‖V′

b←↩Ub‖Ihu− u‖U . �

We formulate for later reference the changes for nonconforming FEMs:

Remark 4.55. Nonconforming variational and classical consistency errors:

1. Nonconforming FEMs applied to linear and nonlinear problems, Au− f = 0 andGu = 0, yield nontrivial variational consistency errors, since A �= Ah, G �= Gh,cf. (4.149).

2. The nonconforming classical consistency error is estimated as the sum of thesenontrivial variational consistency errors and with U ,V,Uh

b ,Vhb , cf. (4.35), (4.36),

‖Q′hAh(Ihu− u)‖V′b≤ C‖Ah‖V′∪V′

T h←↩U∪UT h‖Ihu− u‖UT h

(4.158)

or ‖Q′hGhIhu−Ghu)‖V′

b≤ CL‖Ihu− u‖UT h

,

with a Lipschitz constant L for Gh : D(Gh) ⊂ U ∪ UT h → V ′ ∪ V ′T h

3. The classical consistency error tends to 0, with the variational consistency andthe interpolation error.

To admit nonconforming methods we again consider U ,V,Uhb ,Vh

b , the operatorsPh = Ih and A,Ah are already defined in (4.109), (4.117), (4.130), (4.133), (4.137),(4.140).

So Q′h in (4.117) and Proposition 4.50 have to be generalized. The arguments

(4.113)–(4.113) are still valid for f ∈W−m,p′

0 (Ω), 1/p + 1/p′ = 1, so for

〈f, vh〉 = 〈f, vh〉V′b×Vh

b:=∫

Ω

∑|α|≤m

(fα, ∂αvh)dx ∀vh ∈ Vh

b ⊂ Vb. (4.159)

The intended Q′hf ∈ Vh′

b implies that 〈Q′hf, vh〉V′b×Vb

is only defined for vh ∈ Vhb :

Q′h ∈ L(V ′,Vh′

) with 〈Q′hf − f, vh〉V′b×Vh

b= 0∀vh ∈ Vh

b ⇔ Q′hf − f ⊥ Vh

b . (4.160)


Lemma 4.56. For f ∈ V ′b in (4.159), and the projector Q

′h in (4.160), induced bythe approximating spaces Vh

b for Vb, we get, ∀ 0 ≤ m, 1 ≤ p′ ≤ ∞, 1/p + 1/p′ = 1,

‖f‖Vh′b≤ ‖f‖V′

b= ‖f‖Vh′

b(1 + o(1)), V ′

b = W−m,p′

0 (T h) and (4.161)

Q′hf = f |Vh

b, ‖f‖Vh′

b= ‖Q′hf‖Vh′

b≤ ‖f‖V′

b, limh→0

‖Q′hf‖Vh′b

= ‖f‖V′b,

‖Q′h‖Vh′b ←↩V′

b= 1 + o(1), lim

h→0‖Q′h‖Vh′←↩V′

b= 1, hence (4.146). (4.162)

These results remain correct for smooth f ∈ W−m,p′

0,k (Ω), k > 0, see (4.137) andessentially for non-conforming FEMs as well, cf. Proposition 5.55.

After these preparations, we re-interpret (4.143) as⟨(Auh

0 − f), vh⟩V′×V = 0∀ vh ∈ Vh

b ⇐⇒ Ahuh0 = Q

′hf withAh := Q′hA|Uh

b. (4.163)

As a consequence of Theorem 3.21, the convergence for conforming FEMs forthese linear elliptic boundary value problems is proved by the following summarizingtheorem. The Aubin–Nitsche improvements are formulated in Theorem 4.60.

We study the FEMs in (4.130), (4.133), (4.137), (4.140), with piecewise polynomialsof degree ≥ d− 1, and the spaces V,Vb,Vh

b . We assume the conditions (4.131), (4.133),(4.134) for equations, and (4.137), (4.138), (4.141) for systems of orders 2,m = 1 and2m. Note that conforming FEMs for order 2m require the smooth FEs in Subsection4.2.6. The special (4.109) always satisfies (4.131).

Theorem 4.57. Stability and convergence for linear and quasilinear problems:

1. Under these conditions the principal part ap(·, ·) is Uhb -coercive, and the original

form a(·, ·) is Uhb -elliptic. For a boundedly invertible A : Vb → V ′, and under

Condition 4.16, Ah = Q′hA|Ub

is stable by Theorem 3.29.2. Assume an exact solution u0 ∈ Hs(Ω,Rq), q ≥ 1,m ≤ min{s, d}, and V =

Hm(Ω,Rq). Then the FEM (4.163) has unique solutions uh0 ∈ Uh

b , converging as∥∥u0 − uh0

∥∥V ≤ C dist

(u0,Uh

b

)≤ Chmin{s,d}−m‖u0‖Hs(Ω,Rq). (4.164)

3. This remains correct for quasilinear problems Gu0 = 0 and their FEMs for aboundedly invertible G′(u0), satifying the above conditions for A.

Remark 4.58. These convergence estimates, (4.164) in Theorems 4.57, and 5.71,5.66, are formulated for functions, e.g. u ∈ Hs(Ω), s ≥ d smooth enough to yield thebest order of accuracy. If this smoothness is reduced, then the interpolation errors aremodified according to

u ∈ Hd−p(Ω) =⇒ ‖Ihu− u‖Hm(Ω) ≤ Chd−m−p|u|Hd−p(Ω) (4.165)

or, in (5.348) for m = 1 : ‖AhIhu−Q′hAu‖Hm(Ω) ≤ Chd−1−p|u|Hd−p(Ω).

This is a consequence of the fact that these results, including the interpolation errors,are proved with the Bramble–Hilbert Lemma 4.15. Analogous reductions are valid forall the other cases as well. We do not always refer to this remark.


Proof. Now we have available all the necessary results to apply the general convere-gence theory summarized in Theorems 4.53, 4.54. In fact, the conditions (4.144)–(4.151) are verified, ecxcept (4.151): The Vh

b are admissible Galerkin approximatingspaces for Vb. Their Φh is applicable to all the above problems yielding the FEMs listedpreceding Theorem 4.57. By (4.150), (4.156), the consistency errors vanish. The aboveu0 ∈ Hs(Ω), s > m, yields a consistency error as in (4.164). The discrete operator Ah

defined in (4.163) is continuous in Uhb . So the only missing condition, (4.151), for

Theorem 4.53 is the stability of Ah. We have seen above that in all cases the principalparts ap(., .) are Vb- and Vh

b -coercive by the inf–sup condition and Theorem 2.12. Sothese principal parts generate an invertible Ap with stable Ah

p . Theorem 3.29 impliesthen, for a consistent Ah in u0, and an invertible A : Ub → V ′, in fact a compactperturbation of Ap, a stable Ah. So these FE equations have a unique solution, uh

0 ,converging as in (4.164). �

Under conditions, which usually are satisfied, it is possible to increase the orderof convergence with respect to a weaker norm. In addition to the original problema(u0, v) = (f, v)∀v ∈ Vb we need the solution, φg, of its dual problem, see (5.352),

φg ∈ Vb s.t. a(w, φg) = (g, w)∀w ∈ Ub.

The proof of this lemma and the following theorem is presented for the more generalcase of FEMs violating continuity in Lemma 5.73 and Theorem 5.74.

Lemma 4.59. Aubin–Nitsche lemma: Let a(·, ·) : Ub × Vb → R be a continuous bilin-ear form, and let the conditions of Theorem 4.57 be satisfied. Then the original and itsdual problem have unique exact and FE solutions u0, u

h0 and φg, φ

hg . For the Banach

space W ⊃ Vb with inner product and norm (g, w) and ‖g‖W , respectively, let theembedding Vb ↪→W be continuous. Then the error for the FE solution can be estimatedin the W norm as∥∥u0 − uh

0

∥∥Wh ≤ C

∥∥u0 − uh0

∥∥Uh sup

0�=g∈W

∥∥φg − φhg

∥∥Vh /‖g‖W . (4.166)

Theorem 4.60. Aubin–Nitsche convergence for ‖u0−uh0‖L2(Ω): Under the conditions

of Theorem 4.57 and Lemma 4.59 and for an exact solution u0 ∈ Hd(Ω), the solutionof the FEM equation satisfies∥∥u0 − uh

0

∥∥L2(Ω)

≤ Ch∥∥u0 − uh

0

∥∥V ≤ Chd‖u0‖Hd(Ω). (4.167)

Instead of the Cea lemma the discrete analogue of the inf–sup Theorem 2.12 can beemployed, an important tool for proving the stability of the discrete Ah.

Theorem 4.61. Discrete Brezzi–Babuska or discrete inf–sup condition: Let Vhb be

finite dimensional Banach spaces, Ah ∈ L(Uhb ,V

′hb ), and ah(·, ·) : Uh

b × Vhb → R the

associated operator and continuous bilinear form, defined

for fixed uh ∈ Uhb by 〈Ahuh, vh〉V′h

b ×Vhb

= ah(uh, vh) ∀vh ∈ Vhb . (4.168)

4.4. FEMs for divergent quasilinear elliptic equations and systems 273

Then the four statements (4.169), (4.170), (4.171) and (4.172) are equivalent.

(Ah)−1 ∈ L(V ′h

b ,Uhb

)and C exists, s.t.‖(Ah)−1‖V′h

b ←Uhb≤ C, (4.169)

∃ ε, ε′ > 0 s.t sup0�=vh∈Vh

b

|ah(uh, vh)|/‖vh‖Vh ≥ ε‖uh‖U∀uh ∈ Uhb (4.170)

and sup0�=uh∈Uh

b

|ah(uh, vh)|/‖uh‖Uh ≥ ε′‖vh‖Vh ∀vh ∈ Vhb ,

∃ ε > 0 s.t sup0�=vh∈Vh

b

|ah(uh, vh)|/‖vh‖Vh ≥ ε‖uh‖U ∀uh ∈ Uhb (4.171)

and sup0�=uh∈Uh

b

|ah(uh, vh)|/‖uh‖Uh > 0 ∀vh ∈ Vhb ,

∃ ε > 0 s.t. sup0�=vh∈Vh

b

|ah(uh, vh)|/‖vh‖Vh ≥ ε‖uh‖Uh ∀uh ∈ Uhb (4.172)

and ∀ vh ∈ Vhb ∃ uh ∈ Uh

b : ah(uh, vh) �= 0.

For stability arguments the following constants C, ε, ε′ have to be valid only forsufficiently small h < h0, h0 > 0 and have to be independent of h, but uniformly withrespect to h.

Remark 4.62. We want to point out that the inf−sup conditions for A, a(·, ·),U ,V, . . . in Theorem 2.12 in Chapter 2 do not imply the corresponding inf−sup condi-tions for the corresponding Ah, ah(·, ·). However the equivalence of the four statementsis still correct. The original a(·, ·) : Uh → Vh or later generalizations ah(·, ·) : Uh → Vh

have to satisfy these equations uniformly with respect to h. Hence, ε > 0, ε′ > 0, haveto be independent of h. For stability arguments the constants C, ε, ε′ have to be validonly for sufficiently small h < h0, h0 > 0 and have to be independent of h in thediscrete counterpart of Theorem 2.12. Then the previous references (4.169), (4.170),(4.171) and (4.172) in Theorem 4.61 correspond to (2.51), (2.52), (2.53) and (2.54)in Theorem 2.12.

4.4 Finite element methods for divergent quasilinear ellipticequations and systems

For divergent semilinear and quasilinear elliptic equations, FEMs are known, see,e.g. Zeidler [678]. We extend them to systems. Many of the existence, uniquenessand regularity results are only known in Hm

0 (Ω). Generalizations to the Wm,p0 (Ω,Rq)

situation are possible and worthwhile. In fact linearization results as in Theorem 2.122form the basis for numerical methods for the Wm,p

0 (Ω,Rq) setting for all types ofnonlinear and even fully nonlinear problems. This opens the possibility for researchinto new problems outside of the previous scope.

We use the standard notation (4.129), and again the same symbol

u, uh for functions �u ∈Wm,p0 (Ω,Rq), �uh ∈ Vh

b ⊂Wm,p0 (Ω,Rq),m, q ≥ 1, p ≥ 2,


of equations and systems. On the basis of Section 2.7 we restrict the discussionto 2 ≤ p. But in Section 4.5 we will study these quasilinear problems as monotoneoperators. Then we prove existence and convergence for Wm,p

0 (Ω,Rq) for 1 < p <∞.To emphasize the similarity between orders 2 and 2m, we we use the notation (4.173)with |α| ≤ 1 for second order equations. The necessary existence, regularity andlinearization results are quoted before we formulate Theorem 4.63. This includesthe conditions for the Wm,p

0 (Ω,Rq)-coercivity of the principal part. The divergentquasilinear elliptic equation or system with Dirichlet boundary conditions in the tracesense and Ω in (4.5), see Skrypnik [591], Zeidler [678] and Bohmer [117], is definedand solved by u0 ∈ Ub such that with 〈·, ·〉 := 〈·, ·〉V′

b×Vband V := Wm,p(Ω,Rq)

m, q ≥ 1 : Dirichlet condition (∂ju/∂νj)|∂Ω = 0, j = 0, · · · ,m− 1, hence (4.173)

G : D(G) ⊂ Vb := Wm,p0 (Ω,Rq) → V ′ = W−m,p′

0 (Ω,Rq), 1/p + 1/p′ = 1,

∀v ∈ Vb : 〈Gu0, v〉 = a(u0, v) :=∫

Ω

∑|α|≤m

(Aα(·, u0, . . . ,∇mu0), ∂αv)qdx = 〈f, v〉.

The Aα(·, u, . . . ,∇mu) : Vb → V ′ are so-called Nemyckii operators. a(u, v) is welldefined under the conditions discussed in Theorems 2.63, 2.73, 2.75 and 2.76.

In Section 2.7, Theorems 2.122, 2.124–2.126, we have formulated conditions forbounded G′(u0) : Vb → V ′. In particular, G′(u0) : Hm

0 (Ω,Rq) → H−m(Ω,Rq) yields aHm

0 (Ω,Rq)-coercive principal part. This implies, for 2 ≤ p <∞, only a Hm0 (Ω,Rq)-

coercive principal part, cf. Theorem 2.122. For 1 ≤ p < 2, in Wm,p0 (Ω,Rq) the bilinear

a(uh, vh) are not even defined, hence 1 ≤ p < 2 has to be excluded. This differencewill be reflected in the convergence results in Theorem 4.63. However, we will developin Section 4.5 a full convergence theory for quasilinear problems in Wm,p

0 (Ω,Rq) forall cases 1 < p <∞.

However, there is an essential difference between equations and systems. Forq = 1 analytic results for the Banach space setting, G : D(G) ⊂Wm,p

0 (Ω,Rq) →W−m,p′

(Ω,Rq), are available, see Subsections 2.5.3, 2.5.4, 2.5.6. One of the condi-tions is uniform monotonicity for the quasilinear operator or its principal part,see (2.289) or (2.294). This is very loosely related to the Wm,p

0 (Ω,Rq)-coercivityof the principal part of G′(u0), the basic result for the stability of the FEMfor the quasilinear systems. We have discussed this relation in Subsections 2.7.3and 2.7.4.

For q > 1 = m analytic results are available only for the Hilbert space setting, G :D(G) ⊂ Hm

0 (Ω,Rq) → H−m(Ω,Rq); for q,m > 1 only little seems to be known, seeSubsections 2.6.4, 2.6.6.

The proof of numerical stability follows our standard lines. Combining the differenttypes of coercivity for the principal part with the compact perturbation argumentin Theorem 3.29 implies the corresponding stability of the FEM for a boundedlyinvertible G′(u0).

For this weak form (4.173) it is straightforward to formulate a conforming FEM.As test and approximation spaces Vh

b we use the standard continuous FEs vanishing

4.4. FEMs for divergent quasilinear elliptic equations and systems 275

along ∂Ω. Since Vhb ⊂W 1,p

0 (Ω,Rq), q ≥ 1, we simply replace in (4.173) the u, v ∈ Vb

by uh, vh ∈ Vhb . In this conforming FEM we determine, for m, q ≥ 1

uh0 ∈ Vh

b s.t. a(uh

0 , vh)

=⟨Guh

0 , vh⟩V′

b×Vb

:=∫

Ω

∑|α|≤m

(Aα

(x, uh

0 ,∇uh0

), ∂αvh)qdx

= 〈f, vh〉V′b×Vb

=∑

|α|≤m

∫Ω

fα∂αvhdx∀ vh ∈ Vh

b ⊂ V = Wm,p0 (Ω,Rq), (4.174)

cf. Theorems 2.63, 2.73, and 2.76. Again, the variational consistency error vanishes byTheorem 4.54.

The generalization of Q′h ∈ L(V ′,Vh′

) in (4.160) and of Lemma 4.56 to the presentsituation is obvious. This includes the necessary limits of the norms and Summary4.52 applies. So we mainly need the results guaranteeing the unique existence andregularity of solutions, and the bounded linearization for the original problem, forthe many different cases included in (4.174). These results are available in the uniqueexistence in Theorems 2.55, 2.57, 2.58, 2.61, 2.62, 2.73, 2.75, 2.76, 2.96, 2.115 or forthe regularity of solutions, cf. Theorem 4.57, in Theorems 2.56, 2.60, 2.63, 2.64,2.99,2.91, 2.101, 2.109, 2.116, and finally the bounded linearization in Theorems 2.119,2.122, 2.99, 2.125, 2.126. Similarly we have summarized in Theorems 2.122, 2.124–2.126 the conditions for Wm,p

0 (Ω,Rq), the Hm0 (Ω,Rq)-coercive principal part, and have

to modify it as above. The FE solutions, with the same symbol u0, uh0 ∈ Hs(Ω,Rq),

converge according Theorems 4.17, 4.54; compare Theorem 4.57 and (4.164). However,we have to choose the appropriate norms.

Theorem 4.63. Unique uh0 converge for quasilinear problems and 2 ≤ p <∞ : We

assume for the different cases the conditions in the previous lines, 2 ≤ p <∞, and alocally unique exact solution u0 ∈W s,p(Ω), s ≥ m, for the original problems (4.173),with a bounded and boundedly invertible linearization G′(u0) ∈ L (Vb,V ′

b). Again wehave to exclude 1 ≤ p < 2.

1. For the conforming FEMs in (4.174), with d ≥ m the variational consistencyerrors for u0 vanish, Q

′hGu0 −Ghuh0 ≡ 0.

2. The principal part G′p(u0) of G′(u0) ∈ L (Vb,V ′

b) in Vb = Wm,p0 (Ω) is Hm

0 (Ω)-

coercive, by Theorem 2.122. So the discrete(G′(u0)

)h is stable and consistencyand convergence have to be and are formulated with respect to this V = Hm(Ω)norm.

3. For G ∈ CL(D(G)), with the Lipschitz constant L for G, and u, Ihu, u0, Ihu0

∈ D(G) the classical consistency or local discretization error is estimated by

‖(ΦhG)(Ihu)−Q′hGu‖V′ = ‖(ΦhG)(Ihu0)‖V′ ≤ CL‖Ihu− u‖V .


4. By Theorem 4.54 the FE equations (4.174) are stable and have unique solutionsuh

0 ∈ Vhb . These uh

0 converge as∥∥u0 − uh0

∥∥Hm(Ω)

≤ C dist(u0,Vh

b

)≤ Chmin{s,d}−m‖u0‖Hs(Ω). (4.175)

Proof. G, hence Gh, is continuous near Ihu0. For the classical consistency error wefind, with Vh

b ⊂ Vb, by the mean value Theorem 1.43,

(ΦhG)(Ihu)−Q′h(Gu) = Q

′h(G(Ihu)−G(u)) (4.176)

= Q′h

1∫0

G′(u + t(Ihu− u))dt(Ihu− u)

= O(‖Ihu− u‖V).

The stability, and consequently consistency and convergence, is for 2 ≤ p <∞, guar-anteed by the above conditions for the linearization G′(u0), see Theorems 3.16 and3.23, hence the convergence by Theorem 3.21. �

Pousin and Rappaz [534] study from another point of view consistency, stability,convergence and a priori and a posteriori errors for Petrov-Galerkin methods appliedto other nonlinear problems.

The conforming FEMs are linear, cf. (4.154). Hence all the quasilinear equationsin (4.174) can be solved with the methods in Section 3.7: Continuation methodsyield good approximations uh

1 for uh0 . We reformulate the Newton method for the

discrete problem, Gh(uh

0

)= 0, based upon the mesh independence principle in The-

orem 3.40 and Corollary 3.41. We start with a moderately good approximation foruh

0 obtained, e.g. by continuation. Let the Newton method for the exact problemui+1 := ui − (G(ui)′)−1G(ui), start with u1. The Newton method for the discreteproblem, starting with uh

1 := Phu1, yields for i = 1, . . . ,

uhi+1 := uh

i −(Gh(uh

i

)′)−1

Gh(uh

i

), with

∥∥uh0 − uh

i+1

∥∥V ≤ Ch

∥∥uh0 − uh

i

∥∥2V . (4.177)

We require the following Lipschitz-continuity for G′,

G′(·) ∈ CL(Br(u0) ∪Br(Ihu0)), (4.178)

G|Vs: D(G) ∩

(Vs := Hs(Ω)

)→ Hs−m(Ω).

Then Theorem 3.40 shows, for the iterates uhi of the Newton method, a locally

quadratic convergence, essentially independent of h for small enough h. For u0 ∈Hs(Ω,Rq), s > m ≥ 1, we obtain convergence of the discrete to the exact iterates oforder min{s, d} −m > 0 by (4.175).

Theorem 4.64. Newton’s method for the discrete problem and mesh independenceprinciple: In addition to the conditions in Theorem 4.63 we assume (4.178). Startthe Newton process for G and Gh in (4.177) with u1 ∈ Br(u0) ∩ D(G) ∩ Vs and uh

1 :=Ihu1. Then, for small enough ‖u0 − u1‖V and h, both methods converge quadraticallyand the sequences ui, u

hi satisfy for i = 1, . . . ,

4.5. General convergence theory for monotone operators 277

∥∥uhi+1 − uh

i

∥∥V ≤ C

(∥∥uhi − uh

i−1

∥∥V)2

and∥∥uh

i − Ihui

∥∥V ≤ Chmin{s,d}−m‖ui‖Vs

.

(4.179)

4.5 General convergence theory for monotoneand quasilinear operators

The linearization technique for the previous FEMs and all the following methodsdoes not allow convergence results for monotone operators and quasilinear operatorsin Wm,p(Ω) for 1 < p < 2, cf. Subsection 4.3.3 and Lemma 2.77, Section 2.7, andChapters 4 and 5. However it yields, for 2 ≤ p <∞, convergence of the expectedorder, e.g. hmin{s,d}−m, with respect to the discrete Hm(Ω) norms for all types ofelliptic problems. The conditions for the analytical approaches strongly vary, e.g. thecondition of a strongly monotone or an elliptic operator, cf. Chapter 2.

On the other hand, the monotone operator techniques yield convergence resultsfor 1 < p <∞ with respect to discrete Wm,p(Ω) norms, with different orders ofconvergence, cf. Theorem 4.67. Strongly monotone operators or quasilinear operatorsin H2(Ω) allow specific results.

So we do not want to miss the chance for embedding the monotone operatorapproach into our general discretization theory, cf. Chapter 3. We do not aim to give afull proof for this area. Rather we combine the excellent presentation in Zeidler [678],mainly his Theorems 25.B and 26.A, cf. Theorem 2.68, with Chapter 3. Additionally,we formulate the results such that the nonconforming methods, indicated in Summary4.52, are included.

We only formulate results yielding unique existence of discrete solutions, uh0 , their

convergence, and converging iteration methods for computation.The basis for the following results is the obvious

Proposition 4.65. All approximation spaces Uh,Vh, in particular Uh,Vh ⊂Wm,p(Ω) or ⊂Wm,p(T h), 1 ≤ p <∞, in this and the next book [120], are reflexiveBanach spaces.

We need the operators Ph : Ub → Uhb and the Q

′h such that for f ∈ V ′

b and f ∈ V ′

b ∪ V′

T h

Q′h : V ′

b → V ′hb :

⟨Q

′hf − f, vh⟩V′h×Vh

= 0 ∀vh ∈ Vhb , for conforming, and

Q′h : V ′

b ∪ V ′T h → V ′h

b :⟨Q

′hf − f, vh⟩V′

T h×VT h

= 0 ∀vh ∈ Vhb nonconf. meth. (4.180)

In Summary 4.52 we had introduced FEMs by applying a mapping Φh : D(Φh) ⊂ (U →V ′) → (Uh → Vh′

). It transforms Au0 = f or Gu0 = 0 into Ahuh0 = fh or Ghuh

0 = 0.For nonconforming FEs, we have to extend the A,G : U → V ′ to the correspondingspaces, UT h ,V ′

T h , piecewise defined on the subdivision, T h, cf. (4.35), (4.36), and(5.251) ff., (7.57) ff.

extend A,G : U → V ′ into Ah, Gh : U ∪ UT h → V ′ ∪ V ′T h piecewise with (4.181)

Ah, Gh|U = A,G, and Ah, Gh = A,G, for conforming FEMs.


These A,G,Ah, Gh, Ah, Gh do not, but the unique solvability does need the boundary

conditions. With the previous Qh′, Ah, Gh,Uh

b , the Φh is defined as

Ah := ΦhA := Qh′Ah|Uh

b, and fh := Φhf := Qh′

fh|Uhb, hence (4.182)

Φh(A · −f) = Qh′(Ah · −fh)|Uh

bor Gh := ΦhG := Qh′

Gh|Uhb.

Now we restrict the discussion to the case

Uhb = Vh

b . (4.183)

We modify the results in Subsection 2.5.5 to show that Theorem 2.68 in Chapter 2applies to Gh. We formulate the properties for Gh : D(Gh) ⊂ X h = Uh

b → U ′hb , with

X h to emphasize the similarity to (2.242)–(2.253) The definition of Q′h in (4.180) is

combined with the essential properties of Definition 2.65, cf ( 2.241), and the function

a : R+ → R+, a(0) = 0, limt→∞

a(t) =∞, e.g. a(t) = c|t|p−1, 0 < c, 1 < p. (4.184)

So Gh : D(Gh) ⊂ Uhb → U ′h

b is an operator, defined on a Banach space, X h = Uhb .

The following conditions are imposed ∀uh, vh ∈ X h = Uhb . Then Gh is, cf. (2.242)–

(2.253),

monotone ⇔ 〈Ghuh −Ghvh, uh − vh〉 ≥ 0, ∀uh, vh ∈ X h = Uhb , . . . , (4.185)

strictly mon. ⇔ 〈Ghuh −Ghvh, uh − vh〉 > 0 ∀uh �= vh, (4.186)

uniform. mon. ⇔ 〈Ghuh −Ghvh, uh − vh〉 ≥ a(‖uh − vh‖Xh)‖uh − vh‖Xh , (4.187)

e.g. ≥ c‖uh − vh‖pXh ,

strongly mon. ⇔ 〈Ghuh −Ghvh, uh − vh〉 ≥ c‖uh − vh‖2Xh , c ∈ R+, (4.188)

coercive ⇔ lim‖uh‖Xh→∞

〈Ghuh, uh〉‖uh‖Xh

= ∞, (4.189)

continuous ⇔ ∀ε > 0∃δ > 0 : ‖Ghuh −Ghvh‖X ′h < ε∀‖uh − vh‖Xh < δ, (4.190)

hemi cont. ⇔ t→ 〈Gh(uh + tvh), w〉 is continuous on [0, 1], (4.191)

stable ⇔ ‖Ghuh −Ghvh‖X ′h ≥ a(‖uh − vh‖Xh). (4.192)

We claim that the properties (4.185)–(4.192) for the discrete Gh are inherited from(2.242)–(2.253) for the original G. This is a consequence of (3.18), (4.147). Combinedwith (2.242), we obtain for conforming methods, hence G = Gh, uh, vh ∈ Uh

b ∈ Ub

∀uh, vh ∈ Uhb : 〈Ghuh −Ghvh, uh − vh〉 = 〈Q′hGuh −Q

′hGvh, uh − vh〉= 〈Guh −Gvh, uh − vh〉 ≥ 0, (4.193)

with the same relation for all the other (4.185)–(4.192).For extending this argument to nonconforming methods, cf. Definition 3.5 and

Summary 4.52, we distinguish the cases of Uhb �⊂ Ub violating the boundary conditions,

4.5. General convergence theory for monotone operators 279

or Uh �⊂ U . The first case can be handled by combining Theorem 2.128 with thetechnique applied in Section 5.2.6. We do not give further details.

For Uh �⊂ U , we refer to the conditions for the coefficient functions of G, simultane-ously of Gh, e.g. in (2.262)–(2.267) and (2.284)–(2.289). Then Theorems 2.71 and 2.73can be immediately applied to the broken Sobolev spaces, Wm,p(T h) ⊃ Uh, cf. (4.35)ff. They yield the desired properties for Gh by Theorem 2.68, e.g. (4.185)–(4.192). Theprevious ellipticity conditions for the different equations and systems of orders 2 and2m imply strong monotonicity. We summarize this discussion in

Theorem 4.66. Existence of (Gh)−1 and converging discrete solutions:

1. The properties of monotony-stability of G are inherited by Gh for conformingand nonconforming methods, cf. (4.185)–(4.192).

2. The previous properties of monotony-stability for G and Gh, (4.185)–(4.192), arevalid for equations and systems, where the uh, vh have to be replaced by �uh, �vh.

3. If Gh is strictly monotone, coercive and hemicontinuous, cf. (4.186), (4.189),(4.191), Theorem 2.68 3., implies the existence of the inverse operator (Gh)−1 :U ′h → Uh and ∀fh ∈ U ′h of unique solutions uh

0 for Gh(uh

0

)= fh.

4. If Gh is additionally uniformly and strongly monotone, respectively, cf. (4.187)and (4.188), then the inverse (Gh)−1 is continuous and Lipschitz-continuous.

5. A strongly monotone operator is coercive. Hence, if Gh is strongly monotone andLipschitz-continuous, then the inverse (Gh)−1 exists and is Lipschitz-continuous.

The classical consistency in u0 with Gu0 = 0 implies, see (3.35), (3.42)–(3.43),

GhPhu0 −Q′hGu0 = GhPhu0 −Q

′h0 = GhPhu0 → 0 ∀ h → 0, h ∈ H. (4.194)

By Theorem 4.66 (Gh)−1 is defined in and close to 0 = Ghuh0 . So we obtain

Phu0 − uh0 = (Gh)−1GhPhu0 − (Gh)−10.

For a vanishing consistency error in u0 and a continuous or even Lipschitz-continuous(Gh)−1 this implies the following estimate and theorem:∥∥Phu0 − uh

0

∥∥Uh → 0 or ≤ L‖Gh(Phu0)‖U ′h

b→ 0 ∀ h → 0, h ∈ H. (4.195)

Theorem 4.67. Unique solution uh0 and convergence for all monotone operators and

quasilinear equations and systems of order 2m,m ≥ 1, in Wm,p0 (T h), 1 < p <∞, and

all discretization methods in this and the next book [120]: Let (2.242)–(2.253) or(2.284)–(2.288) be satisfied for X h, G : X h→ X ′h, consequently for Uh

b , Gh : Uh

b → U ′hb

as well. Here Uh = X h is a sequence of approximating subspaces for monotone opera-tors and Uh

b ⊂Wm,p0 (Ω) for quasilinear equations and systems or ⊂Wm,p

0 (T h) for thenonconforming cases. Choose any of the conforming or nonconforming discretizationmethods in this or the next book [120], cf. (4.181), (4.182), yielding Gh for G.

1. Then both G(u) = 0 and Gh(uh) = 0 have unique solutions u0 and uh0 ,

respectively.2. These solutions converge according to limh→0,h∈H

∥∥Phu0 − uh0

∥∥Uh

b

= 0 for con-forming methods. For nonconforming methods with m = 1, they converge as


‖GhPhu0‖U ′hb

. These classical consistency errors ‖GhPhu0‖U ′hb

are estimated inChapters 5 and 7.

3. If the function a in (4.184) has the form a(t) = c|t|p−1, 0 < c, 1 < p, then∥∥Phu0 − uh0

∥∥Uh

b

≤(L‖Gh(Phu0)‖U ′h

b/c)1/(p−1)

. (4.196)

4. For FEMs of local degree ≥ d− 1, a Lipschitz-continuous G, hence Gh, and u0 ∈W d,p

0 (Ω), d > m, cf. (4.175), (4.41), we estimate for conforming methods∥∥Phu0 − uh0

∥∥Uh

b

≤(hd−mL‖u0‖W d(Ω)

)1/(p−1). (4.197)

For the nonconforming methods in Chapters 5 and 7 this holds for m = 1.5. If, additionally, G is strongly monotone, Uh ⊂ U = Hm

0 (Ω), and u0 ∈ Hd0 (Ω),

d > m, we obtain ∥∥Phu0 − uh0

∥∥Uh ≤

(hd−mL‖u0‖Hd(Ω)

). (4.198)

6. The case Uhb �⊂ Ub, violating the boundary conditions, can be handled by combining

Theorem 2.128 with the technique applied in Section 5.2.6.

Remark 4.68. Quasilinear PDEs cannot yet be evaluated for wavelets, hence waveletmethods are impossible. For all other methods, these results remain valid with slightmodifications as well. For difference methods this is formulated in Remark 8.27.

We finish with formulating the following generalized gradient iteration method forcomputing the uh

0 , cf. Zeidler [678], Theorem 26.B. For the above

Gh : D(Gh) ⊂ Uhb → U ′h

b , Uhb a Hilbert space, and (4.199)

Gh(uh

0

)= 0, uh

0 ∈ Uhb define

uhk+1 := uh

k − tLG(uh

k

), k = 1, 2, . . . .

The following assumption Gh(0) �= 0 makes sense, since otherwise 0 would be thedesired solution.

Theorem 4.69. Convergent gradient iteration: In (4.199) let Gh(0) �= 0, and Gh

be uniformly monotone, cf. (4.187), locally Lipschitz-continuous, hence for uh, vh ∈D(Gh),

∀r > 0 ∃M(r) : ∀‖uh‖Uh , ‖vh‖Uh ≤ r : ‖Ghuh −Ghvh‖U ′hb≤M(r)‖uh − vh‖Uh .

Furthermore, let L be linear, bounded, self-adjoint, and strongly monotone with(Luh, uh) ≥ b‖uh‖2Uh ,∀uh ∈ Uh

b , b > 0, and define the numbers with a in (4.187),

c := ‖L‖Uhb →U ′h

b, r0 :=

(1 +√c/b)‖G(0)‖/a, t0 := 2a/

(cM(r0)2

).

Then for every t ∈ (0, t0), the sequence uhk in (4.199) converges to uh

0 according to

limk→∞

∥∥uhk − uh

0

∥∥Uh

b

= 0 and∥∥uh

k − uh0

∥∥Uh

b

≤∥∥Ghuh

k

∥∥U ′h

b

/a. (4.200)

4.6. Mixed FEMs for Navier–Stokes and saddle point equations 281

4.6 Mixed FEMs for Navier–Stokes and saddle point equations

4.6.1 Navier–Stokes and saddle point equations

In Section 2.8 we presented the necessary theoretical results for the Navier–Stokes andsaddle point equations, cf. (2.320)–(2.522) and (2.501). So we repeat here only thoseaspects necessary for the mixed FEMs in Subsection 4.6.2. This area has collected ahuge amount of very substantial work. For examples, the numerical methods for theNavier–Stokes equations are listed 3,731 times in the Zentralblatt in late 2009. So theauthor has decided to present one of the classical approaches. This shows many of theessential ideas for these methods very clearly.

The stationary Navier–Stokes equation has the form, cf. (2.320),

G(�u, p) : =

⎛⎝−νΔ�u +n∑

i=1

ui∂i�u + grad p

div �u

⎞⎠ =(�f0

)in Ω

�u = 0 on ∂Ω,∫

Ω

pdx = 0, Ω ∈ C1L bounded and p : Ω → R, (4.201)

�u = (u1, . . . , un), �v = (v1, . . . , vn), �f = (f1, . . . , fn) : Ω ⊂ Rn → Rn, n ≤ 3.

The �u, p and �f denote the velocity, pressure and forcing terms of an incompress-ible medium, �v is a test function with div �u =

∑ni=1 ∂ui/∂xi, and ν = 1/R with

the Reynolds number, R. Its linearization in (�u0 ≡ 0, p0), applied to an increment(�u, p), �u �= 0, and for ν = 1, is the Stokes operator, S, cf. (2.496),

S(�u, p) :=(−Δ�u +∇p−div �u

)=(�f0

)in Ω, �u = 0 on ∂Ω,

∫Ω

pdx = 0. (4.202)

For proving convergence for our conforming FEMs, we recall that the variationalconsistency errors vanish, hence the classical consistency error can be estimated by theinterpolation errors. The stability results are obtained by applying the results in Chap-ter 3, cf. Summary 4.52 to S and its generalization, a saddle point problem, see (4.208).In Theorems 2.129, 4.70, this is shown to be stable, essentially if the well-knownBrezzi–Babuska conditions are satisfied, see (4.214). We show that, for moderate ν,the linearized Navier–Stokes operator represents a compact perturbation of S, hencesatisfies the conditions of Theorem 3.29. So we obtain convergence for the stationaryNavier–Stokes equations as well. This seems to be the state of the art anyway.

We have introduced the spaces and continuous bilinear forms, cf (2.497),(2.498),

Vb =(H1

0 (Ω))n

, W = L2∗(Ω) =

{p ∈ L2(Ω) : ∫

Ωp(x)dx = 0

}, X = Vb ×W (4.203)

a(�u,�v) =∫

Ω

(∇�u(x),∇�v(x))ndx with (∇�u(x),∇�v(x))n =n∑

i,j=1

∂ui

∂xj

∂vi

∂xj(4.204)

and b(p,�v) = −∫

Ω

p(x)div �v(x)dx for p ∈ L2∗(Ω), �u,�v ∈ Vb,

with bounded |a(�u,�v)| ≤ Ca‖�u‖V · ‖�v‖V , |b(p,�v)| ≤ Cb‖�v‖V · ‖p‖W . (4.205)


We generalized this weak Stokes form (4.203)–(4.205), cf. (2.500), to a saddle pointproblem by admitting continuous linear and bilinear forms, �f1(�v), f2(�v), a(�u,�v), b(p,�v),and Hilbert spaces Vb,W: For

continuous a(·, ·) : Vb × Vb → R, b(·, ·) : W ×Vb → R, �f = (�f1, f2) : Vb ×W → R2,

determine �u0 ∈ Vb, p0 ∈ W s.t. a(�u0, �v) + b(p0, �v) = �f1(�v) ∀ �v ∈ Vb,

b(q, �u0) = f2(q) ∀ q ∈ W. (4.206)

To obtain the situation in Chapter 3 and Summary 4.52, we introduce X := Vb ×W, �x := (�u, p), �y := (�v, q) ∈ X and the continuous bilinear form c(�x, �y) = a(�u,�v) +b(p,�v) + b(q, �u). Let A ∈ L (Vb,V ′

b) , B ∈ L (W,V ′b) and C ∈ L(X ,X ′) be the linear

operators induced by a(·, ·), b(·, ·) and c(·, ·), respectively. Then, cf (2.502),

C :=(

ABd

B0

)∈ L(X ,X ′), C�x =

(ABd

B0

)(�up

)=(A�u + BpBd�u

)

c(�x, �y) = 〈C�x, �y〉X ′×X =⟨(

A�u + BpBd�u

),

(�vq

)⟩X ′×X

(4.207)

= 〈A�u,�v〉V′b×Vb


+ 〈Bd�u, q〉W′×W

= a(�u,�v) + b(p,�v) + b(q, �u) = c(�x, �y).

For solving the saddle point and the Stokes problem, cf. (2.503) and Chapter 9,

determine �x0 = (�u0, p0) ∈ X for given �f ∈ X ′ such that (4.208)

c(�x0, �y) = 〈�f, �y〉X ′×X for all �y ∈ X ⇐⇒ C�x0 = �f ∈ X ′.

In Theorem 2.131 we formulated results for the bounded invertibility of C and theexistence, uniqueness and regularity of solutions �x0.

4.6.2 Mixed FEMs for Stokes and saddle point equations

We formulate the corresponding stability, consistency and convergence properties inthis subsection, based upon the results in Chapter 3. In the next subsection we willcombine these results with Summary 4.52. The C−1 ∈ L(X ′,X ), and the stabilityof Ch in (4.211) yield the stability for the (linearized) Navier–Stokes operator, cf.Theorems 2.131 and 3.22.

By (4.201) the orders of the highest derivatives of �u and p are 2 and 1, respectively.This is one of the reasons for choosing the FE spaces approximating �u and p differently.The corresponding FEMs are called mixed FEMs. Obviously, approximating spaces(Vh

b

)h∈H

, (Wh)h∈H for Vb,W yield(Vh

b ×Wh)h∈H

, again approximating Vb ×W.

They are Petrov–Galerkin approximations for X simultaneously with Vhb and Wh

for Vb and W. One can show that dimVhb ≥ dimWh is a necessary criterion for the

stability of the discrete operator, cf. [387]. According to [374, 387], we formulate aconforming FEM for (4.206) and (4.208): we choose approximating subspaces Vh

b ,Wh


and a right-hand side �f :(Vh

b

)h∈H

, (Wh)h∈H∀h ∈ H : dimVhb ≥ dimWh and Vh

b ⊂ Vb,Wh ⊂ W (4.209)

for �f = (�f1, f2) ∈ V ′b ×W ′ = X ′ determine �xh

0 =(�uh

0 , ph0

)∈ Vh

b ×Wh = X h

by a(�uh

0 , �vh)

+ b(ph0 , �v

h)

= �f1(�vh) ∀ �vh ∈ Vhb , (4.210)

b(qh, �uh

0

)= f2(qh) ∀ qh ∈ Wh ⇐⇒

c(�xh

0 , �yh)

= �f(�yh) ∀ �yh = (�vh, qh) ∈ X h ⇐⇒ Ch�xh0 = �fh. (4.211)

In Section 2.8 we introduced in (2.507) the closed subspace V0 ⊂ Vb. This V0 and thecorresponding discrete spaces Vh

0 are

V0 = {�v ∈ Vb : b(q,�v) = 0∀q ∈ W},Vh0 =

{�vh ∈ Vh

b : b(qh, �vh

)= 0∀qh ∈ Wh

}�⊂ V0.

For f2 ≡ 0 every solution uh0 of (4.209)–(4.211)

�uh0 ∈ Vh

0 solves a(�uh

0 , �vh)

= �f1(�vh) ∀ �vh ∈ Vh0 . (4.212)

Now �vh ∈ Vh0 �⊂ V0 usually violates div �vh = 0. So this form (4.212) is, cf. [386], a

nonconforming FEM for

�u0 ∈ V0 solves a(�u0, �v) = �f1(�v) ∀ �v ∈ V0. (4.213)

This observation is the starting point for intensive research combining mixed FEMswith nonconformity.

Here we use (4.209)–(4.211) directly for the conforming FEMs for our saddle pointproblems. According to Theorem 4.53 we need C−1 ∈ L(X ′,X ), see Theorem 3.29,for the unique existence of an exact solution, u0, the consistency, a consequence ofTheorem 4.54 and the approximating subspaces Vh

b ,Wh,X h for Vb,W,X , and thestability of Ch in (4.211). The latter is related to the Ladyzenskaja–Brezzi–Babuskacondition. It has the form, [387], Theorem 12.3.6,

inf{sup{|a(�uh, �vh)| : �vh ∈ Vh

0 , ‖�vh‖V = 1}

: �uh ∈ Vh0 , ‖�uh‖V = 1

}= αh > 0,

inf{sup{|b(qh, �vh)| : �vh ∈ Vh

b , ‖�vh‖V = 1}

: qh ∈ Wh, ‖qh‖W = 1}

= βh > 0. (4.214)

For these saddle point problems, Hackbusch’s Theorem 12.3.6 [386] yields:

Theorem 4.70. Unique discrete solutions and their convergence for saddle pointproblems:

1. Let Ω ∈ C0,1, let the a(·, ·), b(·, ·) and �f1, f2, in (4.206) be continuous bilinearand linear forms on Vb,W, choose the approximating spaces (X h)h∈H and themixed FEM as in (4.209)–(4.211), and let (4.214) be satisfied. Then the discreteproblem (4.210) is uniquely solvable.


2. If in (4.214) additionally αh ≥ α > 0, βh ≥ β < 0 ∀ h ∈ H, then Ch in (4.211)is stable and, with Ch = Ch(α, β,Ca, Cb) and Ca, Cb in (4.205),(∥∥�uh

0

∥∥2V +

∥∥ph0

∥∥2W

)1/2

≤ Ch

(‖�f1‖2V′

b+ ‖f2‖2W′

)1/2

, and (4.215)

∥∥�u0 − �uh0

∥∥V +

∥∥p0 − ph0

∥∥W ≤ C

(inf

�vh∈Vhb

‖�u0 − �vh‖V + infqh∈Wh

‖p0 − qh‖W)

e.g. ≤ Ch(‖�u0‖H2(Ω) + ‖p0‖H1(Ω))

by (4.220) for the approximations in Examples 4.71 and 4.72

For the specific case of the Stokes operator we only present three examples forX h = Vh

b ×Wh, see [386], satisfying the above conditions. For other examples cf. e.g.Brenner and Scott [141], Brezzi and Fortin [144], Grossmann and Roos [374], Giraultand Raviart [348,349] and Temam [622,623].

Let Ω ⊂ R2 be a polygon, hence Vb = H10 (Ω)×H1

0 (Ω) and T h a quasiuniform trian-gulation for Ω, such that, there exists a fixed σ0 > 0 with the quotient h/dimT ≤ σ0

for every triangle T ∈ T h.For quadrature approximations or curved ∂Ω the modifications from Sections 5.2–

5.5 can be used.

Example 4.71. Piecewise linear elements and bubble functions:We define

Wh :={qh : ∀ T ∈ T h : qh|T ∈ P1,

∫Ω

qhdx = 0}⊂ L2

∗(Ω). (4.216)

To introduce the bubble functions �u on T h let T represent the reference triangle T ={(ξη) : ξ, η > 0, ξ + η < 1} and Φ the bijective affine mapping Φ : T → T. We define,for every T ∈ T h and the components u of �u

u(ξ, η) := ξ · η(1− ξ − η) for (ξ, η) ∈ T , and := 0 otherwise, (4.217)

uT (x, y) := u(Φ−1(x, y)) for (x, y) ∈ T and := 0 otherwise, �uT ∈(H1

0 (Ω))2

and

Vhb1 :=

{vh ∈ H1

0 (Ω) : ∀ T ∈ T h : v|hT linear combinations of P1

and bubble functions on T h}, Vh

b := Vhb1 × Vh

b1, X h = Vhb ×Wh. �

Example 4.72.

1. Piecewise linear elements on T hh/2 and T h: Let T h

h/2 be defined by replacing eachT ∈ T h by four congruent subtriangles. Then

Vhb1 :=

{v ∈ H1

0 (Ω) : linear elements on T hh/2

}, Vh

b := Vhb1 × Vh

b1, (4.218)

Wh :={q ∈ L2

∗(Ω) : linear elements on T h}, X h = Vh

b ×Wh.


2. Piecewise quadratic and linear elements on T h : Let

Vhb,2 :=

{v ∈ H1

0 (Ω) : quadratic elements on T h},Vh

b := Vhb,2 × Vh

b,2, (4.219)

Wh :={q ∈ L2

∗(Ω :) linear elements on T h}, X h = Vh

b,2 ×Wh. �

These approximations (4.217)–(4.219) satisfy

inf{‖v − vh‖V : vh ∈ Vh

b

}≤ h‖v‖H2(Ω)∀v ∈ Vb (4.220)

inf{‖p− ph‖W : ph ∈ Wh} ≤ h‖p‖H1(Ω)∀p ∈ W.

The following theorem, cf. Hackbusch [387], Theorem 12.3.11, needs an Ω, such thatthe Poisson problem is H2(Ω)-regular. That means, the unique solution, u0, satisfies

u0 ∈ H10 (Ω) : a(u0, v) =

∫Ω

(∇u0,∇v)ndx =∫

Ω

fvdx∀v ∈ H10 (Ω) (4.221)

and f ∈ L2(Ω) =⇒ u0 ∈ H2(Ω). (4.222)

By Theorems 2.47, 2.131, this is correct if Ω is convex. We summarize [387], Theorems12.3.11–12.3.15 and Corollary 12.3.16:

Theorem 4.73. Unique discrete solutions and their convergence for the Stokesproblem:

1. Let Ω ⊂ R2 be a bounded polygon, hence Ω ∈ C0,1, let Vb =(H1

0 (Ω))2 and T h be

a quasiuniform triangulation for Ω and let (4.221) be satisfied, e.g. for a convexΩ. Then Examples 4.71, 4.72 satisfy (4.214) with αh > α > 0, βh > β > 0.

2. Then we obtain the following convergence estimates for the discrete solutions�uh

0 , ph0 of (4.210), (4.211), toward the exact solution of (4.202). For ph

0 − p0

we use the dual norm∥∥ph

0 − p0

∥∥(H1(Ω)∩W)∗

with respect to W ∩H1(Ω). Theconvergence is of order 1 or 2, depending upon the smoothness of the solutions:∥∥�uh

0 − �u0

∥∥(L2(Ω))2

+∥∥ph

0 − p0

∥∥(H1(Ω)∩W)∗

≤ C∗h(‖�u0‖(H1(Ω))2 + ‖p0‖L2(Ω)

)≤ C∗∗h2

(‖�u0‖(H2(Ω))2 + ‖p0‖H1(Ω)

)(4.223)∥∥�uh

0 − �u0

∥∥(L2(Ω))2

+ h∥∥ph

0 − p0

∥∥L2(Ω)

≤ C∗∗∗h2(‖�u0‖(H2(Ω))2 + ‖p0‖H1(Ω)

).

Proof. We only show that the additional condition in [387], Theorem 12.3.15, is aconsequence of (4.221). In fact, for f2 ∈ W ∩H1(Ω) we can determine a �u1 ∈ W ∩H2(Ω), such that div �u1 = f2. Then �u− �u1 satisfies �f3 := Δ(�u− �u1) ∈ (L2(Ω))2. Sowe have to discuss for every �f − �f3 ∈ (L2(Ω))2 and f2 = 0 ∈ W ∩H1(Ω) the Stokesproblem Δ�u +∇p = �f, div �u = 0.0 By (4.221) this has a solution �u0 ∈ (H2(Ω))2 ∩(H1

0 (Ω))2, p0 ∈ H1(Ω)2 ∩W, cf. Theorem 2.131, with

‖�u0‖(H2(Ω))2 + ‖p0‖H1(Ω) ≤ C(‖�f − �f3‖(L2(Ω))2

),

and we are done. �


4.6.3 Mixed FEMs for the Navier–Stokes operator

Now we discuss the linearized Navier–Stokes operator again for Ω ⊂ Rn and follow[349, 623, 623]. Again we multiply by the test function �v = (v1, . . . , vn) and use theGreen formula to obtain, with a(·, ·), b(·, ·),Vb,W in (4.204)

For given (�f1, 0) ∈ V ′b ×W ′ determine �u0 ∈ Vb, p0 ∈ W such that∫

Ω

(−νΔ�u0 +

n∑i=1

(�u0)i∂i�u0 + grad p0, �v

)n

dx (4.224)

= νa(�u0, �v) + d(�u0, �u0, �v) + b(p0, �v) = �f1(�v) ∀ �v ∈ Vb and

b(p0, �v) = 0 ∀ �v ∈ Vb; here

d(�u,�v, �w) :=n∑

i,j=1

∫Ω

�ui(∂ivj)wjdx.

For bounded Ω, and n ≤ 4, see Temam [623] Lemma 1.2, Chapter II, Section 1,

d(�u,�v, �w) is a bounded trilinear form on Vb × Vb × Vb.

To linearize we consider for fixed �u,�v and small �w

d(�u + �w, �u + �w,�v)− d(�u, �u,�v) = d(�u, �w,�v) + d(�w, �u,�v) + o(�w).

An analogous results holds for the nonlinear (4.201) with (4.224): We obtain

G′(�u, p)(�w, r) =

⎛⎝−νΔ�w +n∑

i=1

(ui∂i �w + wi∂i�u) + grad r

div �w

⎞⎠ (4.225)

and, with the a(·, ·), b(·, ·), d(·, ·) in (4.204) and (4.224).((G′(�u, p)(�w, r)), (�v, q)

)(L2(Ω))2

:=(νa(�w,�v) + d(�u, �w,�v) + d(�w, �u,�v) + b(r,�v)

b(q, �w)

)∀(�v, q) ∈ Vb ×W = X . (4.226)

Now we consider the d(�u, �w,�v), d(�w, �u,�v)

d(�u, �w,�v) =n∑

i,j=1

∫Ω

ui(∂iwj)vjdx, for �u,�v, �w ∈ Vb (4.227)

d(�w, �u,�v) =n∑

i,j=1

∫Ω

wi(∂iuj)vjdx = −n∑

i,j=1

∫Ω

wiuj∂ivjdx.

For fixed �u, the d(�u, �w,�v) + d(�w, �u,�v) are continuous bilinear forms, correspondingto the (sum of) operator(s)

∑ni,j=1(ui∂i �w + wi∂i�u) in (4.225). This is, for fixed �u,

linear and bounded in �v, �w ∈ Vb. Then the continuous bilinear forms d(�u, �w,�v) and


d(�w, �u,�v) ∈ R define elements

d(�u, �w, ·), d(�w, �u, ·) ∈ V ′b,

hence define linear continuous operators

D1, D2 ∈ L (Vb,V ′b) as D1 �w := d(�u, �w, ·), D2 �w := d(�w, �u, ·), for �u fixed. (4.228)

By (4.224), note that �u is fixed, see (4.227)

〈D1 �w,�v〉V′b×Vb

= d(�u, �w,�v), 〈D2 �w,�v〉V′b×Vb

= d(�w, �u,�v). (4.229)

Now the embedding I : H10 (Ω) → L2(Ω) is continuous and compact, cf. Theorem 1.26,

and (4.229) shows that

D1�v = D1I�v ∀ �v ∈ H10 (Ω).

Hence, as a product of a compact and a continuous operator, D1 = D1I is a compactoperator. The same is correct for D2 as well.

Similarly to the transformation from (4.206) to (4.208) we use here, with the factorν in (4.201),

C :=(νABd

B0

), D :=

(D1 + D2

000

), (4.230)

and, with slightly different �x = (�w, r), �y = (�v, q),

(C + D)�x =(νA�w + Br + D1 �w + D2 �w

Bd �w

),

〈(C + D)�x, �y〉X ′×X = ν〈A�w,�v〉V′b×Vb

+ 〈Br,�v〉V′b×Vb

+〈D1 �w + D2 �w,�v〉V′b×Vb

+ 〈Bd �w, q〉W′×W .

We formulate

for given �f ∈ X ′ determine �x0 = (�w0, r0) such that

〈(C + D)�x0, �y〉X ′×X = 〈�f, �y〉X ′×X ∀ �y = (�v, q) ∈ X

or equivalently (C + D)�x0 = �f ∈ X ′. (4.231)

Again Remark 3.30 applies. So we can guarantee the existence of the inverse ofC + D if −1 is not an eigenvalue of the compact operator C−1D. The correspondingmixed FEMs are then defined by straightforward modification of the above (4.209)–(4.211) for (4.230)–(4.231).

The mixed FEM for the nonlinear Navier–Stokes equation requires: Find a pair(uh

0 , ph0

)∈ Vn

b ×Wh such that

νa(uh

0 , vh)

+ d(uh

0 , uh0 , v

h)

+ b(vh, ph

0

)= 〈f, vh〉 for all vh ∈ Vh

b ,

b(uh

0 , qh)

= 〈g, qh〉 for all qh ∈ Wh.(4.232)

Combining this with Summary 4.52 we obtain the following.


Theorem 4.74. Convergence of mixed FEMs for the (nonlinearized) Navier–Stokesequations, and solution of the nonlinear systems, cf. Theorems 4.70 and 4.73:

1. For Ω ∈ C0.1, the approximating spaces(X h =

(Vh

b ,Wh) )

h∈Hand the mixed

FEM for (4.230)–(4.231) analogous to (4.209)–(4.211), let (4.214) be satisfied,and let G′(�u, p) : X → X ′ be boundedly invertible. Then the mixed FE equations(4.230)–(4.231) are uniquely solvable.

2. If additionally αh ≥ α > 0, βh ≥ β > 0, then (4.230)–(4.231) are stable, consis-tent, and the discrete solutions converge as in Theorem 4.70.

3. Let Ω ⊂ R2 be a bounded polygon, hence Ω ∈ C0,1, let Vb =(H1

0 (Ω))2 and T h be

a quasiuniform triangulation for Ω and let (4.221) be satisfied, e.g. for a convexΩ. Then Examples 4.71, 4.72 satisfy (4.214) with αh > α > 0, βh > β > 0.

4. Then we obtain the following convergence estimates for the discrete solutions�uh

0 , ph0 of (4.210), (4.211), and of (4.232) toward the exact solution of the Stokes,

the linearized Navier–Stokes and the nonlinear Navier–Stokes equations (4.202),(4.231), and (4.224). For ph

0 − p0 we use the dual norm ‖ph0 − p0‖(H1(Ω)∩W)∗

with respect to W ∩H1(Ω). The convergence is of order 1 or 2, depending uponthe smoothness of the solutions:∥∥�uh

0 − �u0

∥∥(L2(Ω))2

+∥∥ph

0 − p0

∥∥(H1(Ω)∩W)∗

≤ C∗h(‖�u0‖(H1(Ω))2 + ‖p0‖L2(Ω)

)≤ C∗∗h2

(‖�u0‖(H2(Ω))2 + ‖p0‖H1(Ω)

)(4.233)∥∥�uh

0 − �u0

∥∥(L2(Ω))2

+ h∥∥ph

0 − p0

∥∥L2(Ω)

≤ C∗∗∗h2(‖�u0‖(H2(Ω))2 + ‖p0‖H1(Ω)

).

5. The numerical solution of the high dimensional nonlinear systems in this theoremis efficiently computed by combining continuation and discrete Newton’s methods,cf. Theorem 3.40.

Proof. We again apply Summary 4.52. The consistency is proved in Theorem 4.54. Thestability follows from Theorem 4.57. Hence, Theorem 4.53 implies the unique existenceof discrete solutions, their convergence and the quadratic convergence of the discreteNewton’s method, cf. Theorem 3.40. �

Other types of FEMs based upon nonlinear stability, bubbles, and a posterioriestimates are studied by Brezzi, Hughes, Marini, Houston, Rannacher, Russo, andSuli. [145,406,612].

4.7 Variational methods for eigenvalue problems

4.7.1 Introduction

Eigenvalue problems play an important role in many scientific and technical applica-tions. Examples are stability, buckling and breaking, bifurcation of, e.g. mechanicalstructures, and originating of periodic solutions and vibration, e.g. in reaction–diffusion models, and biological, geophysical, fluid dynamical, phenomena, cf. Bohmer[120]. A detailed discussion of important applications, emphasizing linear problems,is presented by e.g., Dautray and Lions [256–261], and nonlinear problems by Zeidler[675–678].

4.7. Variational methods for eigenvalue problems 289

We study only eigenvalue problems for linear operators for two reasons. Thenumerical methods for nonlinear eigenvalue problems are far less developed thanfor the linear case. Our main motivation for including eigenvalue problems is thecomputation of a bifurcation as a starting point for some types of bifurcation and/orevolving dynamical scenarios. If G depends upon a parameter, these scenarios arerelated to the eigenvalues of its Frechet derivative G′(u0), evaluated at u0.

In this section we formulate the results for eigenvalue problems in this chapter onconforming FEMs, based upon the variational characterization of the eigenvalues.However this remains correct for the different variational methods, presented inthis book and in [120]. So all the following results remain valid for conformingand nonconforming FEMs, the DCGMs, difference, wavelet, mesh-free and spectralmethods. This is due to the unified treatment of these methods, based upon theirsimilar structure.

In fact Hackbusch’s [387], Chapter 11, proofs for the eigenvalue results remain validwith extremely minor changes. So we only indicate these modifications and omit theproofs. So we generalize his results from elliptic equations of order 2 and conformingFEMs to equations and systems of order 2 and 2m and all the above methods. Fornonconforming FEMs and DCGMs the results are restricted to m = 1.

For finite difference methods, Hackbusch [387], Chapter 11, employed several tech-nical definitions and lemmas for proving convergence for one second order equation.Here we only refer to these results in [387], but do not formulate or even generalizethem. Partly these results are included or modified in our variational methods foreigenvalues and difference methods.

There are many interesting papers and books on problems with higher eigenvaluesas well. Without trying to present a complete list, we give some examples: Ahues [5],Alimov and Pulatov [7], Arias [38], Aslanov [43], Banerjee and Suri [57], Bramble andOsborn [140], Osborn [521], Parlett et al. [514, 523, 524], Chatelin [167, 168], Babuskaand Osborn [46–48], Ciarlet and Lions [177], Davies [262], Delic et al. [279], Denckeret al. [282], Graham et al. [365], Jorgens [427], Levendorskij [474], Mertins [488, 489],Saad [562,563], Schechter [570], Sun [614], Vanmaele and Zenısek [651], Vanmaele andVan Keer [650] Beattie [76], and Toyonaga et al. [630].

We only consider eigenvalue problems with simple eigenvalues. The study of multipleeigenvalues and their numerical methods is much less advanced than for simpleeigenvalues. The main reason for our restriction here is the fact that the numericalstudy here yields the tools for determining bifurcation. In this context, the invariantsubspaces, a generalization of eigenspaces of higher eigenvalues, are important. So thisclass of problems is delayed to Bohmer [120]. Again we quote some papers and books:Beyn [92], Beyn et al. [93,94], Bosek [133],Cliffe et al. [179], Dieci and Friedman [287],Govaerts [357], Joshi [429]. Sebastian [579] and Schwarzer [577].

4.7.2 Theory for eigenvalue problems

The classical EVP for elliptic PDEs is here defined by a linear elliptic operator A oforder 2m,m ≥ 1, and its boundary operators Bj as

determine λ ∈ C s.t.∃e �= 0 : Ae = λe in Ω.Bje = 0, j = 0, . . . ,m− 1, on ∂Ω. (4.234)


So for a linear operator, A, we determine those nontrivial elements 0 �= e ∈ V whichare mapped into a multiple of themselves.

This A represents either one equation or a system of q equations. For a simplifiednotation we usually employ the same symbols e, u, v, . . . for the functions in equationsand systems and avoid the �e, �u,�v, . . . form for systems. Similarly, Wm,p(Ω) indicatesWm,p(Ω) and Wm,p(Ω,Rq) as well. We include the boundary conditions in V, hence∀v ∈ Vb ⇐⇒ Bjv = 0, j = 0, . . . ,m− 1. Again we consider the bounded bilinear forma(·, ·), induced by A as in Section 2.3, and the Gelfand triple Vb ⊂ W ⊂ V ′

b, cf. (2.62).Then we

determine λ ∈ C s.t. ∃0 �= e : a(e, v) = λ(e, v)W ∀ v ∈ Vb ⊂ W ⊂ V ′b, (4.235)

usually with (u, v)W =∫Ωuvdx. For discussing eigenvalue problems appropriately, we

have to complexify V,W, A, cf. Definition 1.21, Theorem 1.22. The adjoint eigenvalueproblem requires

determine λ ∈ C s.t. ∃0 �= e : a(v, e) = λ(v, e)W ∀ v ∈ Vb ⊂ W ⊂ V ′b. (4.236)

The different A and a(·, ·) are introduced for equations of order 2, in (2.16), (2.18), oforder 2m in (2.76), (2.88), (2.102), (2.110), for systems of order 2 in (2.339), (2.340),and order 2m in (2.391), (2.392).

In Theorems 2.20 and 2.21 we have already introduced the concepts of eigenvalues,eigenfunctions, and eigenspaces E(λ) for A and for a(·, ·) , and Ed(λ) for Ad. Note

that we use the notation Ed(λ) in contrast to Ed(λ) in Hackbusch [387].

Example 4.75. Chladny sound figuresThis nonlinear problem satisfies, with a parameter λ, cf. Example 1.3, (1.16),

G(u) = Δu + λ sinu = 0 in Ω = [0, L]× [0, L] and∂u

∂n= 0 on ∂Ω. (4.237)

We linearize this equation with respect to u at u = 0 and apply G′(u) to v, cf. (1.18),obtaining

G′(0)v = Δv + λv = 0 ⇐⇒ Δv = −λv in Ω with∂u

∂n= 0 on ∂Ω. (4.238)

The −λ, allowing nontrivial solutions of (4.238), are the eigenvalues, λ0 = −λ, ofthe Laplacian with the corresponding eigenfunctions, vm,n, vn,m,

the eigenvalues λ0 = −(m2 + n2)π2/L2 ∀ m,n ∈ N0 with the eigenfunctions,

v(x, y) = vm,n(x, y) := cosm

Lπx cos

n

Lπy, and v(x, y) = vn,m(x, y). (4.239)

Therefore, we obtain, e.g. for (m,n) = (1, 1), (m,n) = (2, 3) and (m,n) = (5, 5),(m,n) = (1, 7) one, two and three linearly independent solutions vm,n, vn,m for thesimple, double and triple eigenvalues λ0L

2/π2 = −2,−13, and −50, respectively. �

The relevant theoretical results for eigenvalue problems in elliptic PDEs are sum-marized in the Riesz–Schauder theory in Theorem 2.20 and the Fredholm alternative


in Theorem 2.21, the latter formulated for the operator A. Here, we essentially usethe bilinear form a(·, ·), and recapitulate the most important results.

Theorem 4.76. Alternatives for eigenvalue problems: We assume a Gelfand triple,see (2.62), Vb ⊂ W =W ′ ⊂ V ′

b with a continuous dense and additionally compactembedding Vb ↪→W, a reflexive Banach space Vb and a Hilbert space W. For thebounded elliptic bilinear form a(·, ·) : Vb × Vb → R we get:

1. For every λ ∈ C one of the following alternatives is valid: either the induced(i) (A− λI)−1 ∈ L (V ′

b,Vb); or(ii) λ is an eigenvalue of A or of a(·, ·).

2. λ is an eigenvalue for (4.235) ⇐⇒ λ is an eigenvalue for (4.236). Then the finitedimensional eigenspaces E(λ) and Ed(λ) satisfy dim E(λ) = dim Ed(λ) <∞.

3. The spectrum of a(·, ·), or A, the σ(A) := {λ eigenvalue for A}, is at mostcountable and admits no limit points in C.

4. If a(·, ·) is symmetric, hence a(u, v) = a(v, u)∀u, v ∈ Vb, then the eigenvalues λ =λ in (4.235) and (4.236) are real and equal and E(λ) = Ed(λ) = Ed(λ).

From here on, we proceed in line with the presentation in Hackbusch [386, 387].As indicated above, the reason is the following. His proofs for the results for theeigenvalues of A and their numerical approximation are based upon the structuralresults for Vb-coercive and Vb-elliptic bilinear forms and regularity results for thesolutions of linear elliptic problems. So they remain valid for our more general situationas well. Note that Hackbusch’s definition of Vb-coercive and Vb-elliptic bilinear formsis chosen according to the originally more analytic version. We have exchanged theirrole, according to the usual version in FEMs. So we formulate his results for our moregeneral case of Vb-coercive and Vb-elliptic bilinear forms with Vb = Wm,p(Ω,Rq),m ≥1, q ≥ 1, 2 ≤ p <∞, and again exclude 1 ≤ p < 2, cf. Sections 4.4 and 4.5. Insteadof repeating Hackbusch’s proofs, we indicate which of his tools (exercises, lemmas,theorems) have to be replaced by our results. We only formulate those results, thatare important for the regularity of the exact solutions, and the computation andconvergence of the numerical approximations of eigenvalues and eigenfunctions. ForHackbusch’s other results, necessary for his proofs, we only indicate the correspondingreplacement of his tools by ours. Note that we only formulate these results for Dirichlet,but not for natural boundary conditions. Correspondingly, our results are only validfor these cases, cf. [386,387], Theorem 11.1.5:

Theorem 4.77. Regular eigenfunctions: Assume Vb = Wm,p0 (Ω,Rq),m, q ≥ 1, 2 ≤

p <∞, and the conditions for regular solutions for different types of problems inTheorems 2.28, 2.31, 2.32, 2.33, and 2.38,2.39, and 2.45,2.47,2.49 for equations, and2.91, 2.92, and 2.108 for systems of order 2m and 2 and 2m, respectively Then forappropriate s > 0 we find E(λ) ⊂Wm+s,p(Ω,Rq),m, q ≥ 1, usually omitting ,Rq.

Proof. In Hackbusch’s proof we only have to replace his Corollary 9.1.19 by ourgeneralized versions. Hackbusch’s (obvious) proof of his Corollary 9.1.19 uses his


Theorem 9.1.16, which is our Theorem 2.45. Similarly the other regularity resultscan be employed for proving Theorem 4.77. �

4.7.3 Different variational methods for eigenvalue problems

The usual approach in variational methods, applied to (4.235) and (4.236), yields

determine λh ∈ C s.t. ∃0 �= eh ∈ Vhb : ah(eh, vh) = λh(eh, vh)W ∀ vh ∈ Vh

b , (4.240)

determine λh ∈ C s.t. ∃0 �= eh∗ ∈ Vhb : ah(vh, eh∗

) = λh(vh, eh∗)W ∀ vh ∈ Vh

b ,

with Vhb ⊂ Vb or Vh

b �⊂ Vb and (eh, vh)W , (vh, eh∗)W defined ∀ eh, eh∗

, vh ∈ Vhb ,

with real λh = λh for a symmetric ah(uh, vh). There is one obvious difficulty: Theoriginal eigenvalue problems for elliptic PDEs, (4.235) and (4.236), have infinitelymany eigenvalues. Their discrete counterparts, (4.240), represent (generalized) eigen-value problems of finite dimension, and thus only have finitely many eigenvalues.Furthermore, we cannot expect dim E(λ) = dim E(λh) for an approximation λh ≈ λ.In fact the above Example 4.75 yields for a square and a symmetric difference methodapproximate eigenvalues λh with multiplicities different from the exact eigenvalues λ.So for simple or double eigenvalues λ we get dim E(λ) = dim E(λh) = k = 1, 2, with λh,near λ. For a multiple eigenvalue λ with dim E(λ) = k > 2 we find λh

j ≈ λ, j = 1, . . . , kwith dim E(λ) =

∑kj=1 dim E

(λh

j

). Then only the mean value λh :=

∑kj=1 λ

hj /k con-

verges comparably well to λ as for simple eigenvalues, cf. Babuska and Aziz [45],p. 338.

We start with the conditions in Theorem 4.76, cf. Hackbusch [387], (11.2.2a) ff.We have seen in Sections 4.4, 4.5 that a numerical convergence theory for Vb =Wm,p

0 (Ω), 2 ≤ p <∞, has to employ the embedding Vb ↪→ Hm(Ω), and yields con-vergence with respect to discrete Hm(Ω) norms. We assume,

Vb = Wm,p0 (Ω) ↪→ Hm(Ω), 2 ≤ p <∞, a(·, ·) : Vb × Vb → R is Hm

0 (Ω)-elliptic, so

Vb ⊂ L2(Ω) is continuously, densely and compactly embedded. (4.241)

Then all our FE, DCG, difference, wavelet, mesh-free and spectral approximationsintroduce sequences of approximating spaces with the corresponding conforming ornonconforming norms

Vb approximated by Vhb ⊂ Hm(T h) with the norm ‖v − vh‖Vh

b:= ‖v − vh‖Hm(T h)

and ∀v ∈ Vb : limh→0

inf{‖v − vh‖Vh

b: vh ∈ Vh

b

}= 0. (4.242)

We still use the notation Vhb , standard in this chapter, although for nonconforming

cases we might have Vhb �⊂ H1

0 (Ω). Except for conforming FE, difference and waveletmethods, we have to introduce extensions, ah(·, ·), of the original, a(·, ·),

ah(·, ·) :(Vb ∪ Vh

b

)×(Vb ∪ Vh

b

)→ R s.t. (4.243)

a(·, ·) = ah(·, ·)|Vb×Vband ah(·, ·) = ah(·, ·)|Vh

b ×Vhb.


For conforming FEMs the original a(·, ·) suffices, and for difference methods we directlywork with the discrete ah(·, ·). So for conforming FEMs we simply omit the index h,for difference methods we only discuss the nonobvious modifications.

But for all cases the Vb-ellipticity of a(·, ·) and the Vb-coercivity of its principal partimplies the Vh

b -ellipticity of ah(·, ·) and the Vhb -coercivity of its principal part with

respect to the discrete norms ‖vh‖Vhb

and ‖gh‖Wh . For nonconforming FEMs we haveto add c(uh, vh)L2(ω), for DCGMs the appropriate penalty terms. These are necessaryas well for defining the ‖vh‖Vh

b.

Now we are ready to introduce the aλ(·, ·), aλ,h(·, ·), ahλ(·, ·) as

aλ(·, ·) : Vb × Vb → C by aλ(u, v) := a(u, v) + λ(u, v)Wh (4.244)

ω(λ) := infu∈Vb, ‖u‖Vb

=1sup

v∈Vb, ‖v‖Vb=1

∣∣aλ(u, v)∣∣ and

ahλ(·, ·) : Vh

b × Vhb → C by ah

λ(uh, vh) := ah(uh, vh) + λ(uh, vh)W ,

ωh(λ) := infuh∈Vh

b , ‖uh‖Vhb

=1sup

vh∈Vhb , ‖vh‖Vh

b=1

∣∣ahλ(uh, vh)

∣∣.Note (uh, vh)Wh is well defined, and = (uh, vh)W , except for difference methods.Analogously to ah

λ(·, ·) we define aλ,h(·, ·).Then [387], Exercise 11.2.4. remains correct, since his Lemmas 6.5.3 and 6.5.17

correspond to our Theorem 2.12 and Corollary 2.14 with Theorem 2.20 yielding hisLemma 6.5.17. His Theorem 6.5.15, our Theorem 2.20, allows relating the ω(λ) andωh(λ) as in Remark 11.2.5 and Lemmmas 11.2.6 and 11.2.7. Here Remark 11.2.5 andLemmma 11.2.6 are rather obvious. For the technical proof of his important Lemmma11.2.7 we have to keep in mind that his definition of zh in his (11.2.6b) requiresthe well-defined ah

μ(zh, vh) = (λ− μ)(u, vh)W∀vh ∈ Vhb . The reference to his Theorem

8.2.2 following (11.2.6f) has to replaced by our convergence results for the variationalmethods, mentioned above. His estimate sup |aλ(u− uh, v)| ≤ CS‖u− uh‖Vb

has tobe replaced by sup |aλ,h(u− uh, v)| ≤ CS‖u− uh‖Vh

b. This last inequality remains

correct with aλ,h(u− uh, v) indicating the bounded extension of aλ(·, ·) in (4.243),(4.244), with still coercive aμ,h(uh, vh), and the consistency of the aλ,h(uh, vh) withthe aλ(u, v). The bounds in (11.2.6a)–(11.2.6f) in [387] have to be slightly modified.For difference methods the coercive ah

μ(uh, vh), and the consistency of the ahλ(uh, vh)

with the aλ(u, v) allow maintaining these [387] arguments.Similarly his Theorem 11.2.8, Lemma 11.2.9, Theorems 11.2.10–11.2.11 and Exercise

11.2.12 remain correct. Here the proofs of Theorem 11.2.8, Lemma 11.2.9, and Theo-rem 11.2.10 are simple generalizations. The proof of Theorem 11.2.11 remains verbatimcorrect for conforming variational methods. Only the reference to [387], Theorem 8.2.2,after (11.2.6f) has to replaced by our convergence results for the variational methods.For the nonconforming FEMs and the DCGMs the [387] approximate eigenfunctions eh

are no longer in Vb. Then convergence results for eigenfunctions have to be formulatedwith respect to ‖ · ‖Vh

b. Additionally, the anticrime transformation, Eh : Uh → U ,

cf. Lemma 5.77 for nonconforming FEMs and for the DCGMs, have to be employed.


Then the [387] eh are approximated by Eheh ∈ Vb and again allow the proof in [387],and consequently of Exercise 11.2.12 as well.

We summarize the qualitative convergence results from the [387], Theorems 11.2.8,11.2.10, 11.2.11, Exercise 11.2.12 in.

Theorem 4.78. Convergence of discrete eigenvalues and eigenfunctions: For all caseswe assume (4.241)–(4.242), or the modifications for conforming FE and differencemethods.

1. Choose, for the sequence of Vhb in (4.242), the approximate eigenvalues λh in

(4.240), and assume limh→0 λh = λ0. Then λ0 is an eigenvalue of (4.235).

2. For λ0, an eigenvalue of (4.235), there exist discrete eigenvalues λh0 of (4.240)

with limh→0 λh = λ0.

3. For eh ∈ E(λh) in (4.240) with ‖eh‖Vhb

= 1 and limh→0 λh = λ0, a subsequence

hi exists such that for an eigenfunction e ∈ E(λ) in (4.235), the ehi converge, solimhi→0 ‖ehi − e‖Vhi

b

= 0.

4. If dim E(λ0) = 1, we get limh→0 dim E(λh) = dim E(λ0), for these λh, eh, λ0, e.5. For dim E(λ0) = 1, choose the eh as in (3). and eh := eh/(eh, e)V if (eh, e)V ≥ 1/2

and otherwise eh := eh. Then limh→0 ‖eh − e‖Vhb

= 0.

We turn to the quantitative convergence results. In [387] the purely analytic resultsin Lemmas 11.2.13, 11.2.14 and Exercise 11.2.15 obviously remain valid. Lemmas11.2.16, 11.2.18 and Exercise 11.2.17 remain correct for the conforming method andhave to be modified for the other methods here and below: ‖v‖Vb

for v ∈ Vhb and aλ(·, ·)

have then to be replaced by ‖v‖Vhb

and ahλ(·, ·) or aλ,h(·, ·). In particular for Lemma

11.2.16, the proof of part (1) with its L2(Ω) arguments remain correct, since Vhb ⊂

L2(Ω). For all other cases the previous replacements yield the proofs. In particular,for difference methods the ah

λ(·, ·) and the consistency have to be combined. For thenext three theorems see [387], Theorems 11.2.19, 11.1.20.

Theorem 4.79. Quantitative convergence results for discrete eigenvalues: We assume(4.241)–(4.242), an eigenvalue λ0 for (4.235) with dim E(λ0) = 1 and geometric equalalgebraic multiplicity of λ0, cf. [387], Lemma 11.2.13, hence

dimN ((A− λ0I)) = dimN ((A− λ0I)2). (4.245)

Then span e = E(λ0) and span e∗ = Ed(λ0) can be normed such that (e, e∗)W = 1.Then there exist discrete eigenvalues λh of (4.240), such that

|λh − λ0| ≤ Cdist(e,Vh

b

)dist

(e∗,Vh

b

)with dist

(e,Vh

b

)= inf

w∈Vhb

‖w − e‖Vhb. (4.246)

Theorem 4.80. Convergence estimates for discrete eigenfunctions: Under the condi-tions of Theorem 4.79 and for the above e ∈ E(λ0) and e∗ ∈ Ed(λ0) there exist discreteapproximations eh ∈ Eh

(λh

0

)and eh,∗ ∈ Eh,d

(λh

0

)such that

‖e− eh‖Vhb≤ Cdist

(e,Vh

b

)and ‖e∗ − eh,∗‖Vh

b≤ Cdist

(e∗,Vh

b

). (4.247)


More detailed convergence results are possible for smooth eigenfunctions. In Theo-rem 4.77 we have formulated conditions guaranteeing

E(λ0), Ed(λ0) ⊂Wm+s,p(Ω,Rq),m, q ≥ 1 for Vb = Wm,p(Ω,Rq). (4.248)

Then the error estimates for the different methods, e.g. Theorems 4.17, 4.43, 4.36, 7.7,and Lemma 9.7, with appropriate modifications for mesh-free and spectral methodsin Bohmer [120], imply

dist(u,Vh

b

)≤ Chs‖u‖W m+s,p(Ω,Rq)∀u ∈ E(λ0) ∪ Ed(λ0). (4.249)

Theorem 4.81. Order of convergence for discrete eigenvalues and eigenfunctions:Under the conditions of Theorems 4.79, 4.80 and (4.248), (4.249) there exist λh, eh ∈Eh(λh) and eh,∗ ∈ Eh,d(λh) such that

|λh − λ0| ≤ Ch2s, ‖e− eh‖Vhb≤ Chs, ‖e∗ − eh,∗‖Vh

b≤ Chs. (4.250)

For difference methods this holds for s = 1 and s = 2 for unsymmetric and symmetricformulas.

Sometimes the eigenfunctions have better regularity properties than in Theorem4.77. This allows for m = 1, cf. [387], Theorem 11.2.22, better estimates of the form‖e− eh‖Vh

b≤ C ′h2, ‖e∗ − eh,∗‖Vh

b≤ C ′h2.

Remark 4.82. These results remain valid with slight modifications for the othermethods as well. In particular, we even might consider cases, where Uh

b �= Vhb .

5

Nonconforming finite elementmethods

5.1 Introduction

In Chapter 4 we discussed conforming FEMs and their strong requirements withrespect to ansatz and test functions, exactly evaluated test integrals, and restrictions toweak forms. We avoid this drawback by turning to nonconforming FEMs in this chap-ter. But then the proofs for consistency and stability become much more complicated.

Proceding along the lines of Chapter 3 requires, for nonconforming FEMs, replacingthe variational consistency errors by classical consistency errors. The variationalconsistency error is the usual concept in the finite element community. It is, for alinear operator A, based upon the bilinear form a(., .), and for a quasilinear operatorbased upon its weak form and vanishes for conforming FEMs. This contrasts withthe classical consistency error, directly related to operators and very appropriate forany nonlinear problems as well. We have to discuss both concepts for the followingvariational crimes.

We start with the most convincing example for this general approach. For twodecades FEMs for general fully nonlinear elliptic problems were an open problem.They were, for the first time and on the basis of nonconforming FEMs, published byBohmer [118,119]. Before results for special equations of order 2 in R2 in Fulton [330],Oliker and Prussner [519], Dean and Glowinski [272–276], have been known. The latterformulate for Monge–Ampere and Pucci equations in R2 equivalent variational formsor use regularization techniques, but see below. Linear or quasilinear elliptic problemsin divergence form are studied in the weak form and solved numerically this way.However, fully nonlinear and nondivergent quasilinear elliptic problems do not allowa weak form. So FEMs for these equations and systems, G(u0) = 0, are based uponthe strong form and are studied in Section 5.2.

As a consequence we do need in this chapter the full machinery of discretizationmethods with

G : D(G) ⊂ U → V ′, Gh : D(Gh) ⊂ Uh → V ′h and possibly U �= V,Uh �= Vh. (5.0)

Quite recently the author discovered a totally different approach in a paperwhich seems to be forgotten: Barles and Souganidis [73] prove the convergence fordifferent discretization methods, based upon the concept of viscosity solutions goingback to Crandall and Lions [217–222, 227] and Crandall et al. [215, 216, 228]. [73]


gives completely new results concerning the convergence of numerical schemes forstochastic differential games. This approach is up to now not extendable to higherorder equations and systems.

For general fully nonlinear elliptic problems, conforming FEMs are not (yet?) possi-ble. In fact, we need new regularity results for the FEM solutions of these Gh(uh

0 ) = 0,requiring convex polyhedral domains or Ω ∈ C1 for m = 1. Thus uh

0 ∈ C1(Ω) willviolate the boundary conditions. So we have to turn to nonconforming FEMs. Thestandard techniques for handling violated boundary conditions or continuity of theFEMs, cf. Section 5.5, do not apply in this case. Combining very unusual proofs forstability, consistency and Davydov’s FEs in C1(Ω), cf. Subsection 4.2.6, we are ableto show that for a continuous G with boundedly invertible linearization G′(u0), thediscrete solutions uh

0 for the nonlinear discretization Gh(uh

0

)= 0 converge to u0 in a

discrete H2(Ω) norm. We generalize this result to elliptic equations and systems oforders 2 and 2m.

Fortunately, the numerical solution methods can be reduced in this case to the wellestablished methods for weak forms. However, the standard FEs in C(Ω) there haveto be replaced by FEs in C1(Ω) here.

In many of the preceding and following cases the integrals in (4.3) and theirappropriate generalizations cannot be computed exactly. So quadrature and cubatureformulas have to be employed. These approximate FEMs, the next kind of variationalcrimes, are studied in Section 5.4.

The following Section 5.5 is devoted to other types of nonconformity. The exactboundary conditions and/or the continuity required in conforming FEs, are violatedhere, cf. Feistauer and Zenısek [317]. Since the fully nonlinear problems are discussedabove and FEs �∈ C(Ω) are excluded for them, only linear and quasilinear equations areleft. The convergence proofs for violated boundary conditions and/or the continuity,use, generalizing Brenner and Scott [141], the detour from the weak form to itsstrong counterpart. For FEMs violating these conditions, there are two options:either we use the approach via quadrature approximations along the boundariesof the subtriangles or polytopes for polynomial FEs. This allows piecewise polyno-mial FEs of degree d− 1 in Ω ⊂ Rn with n = 2, and converging with high order.For violated boundary conditions the second alternative is isoparametric FEs inRn, n ≥ 2. For all these cases we extend the previously known results from linearto quasilinear elliptic equations and systems of order 2. As a consequence of theabove violated continuity a direct comparison of a(·, ·) and ah(·, ·) is not possible.So, in contrast to [129, 567], we have to prove stability via coercivity of the principalpart and estimate the consistency error and finally combine both to get the desiredconvergence.

Another possibility for violated continuity and boundary conditions, the discontin-uous Galerkin methods (DCGMs), combine duality arguments with penalty terms, seee.g. Rachford and Wheeler [540], Douglas and Dupont [303], Wheeler [665], Arnold [39],Riviere et al. [554,555,555], Arnold et al. [40]. We present them in Chapter 7 with V.Dolejsı, including hp- methods. DCGMs eliminate some of the above limitations, butimpose others.

298 5. Nonconforming finite element methods

5.2 Finite element methods for fully nonlinear elliptic problems

5.2.1 Introduction

This section is certainly one of the highlights of this book. We present a FEM forthe general case of fully nonlinear elliptic differential equations of second order inRn, n ≥ 2 and its extension to order 2m and to systems of orders 2 and 2m. To theknowledge of many specialists, consulted by the author, his papers [118, 119] seem tobe the first papers solving these general cases, see below.

The basis is the coercivity of the principal part of the linearized operator. Thisis directly available via Legendre and Legendre–Hadamard conditions. So we requirethese conditions. This does not work for the most general cases of nonlinear systemsin the sense of Agman et al. [3]. Here other methods, e.g. as used for the Navier-Stokes equations, cf. Section 4.6, will have to be developed. We apply our generaldiscretization methods in Chapter 3. The necessary quadrature approximations arestudied in Subsubsection 5.4.4 and 5.4.5.

For these “nonstandard” semiconforming FEMs (based upon conforming FEs andstrong forms of the operator) and (fully) nonconforming FEMs, none of the standardproofs for converegence of FEMs work, see below. So we start giving a short summaryof the main ideas behind the formulation and the proofs for our FEMs. For simplifyingthe presentation we start with one equation of order 2:

� Our FEM for general fully nonlinear elliptic differential equations requires abounded, convex, polyhedron with conforming FEs, or a C2 domain in Rn withnonconforming C1 FEs. Both cases are presented.

� Only FEMs for strong, not for weak, forms make sense. This requires FEs in C1,satisfying Dirichlet boundary conditions on polyhedra and violating them on C2

domains.� The classical theory of discretization methods, cf. Chapter 3, has to be applied to

the differential operator and, on C2 domains, to the differential and the boundaryoperator.

� The most difficult claim, the stability of the nonlinear operator, is proved viathe stability of the linearized operator, with coinciding strong and weak bilinearFE forms. Compact perturbation and a new regularity result for FE solutions ofweak linear FE equations lift the “weak stability” back to the “strong stability”,cf. Theorem 3.29 and Summary 4.52.

� For computations the weak linear forms, coinciding with the strong forms, areused. Thus the many known powerful numerical methods apply for our FEM aswell.

� The mesh independence principle (MIP) guarantees a quadratically convergentsolution process via the Newton method for highly nonlinear systems of equations.

Our FEM is a natural translation of the nonstandard situation for fully nonlin-ear elliptic differential equations, including quasilinear equations not in divergenceform; we abbreviate this as nondivergent quasilinear equations. As for fully nonlinearequations, weak forms are not possible. The solution has to be determined from theoriginal strong form with derivatives up to second order. This situation is directly

5.2. FEMs for fully nonlinear elliptic problems 299

translated into our strong form of a FEM; see Subsection 5.2.2 for more details.This FEM, based upon classical discretization theory, strongly differs from the earlierapproaches.

Difference methods are studied by Crandall and Lions [217] and Crandall et al.[216], by modifying their concept of exact viscosity solutions. Kuo and Trudinger [456]employ discrete maximum principles.

For FEMs, there are only a few papers with results for special equations, e.g.the Monge–Ampere equations in R2 or modifications. Oliker and Prussner [519]prove, by convexity arguments, that their sequences converge monotonically to thesolution. Fulton [330] studies multigrid solutions as models of nonlinear balanceequations in meteorology. Glowinski with Caffarelli, Dean, Guidoboni, Juarez, andPan [154, 272–276, 352] reformulate it in R2 as a problem of calculus of variationsinvolving the biharmonic (or a related) operator. A reinterpretation as a saddle-pointproblem for a well-chosen augmented Lagrangian functional leads to iterative methodssuch as that of Uzawa, Douglas, and Rachford with viscosity solutions. This methodcan be applied to a related problem, the Pucci equation. Other techniques are basedupon operator-splitting methods. Similarly, Feng and Neilan [318] solve the Monge-Ampere equations and prove convergence of Galerkin methods via a regularizationwith εΔ2. This yields a fourth order again quasilinear equation with a solution uε

0

and an error estimate, including a term c(ε) ‖uε0‖Hl hl/

√ε. Finally, [155, 230, 671] are

loosely related to our problem: Cai et al. [155] present a FEM for fully nonlinear,three-dimensional water surface waves, described by standard potential theory. Theirmethod is based on a transformation of the dynamic water volume onto a fixed domain.The transformation brings about an elliptic boundary value problem. This equationis solved by a preconditioned conjugate gradient method. Numerical experimentsindicate convergence of the iterative solver as well as convergence of the entire solutiontowards a reference solution computed by an another scheme. Extensions to parabolicequations are possible and studied, e.g. by Yang [671] and Cui [230].

The other FE approaches are based upon either a combinations of the special prop-erties of a specific problem with the proposed numerical method or upon maximumand/or monotonicity arguments. But none of them seems to be applicable nor hasbeen applied to the general fully nonlinear elliptic differential equations and systemsas treated in this paper. This seems to indicate the need for new ideas.

5.2.2 Main ideas and results for the new FEM: An extended summary

We propose an approach totally different from those in the above papers. As men-tioned, a major part of the analysis for fully nonlinear differential equations is basedupon the strong form of these equations; see, e.g. Gilbarg and Trudinger [346],Showalter [589] and Taylor [618–620]. This motivates a FEM, based upon the strongform of the differential equation. For proving existence, uniqueness, and related resultsfor the exact solution, e.g. monotonicity, maximum principles, and the Schauder fixedpoint theorem are needed. We avoid this, but employ, as for many other numericalapproaches, these results as the basis for our method.

We simplify the presentation by starting with semiconforming and the nonconform-ing FEMs for second order equations; cf. Subsections 5.2.4, 5.2.5 and 5.2.8 for the


general case. We prove convergence in the framework of general discretization theory,based upon consistency and stability, cf. Chapter 3, in a first step fitted to the presentproblem. We need the strong form for the FEM and (see below) a convex, bounded(and we assume additionally) polyhedral Ω or an Ω ∈ C2; both cases are discussedhere. The strong form requires C1 FEs for an L2 evaluation of the differential equation.For a polyhedral Ω, (trivial) boundary conditions are satisfied exactly by the FEs, forcurved domains they violate the boundary conditions. For the first case and the FEsin C1 we need only Theorem 4.36, for the second case Theorems 4.35–4.39. Hence,the first case allows the full order of convergence in Theorem 4.36. For Ω ∈ C2, theproved results in Subsection 4.2.6 yield second order in R2. Conjecture 4.40, probablyproved soon, allows an extension to Rn, those in Conjecture 4.41 higher order as well.So we anticipate Conjecture 4.40 and correspondingly formulate the following results,sometimes hinting at Conjecture 4.41.

The second step is the proof of consistency. For polyhedral Ω and their conformingFEs and a continuous differential operator this is a straightforward consequence of atrivial variational error = 0 and a vanishing interpolation error.

For smooth Ω ∈ C2, requiring nonconforming FEs, consistency is more complicated.We formulate the results on the basis of Conjecture 4.40 and indicate modifica-tions by Conjecture 4.41. The earlier approaches for curved boundaries, cf. e.g.Lenoir [471] and Brenner and Scott [142] for linear problems, do not seem to beapplicable to fully nonlinear problems. So other nonstandard, not surprisingly sim-ilarly complicated considerations are necessary. The clue is an operator with twocomponents, the differential and the boundary operator. We will use this idea forother problems as well, e.g. for nonlinear boundary operators in Section 5.3. Thisrequires a more complicated version of general discretization theory. Specifically,we have to carefully introduce the necessary projectors and spaces for includingthe violated differential operator and the boundary conditions into the classicaldiscretization theory. Then the “classical consistency error” for the differential andthe boundary operator is estimated in (5.65) and Theorem 5.4 below. Essentiallywe split this error into the FE error for a semiconforming FEM on an Ωh

c ≈ Ω andits complementary geometric error, due to ∂Ω �= ∂Ωh

c , according to a new idea ofTiihonen [629].

The third step is the most difficult, the proof of stability. We did not see adirect possibility for the strong form. So we combine linearization with regularityof FE solutions. This is another difference of our FEM to the standard methodsand those in the previous papers. Stability is first proved for the linearized weakform. Unfortunately, we lose the powerful machinery of weak bilinear forms and theirFEMs, unless we find a kind of connection between the two forms. This is achievedby Proposition 5.7 and Lemma 5.10. The first shows coinciding weak and strongFE bilinear forms in our case. The latter states, similarly to the exact solution ofa “smooth” linear elliptic problem, a new regularity and stability result for our FEsolutions. This and the available existence, uniqueness, and regularity results requirea convex, bounded, polyhedral Ω implying satisfied or an Ω ∈ C2, implying violatedboundary conditions for our FEM. We generalize our earlier compact perturbationtechniques, cf. Chapter 3, Theorem 3.29, and Bohmer and Sassmannshausen [128] andBohmer [114,115], to the above satisfied or violated boundary conditions. This allows


lifting the stability from coercive to general elliptic bilinear forms. A combination ofthese results lifts the weak back to the strong stability and allows the convergenceproof.

We do need convex, bounded (polyhedral) or smooth Ω ∈ C2 for existence, con-vergence, and the regularity lemma, Lemma 5.10. The smooth Ω ∈ C2 requires andallows the extension operator in Theorem 4.37. We require FEs in C1, since for FEs inC0 we did not see a possibility to handle the interior variational crimes. The standardinterplay between the weak and strong form is impossible for fully nonlinear problemsand excludes, e.g. discontinuous Galerkin methods. Thus we choose the unusual C1

FEs, conforming for polyhedral and nonconforming for C2 domains.Davydov [264], cf. Subsection 4.2.6, has proved complete approximation results on

bounded, polyhedral Ω in Rn. This allows convergence results similarly to the thestandard conforming FEMs, cf. Theorem 5.2.

For Ω ∈ C2 the situation changes. We need the new “stable splitting” of C1-FEsinto interior and boundary elements. Presently, these results in Subsection 4.2.6are available for modified Argyris piecewise polynomials of degree 5 on polygonalΩh ≈ Ω ⊂ R2. This restricts, for Ω ∈ C2, the order of convergence of our FEM to 2.Davydov is extending his results to Rn and to higher polynomial degrees. Our proofis formulated for including the extensions in Conjecture 4.40, and indicating those inConjecture 4.41.

The FE equations have to be set up. They are solved numerically by a quadraticallyconverging discrete Newton method. Its local quadratic convergence is guaranteedby the MIP; see Section 3.7, Subsection 5.2.10 and Allgower and Bohmer [9, 10]and Allgower et al. [17]. It starts with a good approximation, e.g. obtained by acontinuation method. Again linear problems are essential to solve the highly nonlinearsystems of equations. As a welcome consequence of our coinciding weak and strong FEbilinear forms, the standard solution techniques for weak linear problems are applicableto the fully nonlinear equations as well. However, the standard C0 FEs have to bereplaced by C1 FEs.

With the notation introduced below, let G be a uniformly elliptic operator on Ω,see (5.18), (5.20). Homogeneous Dirichlet conditions are obtained by the standardtechniques. Violated boundary conditions for the nonconforming FEM below will playa dominant role. The exact solution, u0, is determined by the differential equationand the boundary condition in the trace sense, cf. Subsections 5.2.4, 5.2.5. Mindthat G(u0) ∈ L2(Ω) is satisfied for a smooth solution u0 ∈ Us, e.g. Us = H4

+(Ω), cf.Example 3.26 and Proposition 3.27. Smoothness will be required for good convergenceanyway.

u0 ∈ D(G) ∩ Us ⊂ H2(Ω) : G(u0)(x) := Gw(x, u0(x),∇u0(x),∇2u0(x)) = 0 (5.1)

∀ a.e. x ∈ Ω, G(u0) ∈ L2(Ω) ⇐⇒ (G(u0), v)L2(Ω) = 0 ∀v ∈ L2(Ω) = (L2(Ω))′,

and u0|∂Ω = 0 ⇐⇒ (u0|∂Ω, vb)L2(∂Ω) = 0 ∀vb ∈ L2(∂Ω),

with Gw : D(Gw) ⊂ Ω′ × R× Rn × Rn2 → R, Ω, Ωhc ⊂ Ω′ (see (5.16)). We assume

G(uh)(x) := Gw(x, uh(x),∇uh(x),∇2uh(x)) to be defined as well, cf. Proposition 3.27.This condition has to be proved for any specific G, e.g. the Monge-Ampere equations.


a.e. for x, u4(x),∇2u4(x) ∈ D(Gw). Since H3/2(∂Ω) is dense in L2(∂Ω) only testing forall vb ∈ L2(∂Ω) is correct. For a systematic study of the differential and the boundaryoperator, we abbreviate (5.1) into an equivalent form, determining

u0 ∈ Us ∩ D(G) : 0 = F (u0) := (G(u0)|Ω = 0, u0|∂Ω = 0) (5.2)

⇐⇒ (G(u0), v)L2(Ω) + (u0|∂Ω, vb)L2(∂Ω) = 0 ∀v ∈ L2(Ω), vb ∈ L2(∂Ω).

The difference between quasilinear and fully nonlinear problems for a second orderequation is discussed in Subsection 5.2.3. This motivated our new FEM. The standardellipticity condition for nonlinear systems via linear systems of order 2 has to bemodified for equations and systems of higher order. They have to imply the coercivityof the linearized principal part, see Subsections 5.2.3, 2.4.4, 2.6.3, 2.6.5. There weformulated existence, uniqueness and regularity results as a basis for the correspondingFEMs for fully nonlinear equations and systems of orders 2 and 2m in Subsection 5.2.8.

We introduce the C1 FE spaces of local degree d ≥ n2 + 1 in Rn, the S1d

(T h

c

), for

polyhedral, and for C2 domains. For a unified notation we assume that they are definedon a triangulation T h

c of the original polyhedral Ω = Ωhc , or a curved Ωh

c ≈ Ω ∈ C2.In this subsection we use the same notation T h

c ,Ωhc for all cases; cf. Subsections 4.2.6

and 5.2.5 ff. for Sh ⊂ S1d

(T h

c

):

Uh := Sh ⊂ S1d

(T h

c

)and Uh

b := Vh :={uh ∈ Sh : uh|∂Ωh

c= 0}⊂ Uh �= Vh. (5.3)

For convex polyhedral Ω, these FEs uh ∈ U0 ⊂ Sh ⊂ H2(Ω) satisfy the trivial bound-ary conditions exactly. So they define a semiconforming FEM. For Ω ∈ C2 they violatethe boundary conditions, so they yield a nonconforming FEM.

For convex polyhedral Ω, the FE solution, uh0 ∈ Uh

b , is determined by testing onlythe differential equation with respect to vh ∈ Vh, again in the L2 sense:

G :Uhb ∩D(G)→L2(Ω) determine uh

0 ∈Uhb =Vh s.t.

(G(uh

0

), vh)L2(Ω)

=0 ∀vh ∈ Vh. (5.4)

For this semiconforming FEM, uh ∈ Uhb = Vh ⊂ H2(Ω) ∩H1

0 (Ω).For smooth Ω ∈ C2 we have to test both operators. For Ω ∈ C2, comparing uh

0 andu0 requires extension operators, e.g. Ec : H2(Ω) → H2

(Ω ∪ Ωh

c

), cf. Subsection 4.2.6,

Theorem 4.37. A simplified notation is possibel by (realistically) assuming

G in (5.1) defined on D(G)⊂Us ∩ Uc→Vc with Uc := H2(Ω ∪ Ωh

c

),Vc := L2

(Ω ∪ Ωh

c

).

(5.5)

So the FEM, cf. (5.2), defines Fh, and determines, under the Condition H, see below,the FE solution uh

0 ∈ Uh, cf. Theorems 4.39, 5.6, from

uh0 ∈ Uh : 0 = Fh

(uh

0

):=(Gh(uh

0

), uh

0 |∂Ωhc

)= 0 ⇐⇒(

G(uh

0

), vh)L2(T h

c )+(uh

0 |∂Ωhc, vh

b

)L2(∂Ωh

c )= 0

∀vh ∈ Vh, vhb ∈ Vh

b :={uh|∂Ωh

c: uh ∈ Uh

}. (5.6)


Our FEM transforms F (u0) = 0 in (5.2) into Fh(uh

0

)= 0 in (5.6). It is a classical

discretization method in the sense of Chapter 3 and Summary 4.52, cf. e.g. Stetter[596], Stummel [607–609], Vainikko [644], Keller [441], Reinhardt [547], Zeidler [676–678] and Bohmer [113–115]. In particular, Theorems 3.21, 3.25 provide the tools forshowing convergence, if consistency and stability can be proved. Thus Subsections5.2.4–5.2.7 are the core of the section.

The test integrals for nonlinear operators, see (5.6), can be exactly computed onlyfor special cases. So usually they have to be approximated. We extend our results to thenecessary quadrature and cubature approximations, see Subsections 5.4.4 and 5.4.5:We prove the corresponding convergence results for quadrature approximated FEMs.

In the general case, we will update the H1(Ω),H2,H10 and C1 FEs into Hm(Ω,Rq),

H2m,Hm0 and C2m−1 FEs, where 2m indicates the order and q the number of

equations of the operators. Accordingly, (5.2) and (5.6) have to be generalized. Finally,the boundary condition u|∂Ω = 0 has to be replaced by (∂ku)/(∂ν)k|∂Ω = 0, k =0, . . . ,m− 1.

For our unusual FEM, we summarize the main ideas. Here, we formulate theresults for the nonconforming FEM for Ω ∈ C2. The corresponding reduction to thesemiconforming FEM for polyhedral Ω is obvious, cf. Subsection 5.2.4.

1. The FEMs, transforming (5.1) into (5.4), and (5.2) into (5.6), are generaldiscretization methods:� The existence, uniqueness, and regularity results for the original problems in

Subsection 5.2.3 are the basic results for a general discretization method.� The fully nonlinear elliptic problem (5.1), (5.2) is approximated by the strong

form (5.4), (5.6), requiring FEs in C1. For (5.4) identical ansatz and testfunctions, Uh

b = Vh, are possible. For (5.6) we choose as ansatz functionsUh = Sh and test the differential operator with vh ∈ Vh = Sh ∩H1

0

(Ωh

c

)and

the boundary operator with the restrictions to the boundary, vhb ∈ Vh

b =Sh|∂Ωh

c; see (5.3) and Subsection 4.2.6 for the relevant extension results in

Theorem 4.37.� The FEM is a general linear discretization method; cf. Chapter 3 and Subsec-

tions 5.2.4 and 5.2.5 ff.� In Subsections 5.2.5–5.2.7 we reformulate the FEM, introduce the necessary

projectors, and verify the required conditions. Thus the FEMs are convergentif they are stable and consistent; see Theorem 5.2.

� For a “smooth” problem, G, the semiconforming FEMs for a polyhedron andthe nonconforming FEMs for Ω ∈ C2 are consistent; see Theorems 4.54 and5.4.

2. Stability for the nonlinear strong equation (5.4), (5.6), is the most difficultproblem; cf. Subsection 5.2.7:� The nonlinear stability is a consequence of the stability for the linearized strong

problem, again under the conditions (5.4), (5.5), Condition H, see (5.7) andTheorem 5.6. For given uh

0 and f, φ, determine the FE solution uh1 from

uh1 ∈ Uh s.t. (Fh)′

(uh

0

)uh

1 =(G′ (uh

0

)uh

1 , uh1 |∂Ωh

c

)= (f, φ). (5.7)


For this linearized problem, two FEMs complement each other; see below:� The first component of the linearized strong and weak bilinear forms coincide:(

G′ (uh0

)uh, vh

)L2(Ωh

c )=⟨G′ (uh

0

)uh, vh

⟩∀ uh ∈ Uh, vh ∈ Vh, (5.8)

with 〈·, ·〉 = 〈·, ·〉H−10 (Ωh

c )×H10 (Ωh

c ); see Proposition 5.7.� For (extended) (f, φ) ∈ H−1

(Ωh

c

)×H1/2

(∂Ωh

c

), the linearized weak problem

(Fh)′(uh

0

)uh

1 = (f, φ) ⇔⟨G′ (uh

0

)uh

1 − f, vh⟩

+((uh

1 − φ)|∂Ωh

c, vh

b

)L2(∂Ωh

c )= 0

for all vh ∈ Vh, vhb ∈ Vh

b is stable, essentially if u0 is an isolated solution ofF (u0) = 0 with boundedly invertible F ′(u0). This is a consequence of thediscussion at the end of Subsection 5.2.3 and the compactness arguments inTheorem 5.9.

� By the new Lemma 5.10 the regularity results for the exact solution ofa linear elliptic problem are valid for the corresponding FE solutions.In the weak linearized problem, again under the conditions (5.4), (5.5),(Fh)′

(uh

0

)uh

1 = (f, φ) ∈ H−1(Ωh

c

)×H1/2

(∂Ωh

c

), we determine uh

1 ∈ Uh forthe smooth (f, φ) ∈ L2

(Ωh

c

)×H3/2

(∂Ωh

c

); see (5.4), (5.6), (5.8). The stability

of the weak problem implies the stability of the strong problem(G′ (uh

0

)uh

1 − f, vh)L2(Ωh

c )+((uh

1 − φ)|∂Ωh

c, vh

b

)L2(∂Ωh

c )= 0

∀vh ∈ Vh, vhb ∈ Vh

b .

This allows estimates for discrete solutions with respect to a piecewise H2 norminstead of the usual H1 norm for a right-hand side in L2

(T h

c

)×H3/2

(∂Ωh

c

)instead of the usual H−1

(T h

c

)×H1/2

(∂Ωh

c

). This requires smooth coefficients,

necessary for good convergence, and a convex, bounded, polyhedral Ω, orΩ ∈ C2.

3. Summary: Essentially simultaneously with a unique u0, the FE solutions uh0 of

(5.4) and (5.6) uniquely exist, for small pnenough h, and converge, under theconditions (5.4), (5.5), and if G : (D(G) ∩ Uh) ⊂ H2(Ω ∩ Ωh

c ) → L2(Ω ∩ Ωhc ) is

continuous; see (5.9), Condition H, and Theorems 5.2 and 5.13.4. Setting up the FE equations (5.4), (5.6), and solving them; see Subsection 5.2.10:

� Setting up the nonlinear system (5.4), (5.6), is formulated in (5.153).� As a consequence of 1.–2. the MIP is valid. So the Newton method for the

discrete equation (5.4), (5.6), with a good enough initial guess uh1 = Phu1, e.g.

obtained by a continuation method, converges quadratically to uh0 equally well

as the Newton–Kantorovich method, started in u1, for the original problem(5.1); see Theorem 5.21 and (5.10).

� The linearization of G is elliptic. Thus the efficient solvers for linear ellipticproblems in weak form (see (5.7)) can be applied in the continuation andNewton processes, replacing C0 FEs by C1 FEs. Results of Davydov [263] andDavydov and Stevenson [268] for Ωh allow extensions to nested sequences oftriangulations T h

1 � T h2 � · · · , hence even multiresolution techniques.


5. Quadrature approximations for (5.4), (5.6), (5.7), see Subsections 5.4.4, 5.4.5 :� The exact equations (5.4), (5.6), (5.7) usually have to be approximated by good

enough quadrature to maintain convergence, see Theorems 5.33 and 5.40.

We formulate the convergence of the FE and the discrete Newton method: let (5.1),for smooth enough G and Ω, have a smooth isolated solution u0 ∈ H�(Ω) with bounded(F ′(u0))−1 and choose FEs of local degree d in C1 with 0 ≤ �− 1 ≤ d, d ≥ n2 + 1; cf.(4.62). Then the discrete solutions uh

0 of (5.6) converge as (see Theorem 5.13)∥∥Ecu0 − uh0

∥∥h

H2(Ωhc )≤ Chmin{�−2,p}‖u0‖H�(Ω) for large enough �, d. (5.9)

For convex, bounded, polyhedral domains and semiconforming FEMs we only needthe results from Theorem 4.36 in Rn, cf. Subsection 4.2.6, and we get p = �− 2. ForΩ ∈ C2 and the nonconforming FEMs we obtain p = 2 for polyhedral Ωh

c in R2, byTheorem 4.39, in Rn, by Conjecture 4.40 and with p > 2 for future Ωh

c by Conjecture4.41 with appropriate curved boundary approximations. Davydov has fully proved thecase R2, d = 5, p = 2, cf. Subsection 4.2.6, and he is working on higher Rn, p > 2. Theproofs in this paper are based upon these generalizations in Conjecture 4.40.

The Newton method for the discrete problems (5.4), (5.6) has, for (5.6), the form(5.10) and converges locally quadratically; see Theorem 5.21:

uhν+1 ∈ Uh : (Fh)′

(uh

ν

) (uh

ν+1 − uhν

)= −Fh

(uh

ν

)=⇒

∥∥uhν+1 − uh

ν

∥∥h

H2(Ωhc )≤ C

(∥∥uhν − uh

ν−1

∥∥h

H2(Ωhc )

)2

. (5.10)

Finally, we indicate, but do not explicitly formulate, two further extensions. InSubsection 5.2.7 the linearized weak problem turns out to be a compact perturbationof a coercive bilinear form. Hence, the usual Fredholm alternative results apply. Theabove FEM can easily be extended to parameter dependent problems and for studyingbifurcation scenarios. Some general results prove the convergence of the bifurcationcharacteristics of the discrete FEM to those of the original problem, starting withBrezzi et al. [147–149], and, e.g. Bohmer [113–115] and Cliffe et al. [180]. Extensions ofthese results to convergence of the characteristics of center manifolds for the discreteFEM to those for the original problem are, for the first time, studied in Bohmer[115,118], cf. [120].

Remark 5.1. Except for DCGMs and wavelet methods these results remain valid withslight modifications for the other methods as well.

5.2.3 Fully nonlinear and general quasilinear elliptic equations

We recapitulate the standard notation for derivatives, with (−1)j>0 := 1, j = 0, and:= −1 for j ≥ 1,

∂iu =∂u

∂xi, ∇u = (∂iu)n

i=1, ∇2u = (∂i∂ju)ni,j=1, ∂

0u := u. (5.11)


We formulate the conditions for the domain and introduce four Hilbert spaces withW0 for trivial Dirichlet boundary conditions:

Ω is an (open) bounded domain in Rn with Lipschitz-continuous ∂Ω and (5.12)

U := H2(Ω) ⊂ W := H1(Ω) ⊂ V := L2(Ω) = V ′,VD := H3/2(∂Ω),WD := H1/2(∂Ω),

W0 := H10 (Ω) := {v ∈ H1(Ω) : v = 0 on ∂Ω}, U0 :=W0 ∩ U ,W ′ := H−1(Ω),

with the standard Sobolev norms, seminorms, inner products for U ,W,V, andSobolev–Sloboditskii norms for VD,WD; cf. (1.60), Theorem 1.30 [387,518].

For a quasilinear elliptic equation in divergence form a weak semiconforming FEMdetermines the FE solution uh

0 ∈ Vh = Uhb as

⟨Guh

0 , vh⟩W′×W :=

∫Ω

n∑j=0

aj

(·, uh

0 ,∇uh0

)∂jvhdx = 〈f, vh〉W′×W ∀vh ∈ Vh. (5.13)

For a general quasilinear or a fully nonlinear G a weak form is impossible:

u0 ∈ U0 : Gu0 − f =n∑

i,j=0

ai,j(·, u0,∇u0)∂i∂ju0 − f(·) = 0 ∈ V, or (5.14)

u0 ∈ U0 : G(u0) = G(·, u0(·),∇u0(·),∇2u0(·)) = 0 ∈ V. (5.15)

Consequently, a weak form of a FEM is impossible, thus motivating our FEM.The general theory of discretization methods used in this section, cf. Chapter 3, can

be applied if the original problem has two properties: It admits a locally unique solutionu0, strongly related to the stability by the condition of a boundedly invertible G′(u0),and u0 is smooth enough, implying the consistency and good enough convergence ofour method, cf. Condition H. So we refer to the existence and regularity results for fullynonlinear elliptic equations in Subsection 2.5.7. We assume (5.12) for Ω ⊂ Rn, a slightlylarger open Ω′ ⊂ Rn with Ω ⊂ Ω′, Ωh

c ⊂ Ω′, where Ωhc indicates later approximations

for Ω for the uh ∈ Uh. A real valued function Gw is defined such that

Gw : w = (x, z, p, r) ∈ D(Gw) = Ω′ × Γ→ R,Γ ⊂ R× Rn × Rn2,Ω′,Γ open, (5.16)

with r restricted to symmetric real valued n× n matrices. We consider the strongelliptic BVP, cf. (5.1), (5.15), (5.16), often with φ = 0 and

w(x) := wu(x) := (x, u(x),∇u(x),∇2u(x)), u→ G(u) := Gw(wu) for (5.17)

u ∈ D(G) := {u ∈ U : wu(x) ∈ D(Gw) ∀x ∈ Ω a.e., and G(u) ∈ V},the solution satisfies u0 ∈ D(G) : G(u0) = 0 a.e. on Ω, u0 = φ on ∂Ω. (5.18)

Ellipticity is defined by the linear G′(u0) in w0 := wu0 applied to u:

G′(u0)u =∂Gw

∂z(w0)u +

n∑i=1

∂Gw

∂pi(w0)∂iu +

n∑i,j=1

∂Gw

∂rij(w0)∂i∂ju∀u, u0 ∈ D(G).

(5.19)


This will be important below for local uniqueness and stability. The operator G iscalled uniformly elliptic in u0 ∈ D(G) and all of D(G) if its linearization G′(u) isuniformly elliptic for u = u0 and for all u ∈ D(G), respectively. In the first case itsprincipal part, ap(u, v) (see (5.21)), satisfies, for 0 < λ < Λ, and w0 = wu0 in (5.17),

0 < λ|ϑ|2 ≤ −n∑

i,j=1

∂Gw

∂rij(w0)ϑiϑj ≤ Λ|ϑ|2 ∀0 �= ϑ = (ϑ1, . . . , ϑn) ∈ Rn, (5.20)

sometimes only for a.e. x ∈ Ω in w0(x). For fully nonlinear second order ellipticequations for the smooth u, u0 ∈ D(G) this implies the W0-coercivity of the bilinearform induced by the principal part, ap(u, v) of G′(u0). By the standard ϑi = ∂iu(x),integrating over Ω, and using partial integration we obtain, with the equivalence inW0

of the energy norm (∑n

i,j=1 ‖∂iu‖2V)1/2 and ‖ · ‖H1(Ω) and again with 0 < λ1 < Λ1 ∈ R,

λ1‖u‖2H1(Ω) ≤ ap(u, u) :=∫

Ω

n∑i,j=1

∂Gw

∂rij(w0)∂iu∂judx ≤ Λ1‖u‖2H1(Ω) ∀u ∈ W0.

(5.21)

Therefore we choose the Hilbert space setting, W = H1(Ω). A Banach space setting,W 1,p(Ω), might be appropriate if in addition to (5.20) some monotonicity conditionswere considered. Furthermore, since W 2,p(Ω) is continuously embedded into H2(Ω),for p ≥ 2, our new FEM even works for fully nonlinear equations in W 2,p(Ω), e.g.G(u) = (Δu)p + f(u); however it yields only discrete H2(Ω) estimates.

In Example 2.78, some of the most important examples are listed, the Monge–Ampere equation and the equation for a surface with prescribed Gauss curvature, see(2.308), (2.310). The existence of surfaces with prescribed Gauss curvature is known forappropriate given curvature. Specific properties would become more flexibly availablevia our FEMs than by our finite difference methods in Chapter 8, and e.g. Crandall andLions [227] and Barles and Souganidis [73]: our FEM allows C1 FEs on quasiuniformgrids and yields, for the forthcoming better boundary approximations, higher orders ofconvergence than the difference methods in [227], which are only defined on equidistantgrids.

We have previously observed that we need the following conditions H:

Condition H: Essential conditions for well-defined and differentiable G: These veryimportant properties (5.1), (5.4), (5.5) have to be discussed in more detail. Wehave realized that for the Monge-Ampere equation, cf. Example 3.26, the G(u) isnot defined for u ∈ U , but only for u ∈ D(G), a subset of smooth functions u ∈ Us

fitting to the growth properties of G: Smooth solutions u0 are necessary anywaysfor good convergence. For the Monge-Ampere equation Proposition 3.27 shows thatG(uh) ∈ L2(Ω), easily extended to L2(Ω ∪ Ωh

c ). More generally, we require, cf. (5.5)

(H1) Assume G in (5.1) is well defined on

G : D(G) ⊂ Us ⊂ Uc = H2(Ω ∪ Ωhc ) → Vc = L2(Ω ∪ Ωh

c ),

so G(uh) is well defined.


(H2) Below, G′(u0) is important for stability. So we will assume

G : Br(u0) ⊂ D(G) ⊂ Us → L2(Ω ∪ Ωhc ), G ∈ C1(Br(u0))

w.r.t. the ‖ · ‖Us, s.t. ∀u ∈ Br(u0) : G′(u) ∈ L(H2(Ω), L2(Ω)), and can be

extended as ∈ L(Uc,Vc) and ∈ L(Wc,W ′c), e.g. by imposing (5.19)–(5.21),

or in more detail (5.73)–(5.75).

These properties have to be verified for each combination of G,U ,V,Uh,Vh.

Existence, uniqueness and regularity results are summarized in Theorems 2.79, 2.80,2.81, and 2.83. They are obtained by methods, usually independent of linearization.However, preparing the next subsections, we assume additionally smooth D(G),compare Condition H

for F : D(G) ⊂ U → V × VD, in (5.2), with a locally unique solution u0 of

F (u0) = (0, φ), let F ′(u0) ∈ L(U ,V × VD) be boundedly invertible. (5.22)

For systems of q fully nonlinear equations of order 2m of the form (5.111), (5.112)below, nearly nothing seems to be known, for the order 2 some results exist, seeSubsection 2.6.8. Now, a combination of our FEMs below and continuation techniquesallows studying this class of problems.

5.2.4 Existence and convergence for semiconforming FEMs

In Subsection 5.2.2 we have seen that the standard FEM approaches, based on weakforms, do not seem to be possible for the general case of fully nonlinear ellipticequations. We formulated the new FEM for fully nonlinear equations of second orderwith trivial Dirichlet boundary conditions, see (5.4), (5.6) and introduced the necessaryapproximating FE spaces in Subsection 4.2.6.

We apply the general theory of discretization methods, cf. Chapter 3 and Summary4.52, with its standard “consistency and stability implies convergence” result to ourFEMs, here the semiconforming and the nonconforming FEMs in the next subsection.Semiconforming FEMs are based upon strong forms and conforming FEs. We use thenotation in (5.11), (5.16), and (5.17). Let (5.18) have a locally unique solution (seeSubsection 5.2.3) with boundedly invertible derivative G′(u0).

We recall that our program for proving convergence requires several steps, summa-rized in Subsection 5.2.2 1.–4.: Until now we have worked through the first two bulletsof 1., and we turn, in this and the next subsection, to the last three bullets of 1.

The conforming FEs, defined on a triangulation with straight edges, T h, require

Ω convex, bounded and polyhedral with Ω = ∪T∈T h T . (5.23)

In Subsection 5.2.5 we will return to nonconforming FEs, defined on Ω ∈ C2. Thisyields the more complicated case of violated differential and boundary equations.

The original problem and its semiconforming discretization

We need the following spaces, note no Gelfand triple:

U = H2 ⊂ V = V ′ = L2, U0 = U ∩H10 all defined on Ω. (5.24)


G and its exact solution u0 ∈ D(G) ∩ U0 satisfy, cf. Condition H

G : D(G) ⊂ U → V and G(u0) = 0 ∈ V ⇐⇒ (G(u0), v)V = 0 ∀v ∈ V. (5.25)

We choose the discrete Gh for G as in (5.4). The sequences of approximatingsubspaces Uh = Vh = Sh

0 ⊂ H2(Ω) = U in (5.3), defined on the original Ω in (5.23),satisfy the boundary conditions on the exact boundary. So we call this FEM asemiconforming FEM.

More precisely we need

{Uh,Vh, Ph, Qh′, Gh}h∈H , inf{0 < h ∈ H} = 0, S1

d(T h) ⊃ Uh = Vh = Sh0 . (5.26)

For these spaces with identical elements we require (asymptotically) nonequivalentnorms and scalar products, defined as those of the original spaces, cf. (5.24), e.g.

‖uh‖Uh := ‖uh‖U , ‖vh‖Vh := ‖vh‖V . (5.27)

So we might call this method a Petrov–Galerkin method as well. By Theorem 4.36these sequences approximate the Banach spaces U ,V.

For the final FE version of (5.4) we need Gw, wuh(x),D(Gw) in (5.16), (5.18), (5.25):

D(Gh) := {uh ∈ Uh ∩ D(G) : wuh(x) ∈ D(Gw) ∀x a.e. in Ω, G(uh) ∈ V}. (5.28)

So we define the FE solution uh0 ∈ Uh ∩ D(Gh) ⊂ U such that

Gh : D(Gh) → Vh, and Gh(uh

0

)= 0 ⇐⇒

(G(uh

0

), vh)V = 0 ∀vh ∈ Vh ⊂ V. (5.29)

The necessary projectors

Hence, we introduce sequences of projectors, {Ph, Qh′}h∈H , as appropriate tools forour FEM. The Ph have to satisfy, cf. Theorem 4.36,

Ph ∈ L(U ,Uh) : ∀u ∈ U limh→0

‖Phu‖Uh = ‖u‖U , we choose Phu := Ihu. (5.30)

Here we have chosen Ih in (4.63) as in Theorem 4.36, cf. Davydov [264]. Hence,interpolating values of functions or derivatives or other kinds of quasi- or averaging-interpolation or best approximation operators could be used, sometimes calledClement interpolation; cf. e.g. Clement [178], de Boor [270], Scott and Zhang [578].

More delicate than Ph are the projectors to the image spaces, the Qh′ ∈ L(V,Vh′),

motivated by Gh. We test G(u0) = 0 as in (5.25) and Gh(uh

0

)= 0 in (5.29). We keep

the concepts for the original and the discrete problem as close as possible and usethe function spaces and norms in (5.24), (5.26), (5.27). In (5.25) and (5.29) we havesatisfied the differential equation and its discrete counterpart by testing with V andVh. We define,35 cf. (4.73)–(4.75) and Theorem 4.38, for all f ∈ V = V ′,

Qh′ ∈ L(V,Vh′) : 〈Qh′

f − f, vh〉(Vh′∪V′)×Vh = 0 ∀vh ∈ Vh; limh→0

‖Qh′f‖Vh′ = ‖f‖V .

(5.31)

35 Systematically we use the following identification: For U ⊂ V = V ′, for all u ∈ U , for all v ∈ V ⊂U ′, and the corresponding discrete terms, we obtain (u, v)V = 〈u, v〉U×U′ = 〈u, v〉U′×U , similarly for

VD ⊂ Vb = V ′b ⊂ V ′

D, . . . , and W ⊂ V = V ′ ⊂ W ′ below. This is applied for the following projectors

Qh′, . . . as well.


U V V

Vh

= V ′

P h Φh Qh ′

Uh Vh ′ ≠Vh ′

G

Uh tested by

tested by

Figure 5.1 Semiconforming FEMs for fully nonlinear problems.

The last limit is obtained as a consequence of (4.64). By (4.82), (4.83) with Vh �= Vh′

it makes sense that for a function, e.g. f ∈ V or G(uh) �∈ Vh, the Qh′f ∈ Vh′

is not afunction. We did not identify Vh �= Vh′

for emphasizing this situation more clearly.The choice of Ph, Qh′

, for the given Uh,Vh, and problem (5.25) certainly isnot unique. The combination U ,V,Uh,Vh, Ph, Qh′

, Gh should be chosen appropri-ately to yield classical consistency (see below) with the highest possible order;see (4.150).

Projectors and discretization

For the nonlinear operator, G : D(G) ⊂ U → V, in (5.25), the previously introducedprojectors Ph, Qh′

allow a shorthand notation of the corresponding discrete operator,Gh : D(Gh) ⊂ Uh → Vh′

; see (5.29). So we determine the approximate solution uh0 ∈

Uh such that⟨G(uh

0

), vh⟩V×Vh = 0 ∀vh ∈ Vh ⇐⇒ Gh

(uh

0

)= 0 with Gh := Qh′

G|Uh . (5.32)

This brings about a mapping

D(Φh) := {G in (5.25)} and Φh defined by Φh(G) := Gh := Qh′G|Uh . (5.33)

This Φh is even linear in the sense

Φh(c1G1 + c2G2) = c1Φh(G1) + c2Φh(G2) ∀c1, c2 ∈ R, ∀G1, G2 ∈ D(Φh). (5.34)

Linearity is essential for the MIP in Subsection 5.2.10, allowing the proof of aquadratically convergent discrete Newton method.

Our general discretization theory in Chapter 3, in contrast to Stetter [596] andVainikko [644], emphasizes the testing with the V, Vh: summarizing yields Figure 5.1with uniformly bounded Ph, Qh′

.In Summary 4.52 we have recalled discretization methods for FEMs for G(u) = 0.

Essential are consistency, consistency of order k and stability in u0. Our semiconform-ing FEM is consistent for a continuous or Lipschitz-continuous G, by Theorem 4.54.For G ∈ CL, the consistency order is the same as the order of the interpolation error,‖Ihu0 − u0‖H2(Ω), cf. Theorem 4.36, (4.66). It is nearly obvious how to simplify thehighly technical proof for the stability of the nonconforming Fh

(uh

0

)in Subsection

5.2.7. So we refer to this modification yielding stability for the semiconforming Gh(uh

0

)in Theorem 5.12. With Theorem 3.21, we get the following Theorem:

Theorem 5.2. Existence and convergence for the semiconforming FEM for convexpolyhedral domains Ω in Rn: Let the original problem (5.25) be defined on Ω in(5.23), satisfy Condition H and (5.22) and have the exact isolated solution u0 ∈H�(Ω), �− 1 ≤ d with a boundedly invertible G′(u0). Let G be Lipschitz-continuous


in Br(u0) ∩ D(G) as ‖G(u)−G(v)‖ ≤ L‖u− v‖U , recall U = H2,V = L2(Ω). Thenits discretization Gh = Qh′

G|Uh : Uh → Vh′in (5.32) with S1

d(T h) ⊃ Uh = Vh = Sh0

in (5.26) has the following properties:

1. The projectors Ph and Qh′are (equi-)bounded and satisfy (5.30) and (5.31).

2. Gh : Uh → Vh′is defined and continuous in Br(Phu0), r > 0, h-independent.

3. Gh is consistent with G in Phu0, so, by (4.66),

‖GhPhu0 −Qh′Gu0‖Vh′ ≤ L‖u− Ihu‖H2(Ω) ≤ KLh�−2|u|H�(Ω). (5.35)

4. Gh is stable for Phu0.

Therefore the discrete problem Gh(uh) = 0 possesses a unique solution uh0 ∈ Uh near

u0 for all sufficiently small h ∈ H, and uh0 converges to u0, according to∥∥uh

0 − Phu0

∥∥Uh ≤ SKLh�−2|u|H�(Ω). (5.36)

We will come back to these conditions 1.–4. in Theorem 5.2 in Remark 5.3 below.

5.2.5 Definition of nonconforming FEMs

Now we return to nonconforming FEs. We choose a triangulation, T hc , possibly with

curved edges near ∂Ω and require and define, for Ω′ cf. (5.16),

Ω ∈ C2, a triangulation, T hc , with curved edges, s.t. Ωh

c := ∪T∈T hcT (5.37)

and a sequence of open Ωh0 s.t. Ω′ ⊃ Ωh

0 ⊃ Ωhc ∩ Ω and dist

(∂Ωh

0 , ∂Ω)≤ Chp,

with p = 2 for the situation in Conjecture 4.40, and possibly p ≥ 2 for Conjecture4.41. As a consequence of Theorems 4.37 and 4.38 for the U ,W,V = V ′, G,B,G′, . . .extensions from Ω to Ωh

0 are possible. The following estimates then have to bemultiplied by the constant C in Theorems 4.37 and 4.38, C independent of h. Wemaintain the notation U ,W,V = V ′, G,B,G′, . . . defined on Ω, and only refer to Ωh

0 ifnecessary.

Again, the first two paragraphs in Subsection 5.2.4 remain valid. The U ,W,V = V ′,defined on the original Ω, are now approximated by sequences of nonconformingsubspaces Uh,Wh,Vh, defined on the approximate Ωh

c . This implies violated boundaryconditions on the exact boundary; cf. (5.38). The FEM changes into a so-callednonconforming Petrov–Galerkin method, requiring a more complicated treatment thanin Subsection 5.2.4.

The original problem and its nonconforming discretization

We need, cf. (5.24),

U = H2 ⊂ W = H1 ⊂ V = V ′ = L2 ⊂ W ′ = H−1, W0 = H10 all defined on Ω (5.38)

and Uc = H2c ⊂ Wc = H1

c ⊂ Vc = L2c ⊂ W ′

c = H−1c , Wc,0 = H1

c,0 all defined on Ωhc .

We have indicated above that the realization of the differential equation and theboundary condition and their violation is a deciding step of this method. Hence, thediscretization method has to care about both aspects. This is achieved by the newoperator F = (G,BD), with the differential operator and the boundary operator as


its two components; see (5.1), (5.2). The boundary operator, BD, maps, by the TraceTheorem 1.38,

BD : U → VD := H3/2(∂Ω) = BDU , BDu := u|∂Ω, VD ⊂ V ′b = Vb := L2(∂Ω). (5.39)

We consider here mainly trivial Dirichlet conditions. Thus we have to characterize

u|∂Ω = 0 ⇔ 〈v′, u|∂Ω〉V′D×VD

= 0 ∀v′ ∈ V ′D ⇔ (u|∂Ω, v)L2(∂Ω) = 0 ∀v ∈ Vb (5.40)

for u ∈ U , with VD = H3/2(∂Ω) ↪→ Vb = V ′b ↪→ V ′

D, a dense (compact) embedding.The transformation from nontrivial to trivial Dirichlet conditions will be combined

in Theorems 5.4, 5.9, and 5.13 obtaining stability and convergence for the inhomoge-neous case as well.F = (G,BD) is called elliptic simultaneously with G, cf. (5.20). F and its Frechet

derivative F ′(u) map D(G) and U into and onto, respectively, the Cartesian productV × VD:

F := (F1, F2) := (G,BD) : D(G) ⊂ U → VΠ := V × VD ⊂ VΠ := V × Vb = V ′Π; (5.41)

see (5.17). Since VD is dense in Vb, the smooth exact solution u0 is hence characterizedby

F (u0) = 0 ∈ VΠ ⇐⇒ (F (u0), vΠ)VΠ= 0 ∀vΠ := (v, vb) ∈ VΠ (5.42)

⇐⇒ (G(u0), v)V = 0 ∀v ∈ V, and (BDu0, vb)Vb= 0 ∀vb ∈ Vb,

abbreviated ⇐⇒ (G(u0), v)V + (BDu0, vb)Vb= 0 ∀v ∈ V, vb ∈ Vb.

For the standard weak form for F ′(u) in Subsection 5.2.7, we need, cf. (5.39), (5.40)

BD : W →WD := H1/2(∂Ω) = BDW, u→ BDu = u|∂Ω, with (5.43)

u ∈ W and BDu = 0⇐⇒ (u|∂Ω, v)L2(∂Ω) = 0 ∀v ∈ Vb dense in W ′D.

BD : U → VD and BD : W →WD are continuous with respect to the correspondingSobolev– Sloboditskii norms, cf. (1.60), Theorem 1.30. Similarly to (5.38), we introduce

Vc,D := H3/2(∂Ωh

c

)⊂ Wc,D := H1/2

(∂Ωh

c

)⊂ Vc,b := L2

(∂Ωh

c

)and (5.44)

Bc,D : Uc → Vc,D, uc → Bc,Duc := uc|∂Ωhc; similarly, Bc,D : Wc →Wc,D.

To apply the general discretization theory in Chapter 3, we introduce the discreteFh for F , and the corresponding projectors Ph ∈ L(U ,Uh), and Qh′

Π ∈ L(VΠ,Vh′Π ),

Qh′c,Π ∈ L(Vc,Π,Vh′

Π ); see (5.62). This yields the standard “consistency and stabilityimplies convergence” result, (5.37); note Sh

0 = Sh ∩H10 (T h

c ) and the stable splittingin Theorem 4.36:

S1d

(T h

c

)⊃ Uh = Wh = Sh = Sh

0 ⊕ Shb , Vh = Sh

0 , VhD =Wh

D = Vhb := Sh

b |∂Ωhc.

(5.45)


Again these spaces require nonequivalent norms and scalar products. With the normsof the original spaces, now defined on Ωh

c and ∂Ωhc , cf. (5.38), (5.44), let

‖uh‖Uh :=‖uh‖Uc, ‖wh‖Wh := ‖wh‖Wc

, ‖vh‖Vh := ‖vh‖Vc,∥∥uh|∂Ωh

c

∥∥Vh

D

:=∥∥uh|∂Ωh

c

∥∥Vc,D

,∥∥wh|∂Ωhc

∥∥Wh

D

:=∥∥wh|∂Ωh

c

∥∥Wc,D

,∥∥vh|∂Ωh

c

∥∥Vh

b

:=∥∥vh|∂Ωh

c

∥∥Vc,b

. (5.46)

By Theorems 4.36–4.38 these sequences approximate the Banach spaces and subsets,cf. (4.73), EcU , EcW, EcV, EcVD, EcWD, EcVb, EcD(G) defined on Ωh

c and ∂Ωhc with

respect to the corresponding norms. The uh ∈ Sh, and vh ∈ Vhb , violate the original

boundary conditions. Since Ec maintains smoothness properties, we can choose thisEc s.t. (5.5) is maintained to these EcU , EcW, . . . .

The product operator F : D(G) ⊂ U → V × VD and the corresponding Fh : D(G) ∩Uh → Vh′ × Vh

D cause some unpleasant technicalities in the definition of Qh′Π , Qh′

c,Π,compared to Subsection 5.2.4. Subsequently, the consistency theory becomes rela-tively simple again; see Figure 5.2 and Subsection 5.2.6. These technicalities are notsurprising compared to some other approaches for violated boundary conditions, e.g.in Lenoir [471] and Bernadou [85–88]. The unavoidable complications are piled up inSubsection 5.2.7.

The terms uh, G(uh), and uh|∂Ωhc

are, by (5.17), defined on Ωhc and ∂Ωh

c insteadof the original Ω and ∂Ω. Thus we start updating the preliminary FE version (5.6).With Gw,D(Gw), wuh(x) in (5.16), (5.17), (5.18), and Uh in (5.45), let

D(Gh) :={uh ∈ Uh : wuh(x) ∈ D(Gw) ∀x ∈ Ω ∪ Ωh

c , G(uh) := Gw(wuh) ∈ Vc

}.

(5.47)

This condition has to be verified similarly to Proposition 3.27, cf. Condition H. Theextended definition of Gw was the reason for introducing the Ω′ in (5.16).

We replace the differential equation G(u0) = 0 ∈ V, u0 ∈ D(G), in (5.42) by the FEequations in equivalent formulations; see (5.6), (5.41), (5.46): define

the sequence Fh1

(uh

0

)= Gh

(uh

0

)= 0 ⇐⇒

(G(uh

0

), vh)Vc

= 0 ∀vh ∈ Vh ⊂ Vc.

(5.48)

Similarly, the boundary conditions (5.40) are tested (see the equivalence (5.6)) as

uh ∈ Uh : Fh2 (uh) = Bh

Duh = uh|∂Ωhc

= 0 ∈ VhD ⇔

(vh

b , uh)Vh

b

= 0 ∀vhb ∈ Sh

b ,

(5.49)with ∪h∈HSh

b dense in V ′c,D. Consequently we introduce, cf. F in (5.41),

Fh =(Fh

1 , Fh2

)=(Gh, Bh

D

)with D(Fh) := D(Gh). (5.50)

The nonconforming projectors

Again, projectors represent an appropriate tool for Fh =(Fh

1 , Fh2

). They have to

handle the discrepancy between uh, G(uh), and uh|∂Ωhc, by (5.17), defined on Ωh

c

and ∂Ωhc instead of the original Ω and ∂Ω. We combine the extension opera-

tor Ec introduced in Theorem 4.37 with inverse estimates and the estimates for


interpolation errors on Ω in Theorems 4.36 and 4.38. The sequence of Ph has tosatisfy

Ph ∈ L(U ,Uh) : ∀u ∈ U limh→0

‖Phu‖Uh = ‖u‖U , we choose Phu := IhEcu. (5.51)

Ec is unnecessary for the Argyris FEs but possibly needed for other schemes.Again more delicate than Ph are the projectors on the product space, Qh′

Π ∈L(VΠ,Vh′

Π ). They live on the original and discrete image spaces, listed in (5.52)–(5.54),with their duals, scalar products, and pairings; cf. (5.41), (5.42). For the reader’sconvenience, we summarize and introduce the spaces defined on Ω and Ωh

c ; cf. (4.82),(4.83), Theorem 4.36, (5.38), (5.40), (5.45), (5.48), (5.49). We mark functions andfunctionals, according to their definition, by obvious indices. For functions defined onΩh

c we use c, for ∂Ω we use b, for ∂Ωhc we use c,b, for the images of BD on ∂Ωh

c weuse c,D, and on the product of the domain and boundary on Ω and Ωh

c we use Π

or c,Π:

VΠ := V × VD ⊂ VΠ := V × Vb, with U = H2(Ω),V = L2(Ω) = V ′, (5.52)

Vb = L2(∂Ω) = V ′b ⊃ VD = BBU = H3/2(∂Ω), defined on Ω, with(

(u, ub), (v, vb))VΠ

:= (u, v)V + (ub, vb)VD,((u, ub), (v, vb)

)VΠ

:= (u, v)V + (ub, vb)Vb.

Similarly, we define the spaces and scalar products on the curved approximations

Vc = L2(Ωh

c

), Vc,D = BBUc = H3/2

(∂Ωh

c

), Vc,Π,Vc,b, . . . , defined on Ωc.

(5.53)

Correspondingly, the discrete analog spaces and their scalar products are

VhΠ := Vh × Vh

D := Sh0 × Sh

b |∂Ωhc, Vh′ �= Vh, Vh′

Π = Vh′ × VhD := Sh′

0 × Shb |∂Ωh

c�= Vh

Π,

VhΠ := Vh × Vh

b ,⟨(

uh′, uh

b

),(vh, vh

b

)⟩Vh′

Π ×VhΠ

:= 〈uh′, vh〉Vh′×Vh +

(uh

b , vhb

)Vh

D

.

(5.54)

In (5.42), (5.48) and (5.40), (5.49) we have satisfied the differential equation andboundary condition or their discrete counterparts by testing with V and Vb or (seethe equivalence (5.6)) with Vh and Vh

b . The two components (5.42), (5.40) and(5.48), (5.49) of F and Fh, defined on Ω and Ωh

c , respectively, are now combinedwith the extension operator Ec in (4.73) for motivating the appropriate definitionsof the projection operators. For the discrete spaces Vh′ �= Vh and Vh′

b = Vhb the

FE equations are always tested with vh ∈ Vh and vhb ∈ Vh

b , but we use for theboundary condition the equivalence (5.49). Thus corresponding to the product spacesVΠ and Vc,Π we have to introduce two projectors Qh′

Π := (Qh′, Qh′

b ) ∈ L(VΠ,Vh′Π )

and Qh′c,Π := (Qh′

c , Qh′

c,b) ∈ L(Vc,Π,Vh′Π ). We start with the component of the dif-

ferential operator, (5.48) (note V = V ′,Vb = V ′b), and define (see (4.73)–(4.75) and

Theorem 4.38)

Qh′ ∈ L(V,Vh′) for f ∈ V by 〈Qh′

f, vh〉Vh′×Vh − (Ecf, vh)Vc

= 0 ∀vh ∈ Vh. (5.55)


The approximate domains require

Qh′

c ∈ L(Vc,Vh′) for fc ∈ Vc :

⟨Qh′

c fc, vh⟩Vh′×Vh

− (fc, vh)Vc

= 0 ∀vh ∈ Vh.

Both projectors yield the norm limits, the last limit as a consequence of (4.64):

limh→0

‖Qh′f‖Vh′ = ‖f‖V , and lim

h→0

∥∥∥Qh′

c fc

∥∥∥Vh′ = lim

h→0‖fc‖Vc

. (5.56)

Since the Sh0 and Sh′

0 form a pair of dual spaces (see Theorem 4.36), we alternativelyformulate corresponding projectors (see Theorem 4.38) and use them, e.g. in (5.103):

Qh′

d ∈ L(V,Vh), by f ∈ V : Qh′

d f :=m0∑j=1

〈Qh′f, sj〉Vh′×Vhs

j ∈ Vh, again with

limh→0

∥∥∥Qh′

d f∥∥∥Vh

= ‖f‖V ; similarly, e.g. limh→0

∥∥∥Qh′

c,dfc

∥∥∥Vh

= limh→0

‖fc‖Vc. (5.57)

We turn to the projectors for the traces on the boundary. We always have torecall that the ub ∈ VD and uh

b ∈ Vhb ⊂ Vb have the forms ub = u|∂Ω, u ∈ U and

uhb = uh|∂Ωh

c, uh ∈ Vh, respectively. It suffices to test vanishing boundary conditions

with respect to the L2 scalar product (·, ·)Vc,b. This and the dense embeddings VD ⊂ Vb

and Vc,D ⊂ Vc,b motivate the following projectors:

Qh′

b ∈ L(VD,Vh

D

)for ub := u|∂Ω ∈ VD ⊂ Vb :

(Qh′

b ub − (Ecu)|∂Ωhc, vh

b

)Vc,b

= 0,

Qh′

c,b ∈ L(Vc,D,Vh

D

), uc,b := u|∂Ωh

c∈ Vc,D :

(Qh′

c,buc,b − uc,b, vhb

)Vc,b

= 0 ∀vhb ∈ Vh

b ,

(5.58)

and by (5.45) we obtain for all vhb ∈ Vh

b = VhD that vh

b = vh|∂Ωhc

for an appropriatevh ∈ Vh and vh

b = Qh′

c,bvh|∂Ωh

c. The convergence results along ∂Ωh

c in (4.66), (5.6), andTheorems 4.36 and 4.38 imply the limits of the boundary norms

limh→0

∥∥∥Qh′

c,buc,b

∥∥∥Vh

D

= limh→0

‖uc,b‖Vc,D, lim

h→0

∥∥∥Qh′

b ub

∥∥∥Vh

D

= limh→0

‖ub‖VD. (5.59)

We summarize the product projectors and norms in (5.55)–(5.59) obtaining

Qh′

Π :=(Qh′

, Qh′

b

)∈ L(V × VD,Vh′ × Vh

D

)= L(VΠ,Vh′

Π

), (5.60)

Qh′

d,Π :=(Qh′

d , Qh′

b

)∈ L(VΠ,Vh

Π

),with ‖(f, u|∂Ω)‖VΠ = ‖f‖V + ‖u|∂Ω‖VD

, and

limh→0

∥∥∥Qh′

Π (f, u|∂Ω)∥∥∥Vh′

Π

= limh→0

∥∥∥Qh′

d,Π(f, u|∂Ω)∥∥∥Vh

Π

= ‖(f, u|∂Ω)‖VΠ .


The approximate domains yield

Qh′

c,Π :=(Qh′

c , Qh′

c,b

)∈ L(Vc × Vc,D,Vh′

Π

)= L(Vc,Π,Vh′

Π

), (5.61)

limh→0

∥∥∥Qh′

c,Π

(fc, Ecu|∂Ωh

c

)∥∥∥Vh′

Π

= limh→0

∥∥(fc, Ecu∣∣∂Ωh

c

)∥∥Vc,Π

.

Again, the choice of Ph, Qh′Π , Qh′

c,Π, . . . is not unique.

Nonconforming projectors and discretization

We need only a few changes compared to the corresponding semiconforming subsubsec-tion. The operator, spaces, and projectors there, G : D(G) ⊂ U → V, Qh′

, have to bereplaced by the product terms, F = (F1, F2) = (G,BD) : D(F ) ⊂ U → VΠ = V × VD,Qh′

Π , Qh′c,Π. Again we obtain a short definition of the corresponding discrete operator,

Fh : Uh → Vh′ × Vhb , and a linear mapping, Φh; see (5.47), (5.48) and (5.40), (5.49),

(5.52)–(5.54), (5.60), (5.33). We finally get

Fh(uh) =(Gh(uh), uh|∂Ωh

c

)=(Qh′

c G(uh), Qh′

c,buh|∂Ωh

c

). (5.62)

We determine the approximate solution uh0 from

uh0 ∈ Uh s.t. Fh

(uh

0

)= 0 ⇐⇒

(F(uh

0

), vh

Π

)Vc,Π

= 0 ∀vhΠ =

(vh, vh

b

)∈ Vh

Π (5.63)

⇐⇒(G(uh

0

), vh)Vh +

(uh

0 , vhb

)Vh

b

= 0 ∀vh ∈ Sh0 = Vh ⊂ Vc ∀vh

b ∈ Shb = Vh

b .

By (5.62), (5.63) the new FEM is defined for the F = (G,BD) in (5.41), cf. (5.44), as

D(Φh) := {F = (G,BD)} : Φh(F ) := Fh = Qh′

c,ΠF |Uh =(Qh′

c G|Uh , Qh′

c,bBc,D|Uh

).

(5.64)

VΠ = V ×VD

Φh

F h

Uh

U

P hΠQh ′

F tested by

tested byVh

Π′ =Vh ′ ×VhD

VΠ =Vh × Vhb cf. (5.62 ),

VΠ =V × Vb cf. (5.42 ),

Figure 5.2 Nonconforming FEMs for fully nonlinear problems.

We have thus defined a nonconforming Petrov–Galerkin method applicable to everyF in (5.41). With uniformly bounded Ph, Qh′

Π , Qh′c,Π we modify the above Figure 5.1

into Figure 5.2. We reformulate only the not quite trivial consistency:∥∥∥FhPhu−Qh′

Π Fu∥∥∥Vh′

Π

=∥∥∥(Qh′

c G(Phu)−Qh′G(u), Phu

∣∣∂Ωh

c− Ecu

∣∣∂Ωh

c

)∥∥∥Vh′

Π

.

(5.65)

Remark 5.3.

1. The above changes will modify Theorem 5.2. We still obtain convergence, andconsistency of reduced order, stability, on modifies domains.


2. We need for our approach only equicontinuous operators and the limits of thenorms as listed in Definition 3.12 and (3.24). They are satisfied according to(5.51) and (5.55) ff. Stetter [596] requires for his theory linear bounded operatorsPh and Qh′

c,Π :=(Qh′

c , Qh′

c,b

)in (5.51) and (5.60). The convergence properties for

the spaces in (5.45) are not required here, since usually the consistency, henceCondition 3. in Theorem 5.2, is impossible without them.

3. Condition 2. is correct if F or G are continuous in Br(Phu0) or Br(u0).4. The consistency condition 3. will be verified in Theorem 5.4 with reduced order.

Due to the careful preparations this is a relatively simple proof.5. Then the stability condition 4. is the only missing and hard condition in Theorems

5.2 and 5.13. In our approach the stability of the linearized problem and itsregularity are the essential tools for the proof of stability; see Theorem 5.6. Thismakes sense only if the original F is continuously differentiable. Then F isLipschitz-continuous as well.

5.2.6 Consistency for nonconforming FEMs

Verifying the missing conditions analogous to Theorem 5.2 we start with the easier con-sistency. Similarly to the Ciarlet, Raviart, Lenoir, Bernadou and Boisserie approaches(see [85–89, 174, 176, 471]) we again use the extension operator, Ec in Theorem 4.37.It allows estimates of the interpolation errors on Ω, ∂Ω, and Ωh

c and ∂Ωhc , cf. (4.76),

(4.77), and thus is essential for the following proof.

Theorem 5.4. Consistency for nonconforming FEM: We require the conditions forTheorems 4.36, with Ω in Cq, q = 2 and 2 < q ≤ 5 for future curved approximations∂Ωh

c for ∂Ω as in Conjecture 4.41, an exact solution u0 ∈ H�(Ω), � > 2, with ConditionH satisfied, a Lipschitz-continuous Gw in (5.16) with a global constant L,

Gw(·) ∈ CL

((Ωh

c ∪ Ω)× Γ′ ) with (Ωhc ∪ Ω)× Γ′ := {(x, z, p, r) : (5.66)

x ∈ Ωhc ∪ Ω, (z, p, r) =

((Ecu)(x),∇(Ecu)(x),∇2(Ecu)(x)) with u ∈ Br(u0) ∩ D(G)},

and Uh ⊂ S1d

(T h

c

)with d ≥ n2 + 1. Then Fh is consistent with F in Phu0 and

∥∥FhPhu0 −Qh′

Π Fu0

∥∥Vh′

Π=∥∥∥(Qh′

c G(Phu0),−Ecu0|∂Ωhc

)∥∥∥Vh′

Π

(5.67)

≤ CLhmin{�−2,q}‖u0‖H�(Ω).

Remark 5.5. d ≥ n2 + 1 (see (5.3)), and �− 1 ≤ d are related by (4.62). Forinhomogeneous boundary conditions as in (5.39), a modified estimate (5.67) remainsvalid.

Proof. We estimate the first component in (5.67) with (5.55), L in (5.66), G(u0) ≡ 0,and (4.76):

318 5. Nonconforming finite element methods∥∥∥Qh′

c G(Phu0)−Qh′G(u0)

∥∥∥Vh′ (5.68)

=∥∥∥Qh′

c G(Phu0)− 0∥∥∥Vh′

≤∥∥∥Qh′

c G(Phu0)−Qh′

c G(Ecu0)∥∥∥Vh′ +

∥∥∥Qh′

c G(Ecu0)∥∥∥Vh′

≤ CL‖Phu0 − Ecu0‖Uh +∥∥∥Qh′

c G(Ecu0)∥∥∥Vh′ by (4.65)

≤ CLh�−2‖u0‖H�(Ω) +∥∥∥Qh′

c G(Ecu0)∥∥∥Vh′ .

We test Qh′c G(Ecu0)−Qh′

G(u0) with vh ∈ Vh, combine it with (5.16), (5.55), and get∣∣∣∣⟨Qh′

c G(Ecu0), vh⟩Vh′×Vh

∣∣∣∣ (5.69)

=

∣∣∣∣∣∫

Ωhc

[G(Ecu0)(x)

]vh(x)dx

∣∣∣∣∣ ∀vh ∈ Vh

≤∣∣∣∣∣∫

Ωhc

[G(Ecu0)(x)

]vh(x)dx−

∫Ω

[0 ≡ G(u0)(x)

]Ecv

h(x)dx

∣∣∣∣∣≤∫

Ωhc \Ω

∣∣G(Ecu0)(x)vh(x)∣∣dx, since

(∫Ωh

c

−∫

Ω

)∣∣∣Ωh

c ∩Ω= 0,

where G(Ecu0), vh ∈ L2(Ωh

c

)are well defined; cf. (5.5), and Condition H, (5.47). By[

0 ≡ G(u0)(x)]Ecv

h(x), we do not need to explicitly define the extension Ecvh. Now

(5.66) and u0 ∈ H�(Ω), � > 2, imply |G(Ecu0)(x)| ≤ CL‖u0‖H�(Ω). This is combinedwith dist

(Ωh

c ,Ω)≤ C ′hq, cf. Conjecture 4.41, and the bounded S∂Ω := measure of the

surface (∂Ω) <∞ yielding

G(Ecu0)(x) satisfies G(Ecu0) ≡ 0 in Ω =⇒‖G(Ecu0)‖L2(Ωh

c \Ω) (5.70)

≤ CL‖u0‖H�(Ω)‖1‖L2(Ωhc \Ω)

≤ CL‖u0‖H�(Ω)S∂ΩC′hq ≤ C∗hq‖u0‖H�(Ω).

We combine (5.68), (5.69), (5.70) with the Cauchy–Schwarz inequality∥∥Qh′

c G(Phu0)∥∥Vh ≤ CLh�−2‖u0‖H�(Ω) +

∥∥Qh′

c G(Ecu0)∥∥Vh (5.71)

≤ CLh�−2‖u0‖H�(Ω) + C ′′hq‖u0‖H�(Ω)

≤ Chmin{�−2,q}‖u0‖H�(Ω);

thus we have estimated the first component in (5.67).To simplify estimating the second component in (5.67), we assume n < 2(�− 1).

Then the Sobolev embedding Theorem 1.26, implies the compact embedding of H�(Ω)


into C1(Ω). Then u0 ∈ H�(Ω) ∩H10 (Ω) implies C(∂Ω) � u0|∂Ω ≡ 0. Now choose any

P ∈ ∂Ωhc , a ray through P ⊥ ∂Ωh

c , the intersection point, Pb of this ray with ∂Ω, andthe directional derivative ∂PEcu0 of Ecu0 from Pb to P . Then

|Ecu0(P )| =∣∣∣∣∣0 +

∫ P

Pb

∂PEcu0

∣∣∣∣∣ ≤ ‖u0‖H�(Ω)C′hq.

So we obtain for the second component in (5.67)

‖Ecu0‖L2(∂Ωhc ) ≤ C ′′hq‖u0‖H�(Ω).

For inhomogeneous boundary conditions, cf. (5.39), we employ the extension of φ,necessarily here φ ∈ H�−1/2(∂Ω), to φ ∈ H�(Ω); we still have G(u0) = 0, and so (5.68),(5.69) remain correct. Boundary conditions for the exact and the discrete problem are(u0 − φ)|∂Ω ≡ 0 and

(uh

0 − Phφ)|∂Ωh

c≡ 0 with Phφ extended to Ωh

c . We replace theexact boundary conditions by (u0 − Phφ)|∂Ω ≡ 0. Then the error estimates in Theorem4.38 and the stability results for linear elliptic operators with respect to the right-handside (f, φ) ∈ V × VD, cf. Theorem 2.45, Hackbusch [387] Theorem 9.1.16, again yield(5.67). �

5.2.7 Stability for the linearized operator and convergence

Fortunately, there is a standard result strongly simplifying the proof of stability tothat for the derivative of an operator; see Theorem 3.23, and e.g. Stetter [596] Theorem1.2.5, Stummel [607–609] and Keller [441]: we impose the condition of a continuouslydifferentiable Fh in Br(Phu), u ∈ D(G). This slightly enforced condition is usuallysatisfied by the equations considered here. It is even simplified in our case, sinceF ∈ C1(Br(u)) implies for smooth u, Fh ∈ C1(Br−δ(Phu)) with a small δ > 0 for h0 >h ∈ H. We formulate it for our Fh.

Theorem 5.6. From linear to nonlinear stability: Assume Condition H, so thenonlinear discrete Fh, is continuously differentiable in Br(Phu) for u ∈ D(G), andlet the sequence of linearized (Fh)′ : Uh → Vh′

Π be stable at Phu, satisfying∥∥∥((Fh)′(Phu))−1∥∥∥Uh←↩Vh′

Π

≤ S uniformly ∀h ∈ H, h0 > h. (5.72)

Then the nonlinear Fh is stable at Phu as well with stability bound and threshold, 2Sand r0 = r/(2S), respectively.

According to Subsection 5.2.2 3., Theorems 5.2, 5.4, and 5.6 and Remark 5.3show that the only missing claim for convergence is the stability for the linearizedoperator for uh = Phu, u ∈ D(G), here u = u0. So we have to elaborate the program inSubsection 5.2.2 in 3. We indicate only the semiconforming FEM on convex polyhedraldomains in (5.23) and mainly discuss the nonconforming FEM on C2 domains in(4.5). For the conforming FEs it suffices to discuss the differential operator G. Thenthe boundary operators, BD, the extension, Ec, and all the product machinery canbe avoided. This would considerably simplify the proofs, presented here only for thenonconforming FEMs. The interested reader can easily verify this claim. For our F (u)


with F ′(u1)u = (G′(u1)u, u|∂Ω) the linearization of G is essential. Thus a major partof the following discussion is concerned with G′(u1). We obtain the claimed stability,‖((Fh)′(uh

1 )|Uh→Vh′Π

)−1‖Vh′Π ←↩Uh ≤ C, by the end of this subsection.

The linearized operator

We start with G : D(G) ⊂ U → V = L2(Ω), assuming a continuously differentiableoperator G or its defining function Gw; see (5.17), (5.18). Here we sometimes integratethe boundary conditions into the spaces U0,W0 and Uh

b ,Whb . In the literature the same

notation G′(u0) : U → V and G′(u0) :W →W ′ is used for the strong and weak formsof the linearized operators, G′

s(u0) = G′(u0) and G′(u0), respectively. For a smoothenough situation, the strong form is just the restriction of the weak form, and soG′(u0)|U = G′

s(u0) : U → V under the condition (5.73). The proofs in this subsectionfundamentally depend upon a systematic interplay between these strong and weakforms of the linearized operators. To avoid misunderstandings, it is necessary todistinguish at least the strong and weak bilinear forms as as(·, ·) and a(·, ·). Lateron, we even use the notation A = G′(u0), As = G′

s(u0) and Ah, Ahs for the weak and

strong linear operators. Correspondingly, we formulate two different FEMs for thelinearized problem. To its strong form, as(·, ·), we apply the FEM (5.48), (5.78);to its weak form, a(·, ·), the standard FEM; see (5.79) below. We will see that forthe linear case both bilinear forms coincide, and so as(uh, vh) = a(uh, vh) for alluh ∈ Uh =Wh ⊃ Uh

b = Vh � vh; see (5.77) and Proposition 5.7.We called the original nonlinear problem G uniformly elliptic in u0 if its derivative

G′(u0)u in (5.19) satisfies (5.20). In contrast to G, it is possible to transform thelinearization in (5.19) into the standard weak form. We start with the strong operatorG′(u0) : U → V (see (5.12)) and introduce new coefficients aij . Condition This yieldsthe strong bilinear form as(·, ·) and conditions for the partials of Gw; see (5.12), (5.16),(5.17). We already have included the (5.73), (5.74) into Condition H.

G′(u0) : U → V and as(u, v) := (G′(u0)u, v)V ∀ u ∈ U , v ∈ V with (5.73)

G′(u0)u =n∑

i,j=0

(−1)j>0∂j(aij∂

i u) and w0(·) = (·, u0(·),∇u0(·),∇2u0(·)) :

aij = aij(u0) =∂Gw

∂rij(w0) for i, j ≥ 1, a00 = a00(u0) =

∂Gw

∂z(w0),

ai0 = ai0(u0) =∂Gw

∂pi(w0) +

n∑j=1

∂j ∂Gw

∂rij(w0), a0i = 0 for i ≥ 1;

and assume aij = aij(w0(·)) ∈W 1−δ0j ,∞(Ω), so this strong form G′(u0) is

bounded.

Partial integration or Green’s formula now combined with v ∈ W ∩H10 (Ω) yields the

(standard) weak problem or the coinciding bounded weak and strong bilinear form,a(·, ·) : W ×W → R, and the induced weak, G′(u0) :W →W ′ (see (5.12), (5.13)), as


a(u, v) :=∫

Ω

n∑i,j=0

aij∂iu∂jvdx =: 〈G′(u0)u, v〉W′×W ∀v ∈ W0 = W ∩H1

0 (Ω)

∀u ∈ W, a(·, ·), G′(u0) bounded for aij = aij(u0) = aij(w0(·)) ∈ L∞(Ω)

for i, j = 0, . . . , n, and with as(u, v) = a(u, v) ∀u ∈ U , ∀v ∈ W0. (5.74)

The principal part ap(u, v) of a(u, v) is W0 coercive, by (5.20), and, since ‖u‖H10 (Ω)

and ‖u‖W are equivalent norms on W0,

ap(u, u) ≥ λ

∫Ω

n∑i=1

(∂iu)2dx = λ‖u‖2H10 (Ω) ≥ α‖u‖2W ∀u ∈ W0. (5.75)

The original a(u, v), inducing A = G′(u0), is the sum of ap(u, v), inducing B, and itscomplement c(u, v), inducing C, a compact perturbation of A = B + C, with A,B,C ∈L(W,W ′):

a(u, v) = ap(u, v) + c(u, v) = 〈G′(u0)u, v〉W′×W = 〈(B + C)u, v〉W′×W . (5.76)

B is boundedly invertible on W0 by (5.75). Thus G′(u0) satisfies the Fredholmalternative on W0. This compact perturbation C ∈ L(W,W ′) will play a dominantrole for the following stability results.

In addition to the above approximating spaces (see (5.45), (5.46), (5.54)), we needfor W0

Uh =Wh ⊃ Vh and Uhb =Wh

b := Vh = Uh ∩H10

(Ωh

c

)for U0 ⊂ W0, (5.77)

again with the corresponding nonequivalent norms ‖ · ‖Uh , ‖ · ‖Wh , ‖ · ‖Vh .We want to contrast the two different FEMs for the linear problem. We start with

(5.48) and then turn to the standard FEM based upon the weak bilinear form a(·, ·).In both cases we have to determine, cf. (5.73), uh

1 ∈ Uhb and uh

1 ∈ Whb s.t.

as

(uh

1 , vh)

=(G′ (Phu0

)uh

1 , vh)Vc

:=

⎛⎝ n∑i,j=0

(−1)j>0∂j(aij(Phu0)∂i uh

1

), vh

⎞⎠Vc

= (Ecf, vh)Vc

∀vh ∈ Vh, with f ∈ V, aij ∈W 1−δ0j ,∞(Ω). (5.78)

Similarly as Gw we assume the aij(Phu0) to be well defined on the extended Ω ∪ Ωhc .

The standard FEM for the weak bilinear form a(·, ·) with its solution uh1 ∈ Wh

b is

uh1 ∈ Wh

b : a(uh

1 , vh)

=∫

Ωhc

n∑i,j=0

aij(Phu0)∂iuh1∂

jvhdx =⟨G′(Phu0)uh

1 , vh⟩W′

c×Wc

= 〈Ecf, vh〉W′

c×Wcor

= 〈Ecf, vh〉Vc

for f ∈ V∀vh ∈ Whb , with aij ∈ L∞(Ω). (5.79)

As in (5.48) for G, we use here the same notation for the discrete as(uh, vh), a(uh, vh)and the original as(u, v), a(u, v). These bilinear forms are, for small enough h,


simultaneously bounded and elliptic. Equations (5.78), (5.79) can and will be similarlyreinterpreted by using the above projectors and defining new ones.

Weak and strong linear operators coincide on Uh × Vh

The techniques in this and the next subsubsections are strongly related to the standardapproach in spectral methods for collocation (see Bohmer [120] and Canuto et al. [158])and to regularity for difference equations; see Chapter 8 and, e.g. Hackbusch [387],Theorem 9.2.26.

Proposition 5.7. Choose the above Uh, . . . ,⊂ S1d

(T h

c

)in (5.77) for d ≥ 2n + 1 as

approximating spaces for U , . . . , and let aij ∈W 1−δ0j ,∞(Ω). Then for all uh ∈ Uh, vh ∈Vh = Uh

b ,

as(uh, vh) = (G′(Phu0)uh, vh)Vc= a(uh, vh) = 〈G′(Phu0)uh, vh〉W′

c×Wc. (5.80)

Proof. We apply partial integration for every T ∈ T hc with Ωh

c = ∪T∈T hcT . We use the

notation ν = νT for the outer unit normal vector of T and Ba and u for the boundaryoperator, induced by A = G′(u0); see, e.g. [387]:

Ba u =n∑

i,j=1

νjaij∂i u +

n∑j=1

νja0j u, e.g. Ba u = ∂u/∂ν for Asu = −Δu. (5.81)

All ∂T have a positive orientation with respect to T . Hence in (5.83) every interior edgee ∈ T will be obtained twice in opposite directions. Edges e ⊂ ∂Ωh

c will appear once.Let ν = νT be one of the normal vectors for an interior edge e ⊂ Tr. It is oppositelyoriented for neighboring Tr , Tl ∈ T h

c with e ⊂ Tr ∩ Tl and ν the outer normal36 for e ⊂∂Ωh

c . To consider the transition from a triangle Tl to its neighboring Tr, we introducethe restriction of v to Tl, Tr, ∂Ωh

c and the standard notation; see [665]:

vl = v|Tl, vr = v|Tr

, [v] := vl|e − vr|e, and {v} := (vl|e + vr|e)/2, (5.82)

for the corresponding jumps and arithmetic means of v across an interior e, and

[v] := {v} := v|∂Ωhc

along e ∈ ∂Ωhc , and v arbitrary in Rn\Ωh

c .

The next transformation via the n-dimensional Green’s formula is the deciding reasonfor the of C1 FEs. Only for this choice do the following jump contributions,

[Bau

h]

=0, vanish (the [vh] already vanish for C FEs vh):

36 We will use the notation e ∈ ∂Ωhc , although mostly e ⊂ ∂Ωh

c ; similarly, e ∈ T hc \ ∂Ωh

c .


a(uh, vh) = 〈G′(Ecu0)uh, vh〉W′c×Wc

=∑

T∈T hc

(∫T

n∑i,j=0

aij∂i uh∂j vhdx

)

=∑

T∈T hc

(∫T

n∑i,j=0

(−1)j>0∂j(aij∂

i uh)vhdx

)+∫

∂Ωhc

vhl Bau

h ds

+∑

e∈T hc \∂Ωh

c

∫e

({vh}

[Bau

h]+ [vh]{Bau

h})ds =

∫Ωh

c

(G′(Ecu0)s uh)vhdx

= as(uh, vh) since uh ∈ Uh ⊂ C1(Ωh

c

), vh ∈ Vh ⊂ H1

0

(Ωh

c

); hence,

a(uh, vh) = as(uh, vh) ∀uh ∈ Uh, vh ∈ Vh. (5.83)

Discontinuous Galerkin methods, cf. Chapter 7, would certainly be possible fora(uh, vh). But they would violate the necessary condition a(uh, vh) = as(uh, vh) bytheir penalty terms. �

Stability of the weak linear operator

The previous subsubsections play the role of preparing this stability proof. We hadessentially confined the discussion there to the differential equation and indicated theboundary conditions either by the index 0 or using vh ∈ Vh. Instead of the previousstrong and weak forms, As : U → V and A : W →W ′ (see (5.38), (5.41)–(5.43)), weneed their two component versions:

AU := (As, BD) : U → V × VD ⊂ VΠ = V × Vb, AU ∈ L(U ,V × VD) and (5.84)

AW := (A,BD) :W →W ′ ×WD ⊂ W ′Π = W ′ × Vb, AW ∈ L(W,W ′ ×WD).

With the dense (compact) embedding WD ↪→ Vb = L2(∂Ω) we test the W ′ ×WD ⊂W ′

Π by the following WΠ. We emphasize that the stable splitting Sh = Sh0 + Sh

b inTheorem 4.36 plays a dominant role. We choose, cf. (4.65), (5.6), (5.52)–(5.54),

W ×WD ⊂ WΠ :=W ×Vb �=W ′Π =W ′ × Vb, with Vb ⊃ WD and (5.85)

〈(u′, ub), (v, vb)〉W′Π×WΠ

:= 〈u′, v〉W′×W + (ub, vb)Vb, and on

Ωhc the Wc,Π =Wc × Vc,b �= W ′

c,Π with 〈(u′, ub), (v, vb)〉W′c,Π×Wc,Π

.

The discrete counterparts are

WhΠ = Wh × Vh

b := Sh × Shb |∂Ωh

c�= Vh

Π, Wh′

Π :=Wh′ × Vhb := Sh′ × Sh

b |∂Ωhc�= Vh′

Π ,

and⟨(

uh′, uh

b

),(vh, vh

b

)⟩Wh′

Π ×WhΠ

:= 〈uh′, vh〉Wh′×Wh +

(uh

b , vhb

)Vh

b

. (5.86)

The exact and discrete linear boundary conditions remain nearly unchanged, exceptreplacing VD by WD, and so they can be treated as in (5.40), (5.49); see (5.58), (5.59).


AW W ′Π=W ′ ×WDtested by WΠ=W×Vb

P h ΦhW ΠQ h ′

W

W h AhW W h

Π′ =W h ′ ×W hD tested by W h

Π=W h×Vhb

Figure 5.3 Nonconforming FEMs for linear problems.

For the weak differential operator, we introduce a modified Ritz operator. The aboveextension operator Ec in Theorem 4.37 is continuously extended to f ∈ Hk(Ω), k < 0.We define the projectors, Qh′

, Qh′c , and so on, indicating the weak form by the exponent

h′, in contrast to the strong projectors Qh′

and so on with the exponent h in (5.60);cf. (4.64), (4.66), (4.65), (5.6):

Qh′ ∈ L(W ′,Wh′) for f ∈ W ′ by 〈Qh′

f, vh〉Wh′×Wh − 〈(Ecf), vh〉W′c×Wc

= 0, (5.87)

Qh′

c ∈ L(W ′

c,Wh′)

: fc ∈ W ′c :⟨Qh′

c fc, vh⟩Wh′×Wh

− 〈fc, vh〉W′

c×Wc= 0 ∀vh ∈ Wh.

This allows formulating the product projectors, and we obtain

Qh′

Π :=(Qh′

, Qh′

b

)∈ L(W ′

Π,Wh′

Π

), Qh′

c,Π :=(Qh′

c , Qh′

c,b

)∈ L(W ′

c,Π,Wh′

Π

). (5.88)

For both types of projectors we obtain the necessary norm limits:

limh→0

‖Qh′f‖Wh′ = ‖f‖W′ , and lim

h→0

∥∥Qh′

c fc

∥∥Wh′ = lim

h→0‖fc‖W′

c= lim

h→0

∥∥Ecfc

∥∥W′ ,

limh→0

‖Qh′

Π (f, u1|∂Ω)‖Wh′Π

= ‖(f, u1|∂Ω)‖W′Π, and

limh→0

∥∥Qh′

c,Π

(fc, (Ecu)|∂Ωh

c

) ∥∥Wh′

Π= lim

h→0

∥∥(fc, (Ecu)|∂Ωhc

)∥∥W′

c,Π

= limh→0

‖(Ecfc, (Ecu)|∂Ω)‖W′Π. (5.89)

Similarly to the above Fh = Qh′c,ΠF |Uh we use the projectors for reformulating (5.78)

and (5.79), cf. (5.84), for aij ∈ CL

(Ω ∪ Ωh

c

). For the strong AU and Ah

U we get

AhU = Qh′

c,ΠAU |Uh : uh0 ∈ Uh : Ah

Uuh0 = Qh′

c,Π

(Asu

h0 , u

h0 |∂Ωh

c

)= Qh′

Π (f, φ). (5.90)

Similarly, the weak linear operators AW and AhW yield, for (f, φ) ∈ W ′

Π,

AhW = Qh′

c,ΠAW |Wh : uh0 ∈ Wh : Ah

Wuh0 = Qh′

c,Π

(Auh

0 , uh0 |∂Ωh

c

)= Qh′

Π (f, φ). (5.91)

We summarize the new situation with uniformly bounded Ah, Ph, Qh′c,Π in Figure 5.3.

For proving the stability for AhW in Theorem 5.9 assume the principal part induces

the (W0-coercive) operator B; cf. (5.76). We use the notation and assume

AW = (A,BD), BW = (B,BD), the principal part B is W0-coercive, (5.92)

CW = (C,BD), C compact, and AhW , Bh

W , ChW are consistent with AW , BW , CW .


Theorem 5.8. Stability and convergence for the principal part: Assume (5.74), theellipticity (5.75), the approximate spaces in (5.77) with the norm limits in (5.89) and(5.60), and BW and Bh

W as in (5.84), (5.92). Then BW is boundedly invertible, andBh

W is stable. The unique solutions u1 of BWu1 = (f, φ) ∈ W ′Π and uh

1 of BhWuh

1 =Qh′

Π (f, φ) ∈ Wh′Π exist and uh

1 converge such that ‖uh1 − Ecu1‖Wh → 0 for h→ 0.

Proof. Equation (5.74) shows that ahp(uh, vh) is bounded on Wh. The ellipticity

condition (5.75) implies with the usual arguments, cf. (5.75), that

ahp(uh, uh) ≥ λ

∫Ωh

c

n∑i=1

(∂iuh)2dx = λ‖uh‖2H10 (Ωh

c ) ≥ α‖uh‖2Wh ∀uh ∈ Whb , (5.93)

hence the stability of BhW in Wh

b . We reduce the in homogeneous problem

BhWuh

1 = Qh′

Π (f, φ) =(Qh′

f ∈ Wh′, Qh′

b φ ∈ Vh′

b

)∈ Wh′

Π

to Whb : As usual, we determine uh

2 ∈ Wh with(uh

2 −Qh′

b φ)|∂Ωh

c= 0 ∈ Vh

b . Then

BhW(uh

1 − uh2

)=(Qh′

f −Bhuh2 ∈ Wh′

, 0|∂Ωhc∈ Vh

b

)∈ Wh′

Π .

This guarantees, by (5.93) and Theorems 5.2 and 5.4, the stability, consistency, andhence existence of the unique discrete solution uh

1 − uh2 and its convergence to the

corresponding exact solution u1 − u2, and hence of uh1 as well. �

Theorem 5.9. Stability of AhW = (A = G′(u0), BD)h ∈ L

(Wh,Wh′ × Vh

b

): Under

the conditions of Theorem 5.8, assume a boundedly invertible AW in (5.84). Then AhW

in (5.91) is stable.

Proof. This is a generalization to nonconforming FEMs of the Bohmer and Sassman-nshausen results (see [114, 115, 567]) for conforming FEMs. For an arbitrary u ∈ Wchoose vw := CWu ∈ W ′

Π = W ′ ×WD. By assumption, unique exact and discretesolutions, u and uh, exist for the equations

BW u = vw and BhW uh = Qh′

c,ΠBW |Wh uh = Qh′

Π BW u = Qh′

Π vw ∈ Wh′

Π . (5.94)

With the notation

Tw := B−1W ∈ L (W ′

Π,W) and Thw :=

(Bh

W)−1

Qh′

Π ∈ L(W ′

Π,Wh).

Theorem 5.8 implies that

‖Ecu− uh‖Wh =∥∥(EcTw − Th

w

)CWu

∥∥Wh → 0 for h→ 0 and ∀u ∈ W.

Since, by assumption, CW is compact and(EcTw − Th

w

)are equibounded, this implies∥∥(EcTw − Th

w

)CW∥∥W←↩W → 0 for h→ 0. (5.95)

Now let uh ∈ Wh ⊂ Wc. Because AW has been extended to yield

AW : Wc →W ′c,Π = W ′

c ×Wc,D still boundedly invertible with Wh ⊂ Wc, (5.96)


we can estimate

‖uh‖Wh ≤∥∥A−1

W∥∥Wh←↩W′

c,Π‖AWuh‖W′

c,Π=∥∥A−1

W∥∥Wh←↩W′

c,Π‖BW(I + TwCW)uh‖W′

c,Π

≤∥∥A−1

W∥∥Wh←↩W′

c,Π‖BW‖W′

c,Π←↩Wh‖(I + TwCW)uh‖Wh ; hence, (5.97)

‖(I + TwCW)uh‖Wh ≥(∥∥A−1

W∥∥Wh←↩W′

c,Π‖BW‖W′

c,Π←↩Wh

)−1

‖uh‖Wh .

The Qh′Π BWu is defined for every u ∈ W. We apply the stability of Bh

W to wh := BhWuh:

‖uh‖Wh ≤∥∥(Bh

W)−1∥∥Wh←↩Wh′

Π· ‖wh‖Wh′

Π.

Furthermore∥∥(I + ThwCW

)uh∥∥Wh =

∥∥(BhW)−1

BhW(I + Th

wCW)uh∥∥Wh

≤∥∥ (Bh

W)−1 ∥∥

Wh←↩Wh′Π

∥∥BhW(I + Th

wCW)uh∥∥Wh′

Π

implies∥∥BhW(I + Th

wCW)uh∥∥Wh′

Π≥ 1∥∥(Bh

W)−1∥∥

Wh←↩Wh′Π

·∥∥(I + Th

wCW)uh∥∥Wh . (5.98)

We combine (5.94)–(5.98) and use the fact that AW , BW , CW are extended to Wh,such that Qh′

c,ΠAWuh is well defined for estimating∥∥AhWuh

∥∥Wh′

Π

=∥∥∥Qh′

c,ΠAW |Whuh∥∥∥Wh′

Π

=∥∥∥Qh′

c,ΠAWuh∥∥∥Wh′

Π

∀uh ∈ Wh

=∥∥∥Qh′

c,ΠBW(I + TwCW)uh∥∥∥Wh′

Π

(5.94)=∥∥Bh

W(I + TwCW)uh∥∥Wh′

Π

≥∥∥Bh

W(I + Th

wCW)uh∥∥Wh′ −

∥∥BhW(EcTw − Th

w

)CWuh

∥∥Wh′

Π

(5.98)

≥∥∥(I + Th

wCW)uh∥∥Wh /

∥∥∥(BhW)−1∥∥∥Wh←↩Wh′

Π

−∥∥Bh

W(EcTw − Th

w

)CWuh

∥∥Wh′

Π

≥(∥∥(I + TwCW)uh

∥∥Wh −

∥∥(EcTw − Thw

)CWuh

∥∥Wh

)/∥∥∥(Bh

W)−1∥∥∥Wh←↩Wh′

Π

−∥∥Bh

W(EcTw − Th

w

)CW∥∥Wh′

Π ←↩Wh ‖uh‖Wh

(5.97)

≥(

1(∥∥A−1W∥∥Wh←↩W′

c,Π

∥∥BhW∥∥Wh′

Π ←↩Wh

∥∥(BhW)−1

∥∥Wh←↩Wh′

Π

)−O

(∥∥(EcTw − Thw)CW

∥∥Wh′

Π ←↩Wh

))‖uh‖Wh .


The last estimate shows the following: the ‖A−1W ‖Wh←↩W′

c,Πis the deciding factor if

BhW has a moderate condition number. Because of (5.95) and the stability of (Bh

W)h∈H

there exists a positive constant K, independent of h, such that for all h ≤ h0,∥∥AhWuh

∥∥Wh =

∥∥∥Qh′

c,ΠAWuh∥∥∥Wh

≥ K‖uh‖Wh ∀uh ∈ Wh.

This and dimWh <∞ show that Qh′c,ΠAW |Wh is invertible for h ≤ h0. Moreover, we

obtain ‖(Qh′c,ΠAW |Wh)−1‖Wh←↩Wh′ ≤ 1/K; i.e. Ah

W |Wh is stable. �

Regularity for FE solutions

Our next task is lifting the preceding stability result with respect to the weak formsback to the strong form. This is achieved by the regularity for FE solutions in thissubsubsection. Until now, regularity results for the solutions of FEMs seem to beknown only as oscillation results in the sense of De Giorgi, Nash, and Moser; seeAguilera and Caffarelli [4]. They are not applicable to our problem of stability. Herewe need, for the solution of a FEM applied to a differential equation of order 2, resultsanalogous to those for the exact solution. These are estimates of the H2(Ω) instead ofthe usual H1(Ω) norm for a right-hand side in L2(Ω)×H3/2(Ω) instead of the usualH−1(Ω)×H1/2(Ω). To the author’s knowledge, the following type of lemma was notknown for FEMs. It is a transformation to our FEM of Hackbusch’s regularity result[387], Theorem 9.2.26, for difference methods. We employ inverse and interpolationerror estimates (4.57), (4.64), (4.76), valid under the conditions of Theorems 4.43,4.36, and 4.38. The lemma is based upon a regularity result for the exact solution u1

of the corresponding linear problem AWu1 = (f, φ) ∈ V × VD ⊂ VΠ, u1 ∈ W0. Indeed,known regularity results, e.g. in Hackbusch [387], Theorems 9.1.22, 9.1.16, showthat, for a convex bounded or smooth domain and smooth coefficients, the restric-tion AU = AW |U ∈ L(U ,V × VD) is boundedly invertible for a boundedly invertibleAW ∈ L(W,W ′ ×WD); hence we assume

Ω bounded, and either convex, polyhedral or Ω ∈ C2, (5.99)

(AW)−1 ∈ L(W ′ ×WD,W), ai,j ∈W 1−δj0,∞(Ω) ∀0 ≤ i, j ≤ n. (5.100)

Lemma 5.10. Regularity for FE solutions:

1. We assume (5.99), the bounded elliptic AW ∈ L(W,W ′ ×WD) and AU ∈L(U ,V × VD) to be boundedly invertible, e.g. by (5.100). Choose the above FEs,(5.90) and (5.91), of local degree d ≥ 2, on a quasiuniform triangulation; see (5.3),(5.77). Then Ah

W is regular with respect to stability and convergence; hence, thereexist unique solutions u1 for AWu1 = (f, φ), uh

1 , and a C ′ > 0, independent ofh, such that (see (4.64), (4.76))

AhWuh

1 = Qh′

c,Π(f, φ), (f, φ) ∈ V × VD ⇒∥∥uh

1

∥∥Uh ≤ C ′(‖f‖V + ‖φ‖VD

) (5.101)

with∥∥Ecu1 − uh

1

∥∥Uh ≤ C ′(‖Ecu1 − Phu1‖Uh + ‖Ecφ− Phu1‖Vh

D). (5.102)

2. AW ∈ L (W,W ′Π) and AU ∈ L(U , VΠ) simultaneously satisfy the Fredholm

alternative.


U

P h

AU

W

P h

AW

VΠ = V ′Π

QhΠ

W ′Π

QhΠ

′

WhΠ′Uh

AhU Ah

W

Wh VΠh ≠ Vh

Π′

Figure 5.4 Strong and weak FEMs for linear problems.

Remark 5.11. Hence, the FE error in the strong norm ‖Ecu1 − uh1‖Uh is estimated

by the interpolation errors of u1 in Uh and VhD. The FE spaces considered here

need a quasiuniform triangulation guaranteeing inverse estimates in the global form,‖vh‖h

Hj(Ωhc ) ≤ C hl−j‖vh‖h

Hl(Ωhc ); see (4.57). In Subsection 4.2.5 we have summarized

inverse estimates for S1d(T h

c ) on specific degenerate triangulations, based upon theresults of Graham, Hackbusch and Sauter [360–364]. The previous lemma remainsvalid for this case. However, since the smooth FEs in Subsection 4.2.6 still neednondegenerate triangulations, we require nondegenerate triangulations in this Section.Under the conditions of the lemma, (5.101) holds even for noninvertible AW with‖(f, φ)‖VΠ

replaced by ‖(f, φ)‖VΠ+ ‖u1‖W ; see [387].

For Figure 5.4 and the following proof we contrast the strong and weak operatorsAU ∈ L(U ∪ Uc, VΠ) and AW ∈ L (W ∪Wc,W ′

Π), extended as in (5.96); see (5.52)–(5.54), (5.85), (5.86) with the FE versions in (5.60), (5.88). The horizontal arrows inthe first line of Figure 5.4 indicate compact embeddings and in the second line thedecreasingly weaker norms in the spaces Uh = Wh and Vh

Π ⊂ Wh′Π . The −→ between

W, VΠ is missing, since only W −→ V would be appropriate.

Proof. For a boundedly invertible AW ∈ L(W,W ′Π), Theorem 5.9 implies the stability

and thus the bounded invertibility of AhW = Qh′

c,ΠAW |Wh ∈ L(Wh,Wh′Π ). This yields,

with the consistency in Theorem 5.4, the existence of a discrete solution uh1 for

AhWuh

1 = Qh′Π (f, φ) ∈ Wh′

Π by Theorem 5.2. The solution uh1 ∈ Wh for (f, φ) ∈ W ′

Π

satisfies the estimate ‖uh1‖Wh ≤ C‖(f, φ)‖W′

Π. Now we use the same technique as in

the proof of Theorem 5.8 for reducing the inhomogeneous to trivial Dirichlet boundaryconditions. We have assumed φ ∈ VD = H3/2(∂Ω) as a restriction of φ ∈ V = H2(Ω).This is possible by combining Theorems 4.37, 1.37, 1.38, cf. [678], p. 1030. The lastlines of the proof of Theorem 5.4 can then be updated to obtain the (5.101), (5.102)results for φ �≡ 0 if they are correct for φ ≡ 0. So we restrict the discussion for theremaining proof to trivial Dirichlet boundary conditions.

We find for Ph = IhEc, AU , and the vhΠ = (vh, 0) ∈ Vh

Π, cf. (5.96), (5.52)–(5.54),

PhA−1U : Vh

Π ⊂ Vc,Π → Uh or PhA−1U vh

Π is well defined ∀vhΠ = (vh, 0) ∈ Vh

Π.

Vh ⊂ C1(Ωhc ) ⊂ Cγ(Ωh

c ) for a 0 < γ < 1 implies that A−1U vh

Π ∈ C2,γ(Ωhc ) by standard

regularity results, cf. Theorems 2.38, 2.39, and, e.g. [677], Chapter 6.3, Problem 6.8,


p. 259. So the PhA−1U vh

Π is well defined for all vhΠ = (vh, 0) ∈ Vh

Π. Alternatively, wechoose interpolation via the Bramble–Hilbert lemma, Theorem 4.15. Thus PhA−1

U vhΠ

is well defined for derivatives up to second order for Ph in (4.83), e.g. for the ArgyrisFEs. Higher derivatives defining Ph require quasi-interpolants as, e.g. in Theorem4.36, Scott and Zhang [578], and Davydov [264].

For the estimate in (5.107) we need the embedding IWU : Wh → Uh, the boundedtransformation, cf. (5.57), and [264]:

IhV0 : Vh′,0

Π := Vh′ × {0} ⊂ Vh′

Π → V0c,Π := Vc × {0} ⊂ Vc,Π, (vh′

, 0) →(Qh′

d vh′, 0).

(5.103)

We assumed a smooth (f ∈ V, φ ≡ 0) ∈ VΠ. By (5.58) and φ ≡ 0 we restrict thediscussion to uh ∈ Wh ∩ Sh

0 = Uhb =Wh

b = Vh. For the equalities between weak andstrong problems we use special spaces and (5.52)–(5.55), (5.80), (5.85)–(5.89):

∀uh, vh ∈ Uhb = Wh

b = Vh, ∀vhΠ := (vh, 0) ∈ Vh,0

Π := Vh × {0}, Vh′,0Π , Wh,0

Π , Wh′,0Π .

We find for f ∈ V and the definition of A,As in Ω ∪ Ωhc that⟨

AhWuh, vh

Π

⟩Wh′,0

Π ×Wh,0Π

= 〈Auh, vh〉W′c×Wc

= (Asuh, vh)Vc

=⟨Ah

Uuh, vh

Π

⟩Vh′,0

Π ×Vh,0Π

,

(Ecf, vh)Vc

=⟨Qh′,0

Π (Ecf, 0), vhΠ

⟩Wh′,0

Π ×Wh,0Π

=⟨Qh,0

Π (Ecf, 0), vhΠ

⟩Vh′,0

Π ×Vh,0Π

.

(5.104)

This implies coinciding weak and strong bilinear forms, since for all vh ∈ Vh,⟨Ah

Wuh1 , v

hΠ

⟩Wh′,0

Π ×Wh,0Π

=(Asu

h1 , v

h)Vc

=⟨Ah

Uuh1 , v

hΠ

⟩Vh′,0

Π ×Vh,0Π

= (Ecf, vh)Vc

.

(5.105)

Hence, a weak solution uh1 solves the strong problem as well:

AhWuh

1 = Qh′,0Π (f, 0), (f, 0) ∈ V0

Π := V × {0} =⇒ AhUu

h1 = Qh,0

Π (f, 0). (5.106)

For proving stability and consistency for the strong form, we start with the identity(Ah

U)−1

Uhb ←↩Vh′,0

Π=[PhUh

b ←↩Uc,0

(A−1

U)Uc,0←↩V0

c,Π− (IWU )Uh

b ←↩Whb

](5.107)

×(Ah

U)−1

Whb ←↩Wh′,0

Π

[(Ah

UPh −Qh

c,ΠAU)]

Wh′,0Π ←↩Uc,0

(A−1

U)Uc,0←↩V0

c,Π

]IV0

c,Π,Vh′,0Π

.

The inverse and the error estimates (4.57) and (4.64), (4.76), the approximationproperty (4.76), for the local degree d ≥ 2, show the three inequalities

‖IWU‖Uhb ←↩Wh

b≤ C1h

−1, ‖Ph‖Uhb ←↩Uc,0

≤ C2, ‖Phu1 − Ecu1‖Wh ≤ C3h‖u1‖U .(5.108)


With (4.76), the solution of AUu1 = (f, 0) ∈ VΠ, equal weak and strong bilinear forms,ai,j , cf. (5.73), defined in Ω ∪ Ωh

c and the special vhΠ = (vh, 0) ∈ Vh,0

Π , we get by (4.75)∣∣∣⟨AhUP

hu1 −QhΠAUu1, v

hΠ

⟩Wh′,0

Π ×Wh,0Π

∣∣∣= |(AsP

hu1 −Qh′f, vh)Vc

|= |(AsP

hu1 − Ecf, vh)Vc

| = |(AsPhu1 − EcAsu1, v

h)Vc|

=

∣∣∣∣∣∣⎛⎝ n∑

i,j=0

aij(∂i Phu1 − Ec∂iu1), ∂jvh

⎞⎠Vc

∣∣∣∣∣∣+ Ch‖u1‖U‖vh‖Wh

≤ (C3 + C)h‖u1‖U‖vh‖Wh ∀vhΠ = (vh, 0) ∈ Vh,0

Π .

With (5.106), this consistency, and the stability of AhW , the ‖(Ah

W)−1

Whb ←↩Wh′,0

Π

‖ ≤ C4,

we estimate the first term in (5.107):∥∥∥(AhU)−1

Whb ←↩Wh′,0

Π

[(Ah

UPh −Qh

c,ΠAU)]

Wh′,0Π ←↩Uc,0

∥∥∥Wh

b ←↩Uc,0

=∥∥∥(Ah

W)−1

Whb ←↩Wh′,0

Π

[(Ah

UPh −Qh

c,ΠAU)]

Wh′,0Π ←↩Uc,0

∥∥∥Wh

b ←↩Uc,0

≤ C4

∥∥(AhUP

h −QhΠAU

)∥∥Wh′,0

Π ←↩Uc,0

≤ C4 sup0�=u∈Uc,0

sup0�=vh∈Wh

b

∣∣∣⟨AhUP

hu−AUu, vhΠ

⟩Wh′,0

Π ×Wh,0Π

∣∣∣‖vh‖Wh‖u‖U

≤ C3C4h.

With (5.108), (5.107) and a boundedly invertible ‖A−1U ‖Uc,0←↩V0

c,Π≤ C5 and a bound

for ‖IV0c,Π,Vh′,0

Π‖V0

c,Π←↩Vh′,0Π

≤ C6, this yields the stability of AhU : Uh

b → Vh′,0Π , since

∥∥∥(AhU)−1∥∥∥Uh

b ←↩Vh′,0Π

≤[∥∥Ph

∥∥Uh

b ←↩Uc,0

∥∥A−1U∥∥Uc,0←↩V0

c,Π+ ‖(IWU )‖Uh

b ←↩Whb

]∥∥∥(Ah

U)−1

Whb ←↩Wh′,0

Π

[Ah

UPh −Qh

c,ΠAU]Wh′,0

Π →Uc,0

∥∥∥Wh

b ←↩Uc,0

∥∥A−1U∥∥Uc,0←↩V0

c,Π×∥∥IV0

c,Π,Vh′,0Π

∥∥V0

c,Π←↩Vh′,0Π

≤[C2C5 + C1h

−1C3C4hC5

]C6 ≤ C.

The combination of these inequalities yields the claim, since∥∥uh1

∥∥Uh =

∥∥∥(AhU)−1

QhΠ(f, 0)

∥∥∥Uh≤∥∥∥(Ah

U)−1∥∥∥Uh

b ←↩Vh′,0Π

∥∥QhΠ

∥∥Vh′,0

Π ←↩V0Π‖f‖V ≤ C‖f‖V .

The convergence in (5.102) is, by Theorem 5.2, an immediate consequence of thestability in (5.101) and the consistency in Theorem 5.4. �


Stability for the strong linear operator and convergence

Combining Theorem 5.9, Proposition 5.7, and Lemma 5.10 we obtain:

Theorem 5.12. Stability for semiconforming and nonconforming FEMs: For thelinearized operator As = G′(u0) ∈ L(U ,V) in (5.73), assume (5.20), a quasiuniformS1

d(T h), d ≥ 2n + 1, cf. (5.3), and let the corresponding As ∈ L(U0,V), cf., for thedomain (5.100), and, hence, AU := (As, BD) ∈ L(U ,V × VD) be boundedly invertible.Then for a convex polyhedral Ω in Rn and linear Ah

suh1 = Qh′

As|U0uh1 = Qh′

f, f ∈ V,the semiconforming FEM (5.78) is stable. For an Ω ∈ C2, the nonconforming FEM(5.90), Ah

Uuh1 = Qh

c,ΠAUuh1 = Qh

Π(f, φ), (f, φ) ∈ V × VD, is stable as well.

We discussed the relation between the bounded invertibility of As = G′(u0) ∈L(U0,V) and AU = (As, BD) ∈ L(U ,V × VD) and the existence of a locally uniquesolution of G(u0) = 0 and F (u0) = 0 at the end of Subsection 5.2.3.

Proof. The boundedness of the strong AU := (As, BD) ∈ L(U ,V × VD) and its weakcounterpart AW ∈ L(W,W ′ ×WD) under the conditions in (5.100), (5.73), (5.74) isstraightforward. We want to show that the bounded invertibility of AU implies that ofAW ∈ L(W,W ′ ×WD) as well. Indeed, by (5.73)–(5.74), the corresponding G′(u0) ∈L(U ,V ′) induces a strong bounded bilinear form, by Proposition 5.7 simultaneously aW-elliptic bilinear form, and thus finally AW ∈ L(W,W ′ ×WD). Thus the Fredholmalternative applies: if AW ∈ L(W,W ′ ×WD) were not boundedly invertible, each ofthe infinitely many solutions of AWu1 = (f, φ) ∈ V × VD for appropriate (f, φ) wouldbe u1 ∈ U by (5.99) and the well-known regularity results; see, e.g. Hackbusch [387],Chapter 9. So AU could not be boundedly invertible either.

Now we apply Theorem 5.9 to guarantee the stability for the strong form: thebounded invertibility of Qh′

Π AW |Wh ∈ L(Wh,Wh′Π ) and Lemma 5.10 guarantee the

stability of Qh′Π AU |Uh ∈ L(Uh,Vh

Π) and thus the claim. �

Theorems 5.2, 5.4 and 5.6 and the last subsection combined with the last proofshow that we have proved stability and convergence for the FEM in (5.62) if G iscontinuously differentiable near a locally unique smooth solution u0, indicated byA = G′(·) ∈ C(Br,W(u0)) with boundedly invertible F ′(u0) cf. Condition H. This isguaranteed, again in the interplay between the weak and strong form of the derivative,e.g. with Br,W(u0) the ball with respect to ‖u− u0‖W < r, by

aij(w(·)) ∈ C1−δ0j ,∞(Br,W(u0)) and aij(w(·)) ∈ C1−δ0j ,∞(Br,U (u0)), (5.109)

yield C(Br,W(u0)) � A = G′(·) :W →W ′ and C(Br,U (u0)) � As = G′(·) : U → V ′,

where i, j = 0, . . . , n, aij(w(·)) are defined in (5.17). Alternatively, we might requireaij(·) ∈ C1−δ0j (D(Gw)), with w,w0 ∈ D(Gw) as in (5.16).

Theorem 5.13. Existence and convergence for the nonconforming FEM: Let thenonlinear elliptic problem in (5.42) satisfy Condition H. In particular, let G(u0) = 0,for u0 ∈ D(G) ∩ U0, or F (u0) = (G,BD)(u0) = 0, for u0 ∈ Br(u0) ∩ D(G) ⊂ U , havean isolated solution, with boundedly invertible G′(u0) : U0 → V, or F ′(u0) : U → VΠ,let G be continuously differentiable in Br(u0) ∩ D(G), e.g. by (5.109), and let the


domain satisfy (5.99) with Ω ∈ C2. Choose Uh = S1d

(T h

c

), d ≥ 2n + 1 (see (5.3)), on

a quasiuniform triangulation.

1. Then the nonconforming FEM (5.62), so

Fh = Qh′

c,ΠF |Uh : D(Fh) ⊂ Uh ∩ D(G) → Vh′ × Vhb , uh

0 ∈ D(Fh) s.t. Fhuh0 = 0,

is stable and consistent in Phu0, and convergent. It has, for small enough h, aunique solution uh

0 near u0. We obtain, cf. (5.67), for u0 ∈ H�(Ω), � > 2, for thenonconforming FEM, and Cp, 2 ≤ p ≤ 5, domains,∥∥Phu0 − uh

0

∥∥Uh ≤ Chmin{�−2,p}‖u0‖H�(Ω), (5.110)

respectively, with p = 2 for polyhedral Ωh and p > 2 for better curved Ωhc approx-

imations for Ω ∈ C2, cf. Conjectures 4.40 and 4.41.2. F ′(u0) ∈ L(W,W ′ ×WD) and F ′(u0) ∈ L(U ,V × VD) simultaneously satisfy the

Fredholm alternative.3. For quadrature approximations, exactly reproducing polynomials of degree < k

with k > 2(d− 1) > d > 2, the solution uh0 of an approximate Fh

(uh

0

)= 0 satis-

fies the same estimate as in (5.110); see Subsection 5.4.4 and 5.4.5 for m > 1 orfor systems.

Remark 5.14. For the above F (see (5.18), (5.20), (5.73)), � and d are related by(4.62). With the ellipticity assumption (5.20), the previous (5.73), (5.74), (5.100) areimplied by (5.109). Ω convex polyhedral or in C2 are necessary for the regularity andthe extension operators.

5.2.8 Discretization of equations and systems of order 2m

The essential ideas have been presented above for equations of second order. So we onlyhave to generalize our FEM formulated in (5.6) to equations and systems of orders 2and 2m. The basic facts for success are the existence of C2m−1 FEs with local support,combined with an unusual splitting. In Theorems 4.36–4.39 we have formulated andcompletely proved that for m = 1 in R2 on polyhedral domains. An extension tosystems q > 1 is obvious. Extensions to m > 1, n ≥ 2, are the goal for future researchin Davydov’s group. For the time being, we have formulated these results for smoothsplines, Sr

d

(T h

c

), needed for m > 1, for the case n ≥ 2, in Conjecture 4.40. Any smooth

Ω ∈ C2m have to be approximated now by polyhedral Ωh in Rn. For future results thisis extended in Conjecture 4.41 to approximating curved boundaries, Ωh

c ≈ Ω ∈ C2m

in Rn.The obvious difference in changing 1 to m > 1, is the change from the H1(Ω), H2(Ω)

setting to the Hm(Ω,Rq), H2m(Ω,Rq) setting and the updated product spaces. Weagain split F into the differential and a more complicated boundary operator.

The less obvious difference is the fact that for elliptic systems and equations of higherorder, the simple equivalence of ellipticity (2.136) for m = q = 1 and W0-coercivityof the principal part, see Theorem 2.43 2., is no longer correct for the general case


m, q ≥ 1. This has been discussed in Subsections 2.4.4, 2.6.3, 2.6.5, Theorems 2.43,2.89, 2.104. We will repeat the essential conditions for ellipticity for m, q ≥ 1 below,when we introduce the derivative of F in (5.128). Secondly, the conditions for theregularity of the exact solutions of linear systems considerably differ for the differentcases, see the summary in (5.140) ff., and Subsections 2.4.4, 2.6.3, 2.6.5, Theorems2.45, 2.47, 2.91–2.93, 2.95, 2.100, 2.101, 2.108.

Finally we generalize the semiconforming FEMs for convex polyhedral domains fromSubsection 5.2.4 to m = 1, q ≥ 1. In fact, for these domains only for m = 1, q ≥ 1 isthe regularity of the FE solutions correct. Compared to (5.32) only the functions uh

0

have to be replaced, for q ≥ 1, by �uh0 . So we reformulate the semiconforming FEM for

m = 1 < q in (5.149). For the other cases we have to impose specific conditions, inparticular Ω ∈ C2m, cf. (5.140) ff.

These three aspects have to be considered in the following discussion.We present the cases m, q ≥ 1 all in parallel. The differences in this and the

beginning of the next subsection are minimal. A direct comparison of the followingdifferences in, e.g. (5.132) and (5.140) ff. is more informative this way, than it wouldbe for separate formulations.

So we only define the updated differential and boundary operators, the originaland approximating spaces, projectors, FEMs in the different equivalent forms andtheir solutions. The proofs of consistency and stability, now based upon the updatedconditions, require only the minor modifications reported below. So we strongly restrictthe presentation here.

Although we have presented the results for second order equations, we include themhere once more as the case m = q = 1. This allows directly comparing all differentcases.

Fully nonlinear elliptic systems and their discretization

In the following we use the notation �u,�v, . . . , for defining Rq valued functions. Weneed the following operators, extensions, cf. (5.5), and spaces defined on Ω and Ωh

c :

G(�u(·)) := Gw(·, �u,∇�u, . . . ,∇2m�u)(·), G : D(G) ⊆ U ∪ Uc → V ∪ Vc, (5.111)

BD : U → VD or BD : W →WD, BD�u(x) :=(∂j�u(x)∂νj

|∂Ω ∀0 ≤ j ≤ m− 1)

(5.112)

with U = H2m(Ω,Rq) ⊂ W = Hm ⊂ V = L2, W0 := {�u ∈ W : BD�u = 0},U0 :=W0 ∩ U , W ′ = H−m,VD = BDU ⊂ WD := BDW ⊂ Vb = (L2(∂Ω,Rq))m

and similarly, defined on Ωhc : Uc = H2m

(T h

c ,Rq)⊂ Wc ⊂ Vc, Wc,0, Uc,0,

W ′c, Vc,D ⊂ Wc,D ⊂ Vc,b =

(L2(∂Ωh

c ,Rq))m

. (5.113)

As a consequence of the consistent notation for m = q = 1 and m, q ≥ 1, the productoperator F = (G,BD), with the differential and the boundary operators, requires onlyobvious changes: There are essentially two differences: Functions u, . . . , have to bereplaced by �u(x) ∈ Rq, . . . , and BD�u(x) ∈ Rm×q. This remains correct for all otheroperators as well, e.g. the projectors in (5.55), . . . F is called elliptic simultaneously


with G, and maps D(G) ⊂ U onto the Cartesian product VΠ, cf. (5.111), (5.112),

F := (G,BD) : D(G) ⊂ U → VΠ := V × VD ⊂ V × Vb;F (�u0) :=(G(�u0), BD�u0

)= �0,

(5.114)

with a locally unique solution, �u0, and a boundedly invertible F ′(�u0) ∈ L(U ,V × VD).We mainly consider trivial right-hand sides, �0 = (�0,�0), characterizing vanishing

G(�u0) and BD�u0 by testing with elements �v ∈ V and �vb ∈ Vb, respectively.For applying general discretization methods, see Chapter 3, we choose the approx-

imating spaces, Uh,VhΠ, . . . , combine Fh with the corresponding projectors Ph ∈

L(U ,Uh), and Qh′Π ∈ L(VΠ,Vh

Π), Qh′c,Π ∈ L(Vc,Π,Vh′

Π ) to get the sequences{Uh,Vh

Π = Vh × Vhb , P

h, Qh′

Π , Qh′

c,Π, Fh}

h∈H,with inf{0 < h ∈ H} = 0 for lim

h→0.

Theorems 4.36 and 4.38 show that the sequences, defined on Ωhc , see (4.77),

Uh = Wh = S2m−1d

(T h

c ,Rq)

= S2m−1d

(T h

c

)= Sh

0 ⊕ Shb ; (5.115)

Vh = Sh0 = S2m−1

d

(T h

c

)∩Wc,0,Vh

b :=(Sh

b |∂Ωhc

)m, or

VhD := Bh

DShb , d ≥ (2m− 1)2n + 1, h ∈ H,

with the corresponding norms, cf. (5.27), approximate the Banach spacesU ,W,V,Vb defined on Ω and ∂Ω. The �vh ∈ Vh

b violate the original boundaryconditions.

Again, the terms �uh, G(�uh), cf. (5.5), and BD�uh are defined on Ωhc and ∂Ωh

c insteadof the original Ω and ∂Ω, cf. (5.5), motivating the choice our projectors.

Ph ∈ L(U ,Uh) : ∀ �u ∈ U limh→0

‖Ph�u‖Uh = ‖�u‖U , we choose Ph�u := IhEc�u. (5.116)

For the discretization we define analogously new Gh, BhD in (5.117), (5.118). This

will be combined in Theorems 5.15, 5.18 to obtain stability and convergence for theinhomogeneous case as well.

More delicate than Ph are the projectors to the image spaces, e.g. the Qh′Π ∈

L(VΠ,Vh

Π

), motivated by Fh. Thus we start updating the previous FE version by

the FE equations in equivalent formulations, see (5.114): define for �uh0 ∈ Uh the

sequence Gh(�uh

0

)= �0 ⇐⇒

(G(�uh

0

), �vh)Vc

= 0 ∀�vh ∈ Vh ⊂ Vc (5.117)

⇐⇒ Vh ⊥VcGh(�uh

0

)�= G

(�uh

0

)�� Vh with �uh

0 ∈ D(Gh) ⊂ Uc.

Similarly, the boundary conditions (5.114) are tested, see the equivalence (5.6), as

BhD�uh

0 :=(∂j�uh

0

∂νj|∂Ωh

c∀0 ≤ j ≤ m− 1

)= �0 ⇔

(�vh

b , BhD�uh

0

)Vc,b

= 0∀�vhb ∈ Vh

b .

(5.118)

Projectors and discretization

So we update the spaces and norms on Ω and Ωhc , omitting Rq, cf. (5.38), (5.113):


For U ,W,V,U0,VD,Uh,Wh,Vh,Uhb ,Vh

D cf. (5.113)–(5.115), (5.46) for ‖ · ‖Uh ,

VΠ := V × VD ⊂ V × Vb,VD = BBU ,Vb := (L2(∂Ω))m with V = V ′,Vb = V ′ ⊃ VD

Vc,Π := Vc × Vc,D ⊂ Vc × Vc,b,Vc,D = BhDUc,Vc,b :=

(L2(∂Ωh

c

))m,Vc = V ′

c, . . . ,

with((�u, �ub), (�vh, �vb)

)VΠ

:= (�u,�v)V + (�ub, �vb)Vband

((�u, �ub), (�vh, �vb)

)Vc,Π

versus

VhΠ = Vh × Vh

b := Sh0 ×

(Sh

b |∂Ωhc

)m, Vh′

Π = Vh′ × Vhb := Sh′

0 × Vhb �= Vh

Π with

Vh′ �= Vh and⟨(

�uh′, �uh

b

),(�vh, �vh

b

)⟩Vh′

Π ×VhΠ

:= 〈�uh′, �vh〉Vh′×Vh +

(�uh

b , �vhb

)Vh

b

.

(5.119)

The two components (5.117), (5.118) of Fh motivate the appropriate definitions ofthe projectors. Compared to the original form (5.55), . . ., we only replace functionsu, . . . , by �u ∈ Rq, . . . So we do not repeat these formulas, except Qh′

c,b, similarly Qh′

b ,for demonstrating BD�u ∈ Rm×q, e.g. by

Qh′

c,b ∈ L(Vc,b,Vh

b

), �uc,b := Bh

DEc�u or := BhD�uh ∈ Vc,D as

(Qh′

c,b�uc,b, �vhb

)Vc,b

−(�uc,b, �v

hb

)Vc,b

= 0 ∀�vhb ∈ Vh

b , now with Qh′

c,bBhD�uh = Bh

D�uh ∀�uh ∈ Uh (5.120)

and limh→0

∥∥Qh′

c,b�uc,b

∥∥Vh

b

= limh→0

‖�uc,b‖Vc,b, lim

h→0

∥∥Qh′

b �ub

∥∥Vh

b

= limh→0

‖�ub‖Vb. (5.121)

The above changes yield the updated definition of product projectors, see (5.52),e.g.

Qh′

Π :=(Qh′

, Qh′

b

)∈ L

(V × VD,Vh′ × Vh

b

)= L

(V × VD,Vh′

Π

), similarly Qh′

c,Π,

(5.122)

with the corresponding limits of norms. The corresponding discrete operators, Fh :Uh → Vh′ × Vh

b , see (5.48)–(5.50), (5.52), (5.60) define the FE solutions �uh0 ∈ Uh as

�uh0 ∈ D(G) ∩ Uh : Fh

(�uh

0

)= �0 ⇔

⟨Fh(�uh

0

), �vh

Π

⟩Vh′

c,Π×VhΠ

=(Fh

(�uh

0

), �vh

Π

)Vh

c,Π= �0 ∀�vh

Π ∈ VhΠ

⇐⇒(G(�uh

0

), �vh)Vc

+(Bh

D�uh0 , �v

hb

)Vh

b

= 0 ∀�vh ∈ Vh ⊂ Vc ∀�vhb ∈ Vh

b , where

Fh : D(Fh) = D(Gh) ∩ Uh → Vh′ × Vhb with (5.123)

Fh(�uh) =(Gh(�uh), Bh

D�uh)

= Qh′

c,ΠFh(�uh) = Qh′

c,Π

(G(�uh), Bh

D�uh).

With these listed modifications the above Theorem 5.2 and Remark 5.3 remain correct,based upon the following

Condition Hm: Generalized Condition H: In Condition H replace H2(Ω ∪ Ωhc )

by H2m(Ω ∪ Ωhε ,R

2q) and (5.19)–(5.21), (5.73)–(5.75) by (5.128)–(5.132), (5.140)–(5.148), everything else remains unchanged.


5.2.9 Consistency, stability and convergence for m,q ≥ 1

By Theorem 3.21, we have to prove consistency and stability, by Theorem 5.6, onlyfor the linearized operator. The approach and the proofs are nearly identical withm = q = 1 up to the above minor modifications. So we reformulate the theoremsand indicate the changes in the proofs. They are mainly caused by the modifiedconditions, guaranteeing the coercivity of and regularity results for the linearizedoperator. Wherever appropriate we refer to the conforming FEM for m = 1, q ≥ 1,cf. Theorems 5.2 and 2.94, (2.358).

Theorem 5.15. Consistency for semiconforming and nonconforming FEs: Werequire the conditions in Theorem 4.36, and Condition Hm, in particular an exactsmooth solution �u0 ∈ H�(Ω,Rq), � > 2m, a Lipschitz-continuous Gw in (5.16) with aglobal L,

Gw(·) ∈ CL(D(Gw) ∩ (Ω′ ×Br,U (�u0))) and d ≥ (2m− 1)n2 + 1. (5.124)

1. For m = 1, q ≥ 1 and convex, polyhedral Ω, the Gh is consistent with G in Ph�u0

and

‖GhPh�u0 −Qh′G�u0‖Vh′ ≤ KLh�−2‖�u0‖H�(Ω,Rq). (5.125)

2. For m, q ≥ 1 and Ω ∈ C2m, the Fh is consistent with F in Ph�u0 and, with p = 2for the polyhedral Ωh and p > 2 for curved Ωh

c , cf. Conjectures 4.40, 4.41,∥∥∥FhPh�u0 −Qh′

Π F�u0

∥∥∥Vh′

Π

≤ CLhmin{�−2m,p}‖�u0‖H�(Ω,Rq). (5.126)

Proof. Replace h�−2, . . . , by h�−2m, . . . , in the proof of Theorem 5.4. �

The linearized operator

We assume a continuously differentiable Gw. Again we sometimes integrate theboundary conditions into the spaces U0,W0 and Uh

b ,Whb , see (5.113). We distinguish

the strong and weak coinciding bilinear forms as(·, ·) and a(·, ·), see Proposition5.16. We formulate two different FEMs for the linearized problem applying thenew FEM (5.127), to its strong form and the standard FEM (5.128) to the weakform.Equations and systems of order 2m with 1 ≤ m, q : We determine the solutions �u0

(smooth) and �uh0 by

F (�u0) = (G(�u0)(·) = Gw(·, �u0(·), . . . ,∇2m�u0(·)), BD�u0) = 0 ∈ V × VD, and

Fh(�uh

0

):=(Gh(�uh

0

), Bh

D�uh0

)= 0 (5.127)

⇐⇒(G(�uh

0

), �vh)L2(T h

c ,Rq)+(Bh

D�uh0 , �v

hb

)Vh

b

= 0 ∀�vhΠ =

(�vh, �vh

b

)∈ Vh

Π.


The stability uses the linearization (Fh)′(�uh0 )�uh = (G′(Ec�u0)�uh, Bh

D�uh), starting withG′(Ec�u0) and �wh

0 := �wh�uh

0and Ah

αβ = Ahαβ(�wh

0 ), cf. (5.17),

G′(Ec�u0)�uh =∑

|γ|≤2m

(∂ϑγGw(·, �ϑβ , |β| ≤ 2m)

(�wh

0

) )∂γ�uh (5.128)

=:∑

|α|,|β|≤m

(−1)|α|∂α(Ah

αβ∂β�uh), ah

s (�uh, �vh) := (G′(Ec�u0)�uh, �vh)Vcand

ah(�uh, �vh) :=∫

Ωhc

∑|α|,|β|≤m

(Ah

αβ∂β�uh, ∂α�v

)qdx = 〈G′(Ec�u0)�uh, �vh〉W′

c×Wc

∀�uh ∈ Uh, �vh ∈ Vh with Aαβ ∈W |α|,∞ and Aαβ ∈ L∞ (T hc ,Rq×q

).

The imposed conditions Aαβ ∈W |α|,∞ and Aαβ ∈ L∞(T hc ,Rq×q) imply well-defined

continuous bilinear forms as(·, ·) and ah(·, ·), respectively Again we split a(u, v),induced by A = G′(u0), into the sum of its principal part, ap(u, v), inducing B, andits complement c(u, v), inducing C, a compact perturbation as A,B,C ∈ L(W,W ′),

a(u, v) = ap(u, v) + c(u, v) = 〈G′(u0)u, v〉W′×W = 〈(B + C)u, v〉W′×W . (5.129)

Here we come back to an essential difference to the case m = q = 1, with equivalentellipticity and coercivity of the principal part. For m, q ≥ 1, we impose

∀x ∈ Ω, �Θ ∈ Rn×q :∑

|α|=|β|=m

(Aαβ(x)�ϑβ)�ϑα ≥ λ|�Θ|2m, or (5.130)

∀ϑ ∈ Rn, η ∈ Cq : ηTAp(x, ϑ)η = ηT∑

|α|=|β|=m

Aαβ(x)ϑβϑαη ≥ λ|ϑ|2m|η|2. (5.131)

Notice that (5.130), (5.131) reduce to (2.136), for m = q = 1. Now we require

Aαβ ∈ L∞(Ω,Rq×q) for |α|, |β| ≤ m with Aαβ = Aij for m = 1,

and Aαβ = aαβ for q = 1,with Aαβ ∈ C(Ω) for |α| = |β| = m > 1, (5.132)

the strong Legendre condition, (5.130), for m ≥ 1, q = 1 and m = 1, q ≥ 1, and thestrong Legendre–Hadamard condition, (5.131), for m > 1, q > 1.

Then the principal part ap(�u,�v) is W0-coercive and B is boundedly invertible onW0. This is a consequence of (2.22), Theorems 2.43, 2.89, 2.104. Thus G′(u0) satisfiesthe Fredholm alternative on W0.

In addition to (5.115), we need for W0

Uh =Wh ⊃ Vh and Uhb = Wh

b := Sh0 = Vh = Uh ∩Hm

0

(T h

c ,Rq). (5.133)

We contrast the two different FEMs for the strong as(·, ·) and the weak bilinearform a(·, ·) and its standard FEM. We determine �uh

1 ∈ Uhb and �uh

1 ∈ Whb such that

�uh1 ∈ Uh

b : ahs

(�uh

1 , �vh)

=(G′(Ec�u0)�uh

1 , �vh)Vc

= (Ec�f,�vh)Vc

∀�vh ∈ Vh, �f ∈ V, (5.134)

�uh1 ∈ Wh

b : ah(�uh

1 , �vh)

=⟨G′(Ec�u0)�uh

1 , �vh⟩

= 〈Ec�f,�vh〉 ∀�vh ∈ Wh

b ,�f ∈ W ′, (5.135)


with 〈·, ·〉 = 〈·, ·〉W′c×Wc

. Equations (5.78), (5.79) can and will be similarly re-

interpreted by replacing the above projectors, Qh′Π , . . . , by the previous new ones

Q′hΠ , . . . , cf. (5.120).

Coinciding strong and weak bilinear form and stability of the weak form

Proposition 5.16. For Uh,Vh in (5.119) and Aα,β ∈W |α|,∞(T hc ,Rq×q) we obtain

∀�uh ∈ Uh, �vh ∈ Vh = Sh0 = Uh

b

ahs (�uh, �vh) = (G′(Ec�u0)�uh, �vh)Vc

= ah(�uh, �vh) = 〈G′(Ec�u0)�uh, �vh〉W′c×Wc

. (5.136)

Proof. This is generalized by induction of the proof for Proposition 5.7. �

Remark 5.17. The technique of the proof for Proposition 5.7, can be applied toquasilinear equations and systems as well. This yields instead of (5.106), and ∀ �uh ∈Uh, �vh ∈ Vh ⊂ W0,

(G(�uh), �vh)Vc=∫

Ωhc

∑|α|≤m

((−1)|α|∂α(Aα(·, �uh, . . . ,∇m�uh))

)�vhdx

= 〈G(�uh), �vh〉W′c×Wc

:=∫

Ωhc

∑|α|≤1

Aα(·, �uh, . . . ,∇m�uh)∂α�vhdx. (5.137)

From the differential operators, we return to the two-component version of ouroperators

AU := (As, BD) : U → VΠ = V × VD ⊂ V × Vb, AU ∈ L(U ,V × VD) and (5.138)

AW := (A,BD) :W →W ′Π = W ′ ×WD ⊂ W ′ × Vb, AW ∈ L(W,W ′ ×WD),

where the “weak spaces” W, . . . and the following projectors represent obvious gener-alization of (5.85), (5.87), (5.88) to our case. Similarly to the above Fh = Qh′

c,ΠF |Uh

we reformulate (5.134) and (5.135) for the strong and weak linear AU and AW anddetermine, for (�f, �φ) ∈ W ′

Π, the solution �uh1 from

AhU = Qh′

c,ΠAU |Uh : �uh1 ∈ Uh : Ah

U�uh1 = Qh′

c,Π

(Ah

s�uh1 , B

hD�uh

1

)= Qh′

Π (�f, �φ),

AhW = Q

′hc,ΠAW |Wh : �uh

1 ∈ Wh : AhW�uh

1 = Q′hc,Π

(A�uh

1 , BhD�uh

1

)= Q

′hΠ (�f, �φ) ⇐⇒⟨

Q′hc,ΠAW�uh

1 −Q′hΠ (�f, �φ), �vh

Π

⟩Wh′

Π ×WhΠ

= 0∀�vhΠ ∈ Wh

Π, Aα,β ∈ CL

(T h

c ∪ Ωhc

),

see (5.138). Figure 5.2 remains unchanged.For the bounded and equibounded AW = (A,BD), BW = (B,BD), CW = (C,BD),

and AhW , Bh

W , ChW ∈ L(Wh,Wh′ × Vh

b ) we use the notation in (5.79). Theorems 5.8and 5.9, stating stability of the weak form and convergence for the principal part and“weak” stability for an invertible AW , remain correct if the ellipticity is replaced by


(5.132) and the approximation spaces in (5.77) by (5.133). Then both proofs remainunchanged except replacing functions, e.g. u by �u.

Regularity for FE solutions

For lifting the preceding stability result with respect to the weak forms back to thestrong form, we need modified regularity results for the exact solutions of linearequations. These are estimates of the H2m(T h

c ,Rq) instead of the usual Hm normfor the right-hand side in L2 ×H2m−1/2 × · · · ×Hm+1/2(T h

c ,Rq) instead of the usualH−m ×Hm−1/2 × · · · ×H1/2(T h

c ,Rq). The basic assumptions for all cases are thebounded invertibility of AW and the coercivity of its principal part, by the conditionsin (5.132). So we require for A:

A strongly elliptic, W0-coercive principal part, (AW)−1 ∈ L(W ′ ×WD,W0). (5.139)

For the different cases m ≥ 1, q ≥ 1 we impose, cf. Theorems 2.45, 2.46, 2.47, 2.91–2.93,2.94, 2.108

For m = q = 1 letΩ ∈ C2, ai,j ∈W 1−δj0,∞ (T hc

), 0 ≤ i, j ≤ n, (f, φ) ∈ V × VD

(5.140)

m > q = 1 let Ω ∈ C2m,∀α, β with |α|, |β| ≤ m,∂γaα,β ∈ L∞ (T hc

)∀γ with

|γ| ≤ |β|, else aα,β ∈ L∞ (T hc

), (f, φ) ∈ V × VD, (5.141)

m = 1 < q let Ω ∈ C2, Akl ∈W 1−δ0k,∞ (T hc

), �f ∈ V ∀k, l, or (5.142)

m = 1 < q let n = 2, 3, Ω be convex, polyhedral, and satisfy Theorem 2.94,

and Akl satisfy (2.358). (5.143)

For systems of order 2m, we need the modified strong Legendre condition, cf.(2.401),

∃λ > 0 : ∀x ∈ Ω,∀ϑ ∈ Rn :∑

|α|=|β|=m

Aαβϑβϑα ≥ λ|ϑ|2mI, (5.144)

with the identity matrix I.

For m, q > 1, assume (5.144), Ω ∈ C∞, Aα,β ∈ C∞ (T hc

)∀α, β, �f ∈ L2

(T h

c

).

(5.145)

These analytic regularity results show that (5.139)–(5.145) imply

AW�u1 = (�f, �φ) ∈ V × VD ⇒ ‖�u1‖U ≤ C(‖�f‖V + ‖�φ‖VD

)= C‖(�f, �φ)‖VΠ . (5.146)

With these modifications Lemma 5.10 in particular (5.101), (5.102) hold. In the prooffor our general case, we only have to replace h, h−1 in (5.108) by hm, h−m.

Stability for the strong linear operator and convergence

We are nearly done. Collecting the results from the last two subsubsections, we getthe “strong” stability of the linear operator (only formulated analogously) and thenfinally the existence and convergence of discrete solutions for fully nonlinear problems.


After all these preparations we are ready to prove the stability of the linearizedstrong operator and thus the convergence of the method. We had to impose manydifferent conditions for the different areas of ellipticity and W0-coercivity, approxima-tion, coinciding weak and strong bilinear forms and regularity. We recall the notationAαβ = aαβ for m > 1, q = 1, Aαβ = Aij for m = 1 < q, and Aαβ = aij , i, j = 0, . . . , nfor m = 1 = q. For the operator and the domain, we imposed (5.130), the strongLegendre condition for m ≥ 1, q = 1 and m = 1, q ≥ 1, and (5.131) and (5.144), thestrong Legendre–Hadamard and the modified Legendre condition for regularity, form > 1, q > 1, summarized as

∀m, q ≥ 1 : U , Uh = S2m−1d

(T h

c

).., in (5.113), (5.115), d ≥ (2m− 1)2n + 1 (5.147)

and (5.130),Ω ∈ C2m, Aαβ ∈W |α|,∞ (T hc ,Rq×q

)∀|α|, |β| ≤ m, except m, q > 1,

equations : m = q = 1 : aij ∈W 1−δi,0,∞(Ω), ∀0 ≤ i, j ≤ n,

systems : m = 1 < q : Aij ∈W 1−δi,0,∞(Ω,Rq×q),∀0 ≤ i, j ≤ n,

equations : m > q = 1 :,Ω ∈ C2m, aαβ ∈ C(Ω)∀|α|, |β| = m,

systems : m, q > 1 : Ω ∈ C∞, (5.131), (5.144), Aαβ ∈ C∞(Ω,Rq×q)∀|α| = |β| ≤ m,

finally let m = 1 < q, n = 2, 3, Ω be convex, polyhedral,

and satisfy Theorem 2.94, and let Akl satisfy (2.358).

The consistency in Theorem 5.4 needed an exact solution �u0 ∈ H�(Ω,Rq), � > 2m,and a Lipschitz-continuous Gw in (5.16) with a global constant L, see (5.66).

With the notation in (5.73), (5.147), we require continuity with respect tothe variable �u, via �w�u, see (5.17), (5.111), for Theorem 5.13 below, suchthat

Aαβ(�w�u) ∈ C|α|,∞(W ∩Br(�u0)) and Aαβ(�w�u) ∈ C|α|,∞(U ∩Br(�u0)), (5.148)

∀|α|, |β| ≤ m, yield G′(u) :W →W ′, and G′(u) : U → V ′ ∈ C(Br(�u0)).

Theorem 5.18. Discrete solutions uniquely exist and converge for m, q ≥ 1: Let thenonlinear elliptic problem in (5.114), F (�u0) = 0 with F = (G,BD) : U → VΠ = V ×VD, equivalently G(�u0) = 0 with G : U0 → V, have an isolated solution �u0 ∈ D(G) ∩ U0.Let G′ and F ′ satisfy (5.132), (5.139)–(5.145) (5.147)–(5.148), let G′(�u0) : U0 → V,hence F ′(�u0) : U → VΠ be boundedly invertible and G, hence F as well, be continuouslydifferentiable in Br(�u0) ∩ D(G), satisfying Condition Hm. In particular, define theFEM for m = 1, q ≥ 1 and convex, polyhedral Ω ∈ Rn, with n = 2, 3 for q > 1, as, cf.(5.123),

�uh0 ∈ D(Gh) ⊂ Uh

b s.t. Gh(�uh

0

)= 0, Gh = Qh′

G|Uhb

: Uhb → Vh′

, (5.149)

and for the other cases in (5.147) as

�uh0 ∈ D(Fh) ⊂ Uh s.t. Fh

(�uh

0

)= 0, Fh = Qh′

c,ΠF |Uh : Uh → Vh′ × Vhb . (5.150)


1. Then these FEMs are stable in Ph�u0, consistent and convergent. They have, forsmall enough h, a unique solution �uh

0 near �u0.2. We get, see (5.67), for �u0 ∈ H�(Ω,Rq), Pn

�−1 ⊂ Sh, � > 2m, cf. (4.62),∥∥Ph�u0 − �uh0

∥∥Uh ≤ Chmin{�−2m,p}‖�u0‖H�(Ω,Rq), (5.151)

with p = 2 for polygonal Ωh and p > 2 for curved Ωhc .

3. For m = 1, q ≥ 1 and convex, polyhedral Ω we obtain instead∥∥P h�u0 − �uh0

∥∥Uh ≤ KLh�−2‖�u0‖H�(Ω,Rq). (5.152)

4. Simultaneously F ′(�u0) ∈ L(W,W ′ ×WD) and F ′(�u0) ∈ L(U ,V × VD) satisfy theFredholm alternative.

5.2.10 Numerical solution of the FE equations with Newton’s method

We have indicated this problem for the (highly) nonlinear systems, cf. (5.29), (5.63),(5.149), (5.150), already in Subsection 5.2.2 4. We studied it for general operators anddiscretization methods in Section 3.7. So we discuss here only the specifics for the fullynonlinear case. We have to determine the FE solution uh

0 ∈ Uh ⊂ C1(Ωhc ) such that⟨

Fh(uh

0

), vh

Π

⟩= 0 ⇐⇒

(G(uh

0

), vh)Vc

= 0 ∀vh ∈ Vh,(uh

0 , vhb

)Vc,b

= 0 ∀vhb ∈ Vh

b .

(5.153)

We restrict the presentation here to second order equations. Let Condition Hbe satisfied The general notation, introduced in (5.113), allow a relatively easygeneralization to the other cases.

In this context again local and stable bases and stable splittings for Sh are important;see Theorems 4.36 and 4.36:

Sh = Sh0 + Sh

b =

{uh =

m0∑μ=1

cμsμ +

m1∑μ=m0+1

cμsμ

}. (5.154)

We insert uh0 ∈ Sh into (5.153) and test G(uh

0 ) with vh = sj , j = 1, . . . ,m0, and theboundary term with vh

b = sib = si+m0 , i = 1, . . . ,m1 −m0. So we obtain a nonlinear

system of m equations for the m unknowns cμ, μ = 1, . . . ,m1.This allows formulating the following algorithm:

Algorithm 5.1.Nonstandard FEM for elliptic equations

INPUT: G, T hc ,S1

d

(T h

c

)= Sh

0 + Shb (Differential operator, subdivision and a stable

splitting for the FE space of dimension m1 on Ωhc );

COMPUTATION: FOR j = 1, . . . , m0, i = 1, . . . , m1 − m0

/* Determine the m1 equations in (5.153) */

compute the⟨Fh

(uh

0

), vh

Π =(sj , si

b

)⟩Vh′

Π ×VhΠ

=(G(uh

0

), sj)Vc

+(uh

0 , sib

)Vc,b

;

OUTPUT:⟨Fh

(uh

0

), vh

Π

⟩Vh′

Π ×VhΠ

, j = 1, . . . , m0, i = 1, . . . , m1 − m0

(Nonlinear system of m1 equations in (5.63)).


For computing the terms in (5.153) the multilevel results of Davydov [263] andDavydov and Stevenson [268], with multiresolution techniques can be used.

Sparse nonlinear systems

After calculating this system (5.153) has to be solved. The most important toolsare continuation and discrete Newton methods. They require the solution of thecorresponding linear sparse FE equations with the defects for the nonlinear FEequations, Fh

(uh

ν

), uh

ν ≈ uh0 as a special input, see below. For these systems, we can

employ the standard efficient solvers for weak formulations by Proposition 5.7. Butthe usual FEs in C0(Ωh) have to be replaced by FEs in C1

(Ωh

c

).

Nevertheless the question is interesting, whether (5.153) represents a sparse nonlin-ear system, with only a few nontrivial terms for the unknowns cμ in each equation.Special conditions guarantee this, see Proposition 5.19.K has the special k-linear algebraic property if

K(u) := g(a1(u), . . . , as(u)) with k�-linear a�(u1, . . . , uk�) : Ωh

c → R, k� ≤ k′, (5.155)

uj : Ωhc → R, � = 1, . . . , s, a�(u1, . . . , uk�

) = 0 if supp u1 ∩ · · · ∩ supp uk�= ∅,

with k := maxs�=1{k�} ≤ k′, a�(u) := a�(u, . . . , u) and a special algebraic function

g : Rs → R, defined by sums of constants, the a�(u) and roots. (5.156)

For Proposition 5.19 and S1d

(T h

c

)we introduce

C1S := max

μ=1,...,m1{ number of triangles T ∈ T h

c s.t.supp sμ ∩ T �= ∅}

CkS := max

νi=1,...,m1{ number of k − tuples (ν1 . . . , νk) s.t. (5.157)

supp sν1 ∩ · · · ∩ supp sνk �= ∅ for νi �= νj for i �= j, and i, j = 1, · · · , k}

Proposition 5.19. For S1d

(T h

c

)we get the upper bound Ck

S ≤(C1

S)k

. If K has the

special k-linear algebraic property (5.155), (5.156) and if(C1

S)k+1 � m1, then (5.153)

is a sparse nonlinear system. For k = 1, this yields sparse linear systems.

Remark 5.20. In some sense, it is disappointing that this proposition only discussesone special case. This is compensated by the sparse linear FE equations in continuationand discrete Newton methods, obtained by linearizing (5.6) into equations (5.19). Thusit does not matter too much that (5.155), (5.156) is a strong condition nor did wetry to get the sharpest possible results. Products are excluded, since a1a2 would be(k1 + k2)-linear. But Proposition 5.19 admits at least important examples: (2.308),for appropriate f and (2.310) are included. Generalizations are certainly possible.

Proof. Obviously CkS ≤

(C1

S)k and by testing a linear K(uh) with a fixed sj , we find

at most C1S � m1 nontrivial coefficients, thus we obtain a sparse linear system. We

restrict the proof to k = 2; k > 2 is then clear. A bilinear a1 evaluated in x is:

a1(uh)(x) = a1(uh =m1∑μ=1

cμsμ, uh =

m1∑ν=1

cνsν)(x) =

m1∑μ,ν=1

∗

cμcνa1(sμ, sν)(x), (5.158)


where ∗ indicates that, for a fixed x ∈ Ω, we only need those terms with x ∈ supp sμ ∩supp sν �= ∅, so at most C2

S =(C1

S)2 terms. For all other x, the a1(sμ, sν)(x) = 0 by

(5.155), (5.156). Now we test with a fixed sj and obtain∫Ωh

c

a1(uh)(x)sj(x)dx =∫

Ωhc

m1∑μ,ν=1

∗

cμcνa1(sμ, sν)(x)sj(x)dx. (5.159)

Each of the C1S triangles T , with supp sj ∩ T �= ∅, contributes nontrivially at most

for T ∈ supp sμ ∩ supp sν �= ∅. The at most C2S triangles in ∪m1

μ,ν=1supp sμ ∩supp sν �= ∅, yield at most C3

S =(C1

S)3 nontrivial terms in (5.159). �

Solving the nonlinear systems with Newton’s method, MIP

In Section 3.7 we have presented continuation methods and the mesh independencepriniple for Newton’s method. So we give only a very short summary here.

Equation (5.153) is solved by the well-known continuation and Newton methods.We formulate the discrete Newton method and its quadratic convergence as specialcase of the preceding mesh independence principle (MIP), cf. Theorem 3.40 andCorollary 3.41. Both methods require the solution of the corresponding linearizedsparse FE equations, with the defect or residuum, Fh

(uh

ν

), uh

ν ≈ uh0 . The uh|∂Ωh

c= 0

and Proposition 5.7 allow standard efficient solvers with FEs in C0(Ωh) replaced byFEs C1

(Ωh

c

). For using these methods for differential equations and systems of order

2m, they can be transformed into larger equations and systems of order 2. Otherwisethe FEs in C0(Ωh) have to be replaced by C2m−1

(Ωh

c

)and appropriate solvers have

to be used.Efficient strategies combining continuation with the MIP on different discretization

levels have been studied by Allgower and Bohmer [9, 10]. Currently, multiresolutiontechniques are available for polyhedral Ω in Rn and d ≥ 2n + 1; see Davydov’s papers,e.g., [263, 267, 268]. For Ω ∈ C2 and the necessary Ωh

c with curved boundaries forgeneral S1

d

(T h

c

)this is still an open problem. For the specific cases in Proposition

5.19, (5.153) represents a sparse nonlinear system.The MIP, the Newton–Kantorowich method, and the convergence of our FEM

require a unique solution u0, such that F ′(u0) is boundedly invertible. So we assume

F (u0) = 0 with F ′(·) ∈ CL(Br(u0)), (F ′(u0))−1 ∈ L(V,U). (5.160)

Starting near uh0 ≈ u0, we apply Newton’s method to Fh. By Theorem 5.12, the(

Fh(uh

i

))′ : Uh → Vh are equiboundedly invertible; hence

uhi+1 := uh

i −(Fh(uh

i

)′)−1

Fh(uh

i

), i = 1, . . . , is well defined. (5.161)

In addition to (5.160), the MIP, cf. Subsection 3.7.2, and Allgower and Bohmer [9,10]and Allgower et al. [17], requires a linear FEM as in (5.34), satisfied for our FEMs.Then the Newton method converges quadratically; cf. Theorem 3.40.

Equations (5.110) and (5.151) show that for u0 ∈ Us = H�(Ω), � > 2, the FEMconverges with orders p′ = �− 2m > 0 and p′ := min{�− 2m, p} > 0. Then, with the


above projectors Ph = IhEc : U → Uh, Qh′Π : VΠ → Vh′

Π (see (5.51), (5.60)), we imposethe condition

F |Us: D(F ) ∩

(Us := H�(Ω)

)→ H�−2m(Ω). (5.162)

Theorem 5.21. MIP; see Theorem 3.40: In addition to the conditions in Theo-rems 5.13 and 5.18, in particular Condion H, with convergence in (5.110) of orderp′ = �− 2m > 0 or p′ = min{�− 2m, p} > 0 we assume (5.160), (5.162). Start theNewton processes for F , by omitting all upper indices h in (5.161), and for Fh with thesufficiently good starting values u1 ∈ Br(u0) ∩ D(F ) ∩ Us and uh

1 := Phu1. Then, forh ≤ h0 = (min{r, (SCL)−1}/(2Sc0))1/p′

, cf. the following proof, both Newton methodsconverge quadratically to u0 and uh

0 , respectively, and∥∥uhν+1 − uh

ν

∥∥Uh ≤ C

(∥∥uhν − uh

ν−1

∥∥Uh

)2and∥∥uh

i − Phui

∥∥Uh ≤ Chp′‖ui‖Us

, i = 1, . . . ,∥∥∥Fh(uh

i

)−Qh′

Π F (ui)∥∥∥V≤ Chp′‖ui‖Us

, i = 1, . . . , (5.163)∥∥(uhi − uh

0

)− (Ph(ui − u0))

∥∥Uh ≤ Chp′‖ui‖Us

, i = 1, . . . .

Under these conditions there exists h = h(ε, u0) such that for all h ∈ (0, h]∣∣min{i ≥ 1 :

∥∥uhi − Phu0

∥∥Uh < ε

}−min {i ≥ 1 : ‖ui − u0‖U < ε}

∣∣ ≤ 1. (5.164)

This last result is the reason for the notation as MIP and our remark concerningthe essential h independence of the convergence.

Proof. We have to show that the conditions in Theorem 3.40 are satisfied. BySubsection 5.2.5, Ph = IhEc ∈ L(U ,Uh), Qh′

Π ∈ L(VΠ = V ′Π, Vh′

Π �= VhΠ) are bounded

and linear, and Fh = Qh′c,ΠF |Uh : D(Fh) = D(F ) ∩ Uh → Vh′

Π , similar for m > 1. Asa consequence of (5.162) and Uh, we obtain the uniform Lipschitz-continuity of our(Fh)′:

‖(Fh)′(uh)− (Fh)′(vh)‖VhΠ←↩Uh =

∥∥∥Qh′

c,Π(F ′(uh)− F ′(vh))∥∥∥Vh

Π←↩Uh≤ CL‖uh − vh‖Uh

for all uh, vh ∈ Br(Phu0). As a consequence of F ′(·) ∈ CL(Br(u0)) and the isolatedsolution u0 of F (u) = 0 with boundedly invertible F ′(u0) we obtain stability of(Fh(uh))′, and hence ‖((Fh)′(uh))−1‖Uh←↩Vh

Π≤ S for all uh ∈ Br/2(Phu0) with a

slightly reduced r/2; see Theorem 5.6. Finally we need the consistent differentiability

∀u, v ∈ Br(u0) ∩ Us :∥∥∥Qh′

Π F (u)−Qh′

c,ΠF (Phu)∥∥∥Vh

Π

≤ Chp′‖u‖Us≤ c0h

p′, (5.165)

and∥∥∥Qh′

Π F ′(u)v −Qh′

c,ΠF′(Phu)Phv

∥∥∥Vh

Π

≤ Chp′(1 + ‖u‖Us

)‖v‖Us≤ c1h

p′‖v‖Us,

an immediate consequence of F ′(·) ∈ CL(Br(u0)) and of (5.162).

5.3. FE and Other Methods for Nonlinear Boundary Conditions 345

For (5.164), the only missing conditions (see (3.25) in [17]) require the existence ofa constant δ > 0 such that

lim infh>0

‖Phu‖Uh ≥ δ‖u‖U ∀u ∈ Us.

This is an immediate consequence of (4.64). �

5.3 FE and other methods for nonlinear boundary conditions

The techniques in the preceding section open up the possibility for formulating FEMsand prove their convergence for nonlinear boundary conditions in elliptic problems.Compared to the previous situation only few analytical existence and even lessregularity results are available, however, cf. Zeidler [678], p. 537 ff.

A convergence analysis for numerical methods seems to be totally missing. Wedemonstrate this analysis only for FEMs, although all the other methods, consid-ered in this and the next book, allow the same results. We assume the nonlinearboundary conditions, studied here, in a form such that the linearization yields eitherDirichlet boundary conditions or the more general form of a special Dirichlet system,cf. (2.80) ff., (2.137) ff., (2.158), and Theorem 2.50. We formulate a nonlinear versionof a Dirichlet boundary condition; extensions to special Dirichlet systems are obvious

bj(∂j−1u/∂νj−1) = 0 with appropriate nonlinear bj , j = 0, . . .m− 1, (5.166)

with obvious extensions to special Dirichlet systems.For m = 1, we admit natural boundary condition, cf. (2.123). For example, the

boundary operators, induced by quasilinear operators as in (5.286), might be imposedas nonlinear boundary operators

BG uh := BG (x, uh,∇uh) :=n∑

i=1

νiAi(x, uh,∇uh). (5.167)

Generalizations to other boundary conditions are possible, but are not studied here.We formulate the convergence result for problems of order 2m.

Theorem 5.22. FE and other methods for nonlinear boundary conditions: Let thenonlinear elliptic problem in (5.42), F (u0) = (G,B)(u0) = 0, u0 ∈ U , have an isolatedsolution, with boundedly invertible F ′(u0) : U → VΠ. We admit nonlinear G and/orB and assume G and B to be continuously differentiable in Br(u0), e.g. by (5.109),and an analogous condition for B. Let the domain satisfy (4.5). Let Gh and Bh beconsistent of the same order, usually min{s, d} −m, for the following combinations.

1. If G is linear or quasilinear, choose Uh,Vh as one of the above families of con-forming or nonconforming FEs of local degree d− 1, such that dist(∂Ω, ∂Ωh) ≤Chd. Assume an exact solution u0 ∈ Hs(Ω),m < min{s, d}. Then the FEM forG is stable, consistent in Phu0, and convergent. Then the same is valid for theFEM for F with the same order of convergence. So this Fh(uh

0 ) = 0 has, for small


enough h, a unique solution uh0 near u0, usually converging as∥∥u0 − uh

0

∥∥V ≤ Chmin{s,d}−m‖u0‖Hs(Ω). (5.168)

2. For a fully nonlinear G and a convex polyhedral or a C2 domain Ω in (5.100),choose Uh = S1

d

(T h

c

), d ≥ 2n + 1 (see (5.3)), on a quasiuniform triangulation.

Then the nonconforming FEM in (5.47) ff., now for nonlinear B,

Fh = Qh′

c,ΠF |Uh : Uh → Vh′ × Vhb , uh

0 ∈ D(Fh) ⊂ Uh s.t. Fhuh0 = 0,

is stable and consistent in Phu0, and convergent. It has, for small enough h,a unique solution uh

0 near u0. We obtain, cf. (5.67), for u0 ∈ H�(Ω), � > 2,for m = 1, the nonconforming FEM, and convex polyhedral or Cp, 2 ≤ p ≤ 5,domains, ∥∥Phu0 − uh

0

∥∥Uh ≤ Chmin{�−2,p}‖u0‖H�(Ω), (5.169)

respectively, with p = 2 for polyhedral Ω and p ≥ 2 for polyhedral or curved Ωhc ≈

Ω ∈ Cp.3. F ′(u0) ∈ L(W,W ′ ×WD) and F ′(u0) ∈ L(U ,V × VD) simultaneously satisfy the

Fredholm alternative.

Remark 5.23.

1. For the above F (see (5.18), (5.20), (5.73)), � and d are related by (4.62).With the ellipticity assumption (5.20), the previous (5.73), (5.74), (5.100)are implied by (5.109), except Ω convex polyhedral or in C2. The latterare necessary for the regularity and the extension operators. The conformingFEM in (5.26) ff. does not apply, since nonlinear boundary conditions will beviolated.

2. For quadrature approximations, exactly reproducing polynomials of degree < kwith k > 2(d− 1) > d > 2, the solution uh

0 of an approximate Fh(uh0 ) = 0 satis-

fies the same estimate as in (5.110); see Subsection 5.4.4 and 5.4.5 for m > 1 orfor system.

3. Quadrature approximations for these problems follow the lines of Section 5.4.

Proof. The proof follows essentially the lines of the last section with some straightfor-ward modifications. The consistency in Theorem 5.4 has to be modified for quasilinearG if nonconforming FEMs are used. In any case these results have to be applied to Bas well. For the stability we apply the technique to the product operator F = (G,B),but need the regularity result in Lemma 5.10 only for fully nonlinear G. The remainingpart of the proof remains essentially unchanged. �

5.4 Quadrature approximate FEMs

5.4.1 Introduction

In many cases it is impossible to evaluate exactly the linear, bilinear and highernonlinear forms defining the FE equations. So we consider in this section, quadrature

5.4. Quadrature approximate FEMs 347

and cubature or other approximations, e.g. difference approximations, for the differentterms in these FE equations. They are applied to elliptic problems of order 2m,m ≥ 1in Rn. These quadrature approximations will be chosen such that, the order of agiven FEM is maintained in its quadrature approximate form. So the quadratureapproximate forms have to be stable and consistent with the exact forms. These twoconditions will sometimes have different implications, see e.g. Theorem 5.31.

Simplifying the notation, we assume throughout that the functionals, coeffi-cients, nonlinear functions, f, aij , Aαβ ;Gw are already extended to Ω ∪ Ωh

c , so(f, aij , Aαβ , G

w) = (Ecf,Ecaij , EcAαβ , EcGw). e.g. by the extension operator Ec, cf.

Theorem 4.37. Thus, e.g. ah(uh, vh) in (5.181) is defined, in contrast to ah(u, vh) foru ∈ U , where the necessary function values of u might not be available. In particular,for the fully nonlinear problems as in Section 5.2 and Subsection 5.4.4 we sometimesomit the extension operator Ec. This is not always possible for functions, u, and thesolution, u0, e.g. Theorem 5.33.

All the following quadrature formulas are defined piecewise on every Tc ∈ T hc ,

admitting broken Sobolev spaces for the different terms. Extensions to several neigh-boring Tc ∈ T h

c , similarly to Subsections 4.2.5, and 4.2.6 would be possible, but arenot considered here. The FEs studied here are again piecewise polynomials withPn

d−1 ⊂ P ⊂ Pnd−τ , τ ≥ −1. For including the violated boundary conditions, studied

already in Section 5.2 and later on in Section 5.5, we use the notation Tc ∈ T hc

indicating that we admit either a polyhedral or a curved approximation Ωhc ≈ Ω

triangulated by T hc with straight or curved edges along the boundary ∂Ω. For the

isoparametric FEMs in Subsection 5.5.10 we will need other strategies anyway.For the second type of variational crimes, studied in Subsection 5.5.7 and caused by

violated continuity, the a(uh, vh) is not, but the piecewise ah(·, ·) ∀Tc ∈ T hc in (5.179)

is defined, cf. (5.78) above, due to uh �∈ U , vh �∈ V. We will in Subsection 5.5.9 studythe total error introduced by the combination of quadrature and these nonconformityerrors.

We denote the original linear, bilinear forms, linear, nonlinear operators asf(·), a(·, ·), A,G, their extensions and modifications due to variational crimes tobroken Sobolev spaces defined on Ωh

c ∪ Ω as fh(·), ah(·, ·), Ah, Gh, and their restrictionto the approximating spaces Uh,Vh, . . . , as fh(·), ah(·, ·), Ah, Gh. Their quadratureapproximations are denoted as fh(·), ah(·, ·), Ah, Gh, respectively Other, e.g. differenceapproximations of order ka, and combinations with quadrature of order k, we indicateas f h(·), ah(·, ·), Ah, Gh and f h(·), ah(·, ·), Ah, Gh, respectively Explicitly, we willstudy and formulate the quadrature approximations in this Section. For the otherappropriate approximations, ah(·, ·), Ah, Gh with the new order k := min{k, ka}, thequadrature results essentially remain correct.

According to Theorem 3.21, we need two different results for proving convergence.For a linear problem with stable Ah, the inducing ah(·, ·) has to satisfy a discrete inf–sup condition. Usually this is a consequence of the corresponding property for ah(·, ·),see e.g. Theorem 5.30. For nonlinear problems its derivative, G′(u0) = 0, has to satisfythis condition. Further we do need consistency results for the discretized equations.

It causes some problems that the ah(·, ·), 〈f, ·〉h, Ah, Gh, Fh are usually not definedfor the original u ∈ U , v ∈ V, since quadrature formulas require u(Pj) or even ∂iu(Pj),


see (5.170), (5.311). This is sometimes circumvented by comparing the exact solutionu0 with uh

0 , solving the exact FE equation or the interpolant Ihu0. Then the stabilityand estimates for ah(uh, vh)− ah(uh, vh) or nonlinear analogues, see e.g. (5.188),(5.204), allow the final error estimates for ‖u0 − uh

0‖Uh . The standard approachbased upon the Strang lemmas and a neighboring u0 ≈ zh ∈ Uh only works forlinear problems. Alternatively, extensions as discussed in Theorem 4.36 could beused.

For the proofs of the following Theorems 5.30, 5.33, 5.36 5.37 and 5.40, we applythe results of the general discretization theory in Chapter 3, here as Summary 4.52,in Subsection 5.2.5. We will prove in this section that the ah(·, ·), . . . are stable andconsistent in Phu, simultaneously with the original or the modified ah(·, ·), . . . , if thequadrature formulas, introduced in Subsection 5.4.2, are good enough and h is smallenough. By Theorem 3.21, we thus get convergence, via stability and consistency ofthe quadrature approximate FEMs, see Definition 4.46.

The standard argumentation for the following convergence results is: the quadra-ture approximations represent discretizations satisfying all the conditions for thechosen spaces, their approximations, the projectors, the operators and their dis-cretizations, Q

′h, Qh, Ah, Gh, or Fh, guaranteeing a unique solution, uh0 , and its

convergence to the unique exact solution, u0. In Subsection 5.4.3 we start withthe spaces in (5.177), combined with the exact projectors and FEMs in (4.117),(4.132), or (4.135) for linear differential operators of order 2, and extend it to fullynonlinear equations of order 2, see (5.52), (5.60), (5.62) in Subsection 5.4.4. Thenext generalization to linear and nonlinear equations and systems of order 2m asin (4.133)–(4.140), (4.163), and (5.111)–(5.114), (5.123), is presented in Subsection5.4.5. As a consequence of the perturbation or consistency results, e.g. in Theorem5.28, the approximate quadrature projectors, Q

′h, Qh, the linear and nonlinear Ah,Gh, Fh simultaneously satisfy these conditions with the exact Qh, Ah, Gh, Fh.In particular, Ah and Gh, Fh are continuous near Phu0 and consistent and stablein Phu0.

Remark 5.24. These results remain valid with slight modifications for the othermethods as well.

5.4.2 Quadrature and cubature formulas

The main tools in this section are the following quadrature formulas in higherdimensions. Usually these quadrature formulas reproduce the integral of polynomialsof a certain degree. For tensor product FEs, e.g. bilinear FEs, see Subsection 4.2.2, orthe serendipity class, see [141], modifications are possible.

Proposition 5.25. Assume the following quadrature formula∫Tc

f(x)dx ≈ qhTc

(f) :=∑

Pj,Tc∈Tc

wj,Tcf(Pj,Tc

) (5.170)


to be exact for polynomials of degree k − 1, and let diam Tc ≤ h. Then ∀k ∈ N , k >n/p′, there exists C = Ck ∀ 0 < h < 1, 1 ≤ p′ ≤ ∞, f ∈W k,p′

(Tc) ∩ C(Tc), such that∣∣∣∣∣∣∫Tc

f(x)dx−∑

Pj,Tc∈Tc

wj,Tcf(Pj,Tc

)

∣∣∣∣∣∣ ≤ Ck hk+(1−1/p′)|f |W k,p′ (Tc). (5.171)

Proof. By k > n/p′ and the Sobolev embedding Theorem 1.26, f ∈ C(Ω), so allthe f(Pj,Tc

) are well defined. Now we proceed as in Proposition 5.62, adding andsubtracting the averaged Taylor polynomial Qkf ∈ Pn

k−1 for f in (5.171). Then (5.170)implies with f ∈W k,p′

(Ω), the Holder inequality (1.45) and the Bramble–Hilbertlemma 4.15∣∣∣∣∫

Tc

f(x)dx −∑

Pj,Tc∈Tc

wj,Tcf(Pj,Tc

)

∣∣∣∣∣∣≤

∣∣∣∣∣∣∫

Tc

f(x)dx−Qkf + Qkfdx−∑

Pj,Tc∈Tc

wj,Tcf(Pj,Tc

)

∣∣∣∣∣∣≤

∣∣∣∣∣∣∫T

1 · (f(x)−Qkf)dx

∣∣∣∣∣∣ with Holder and Bramble–Hilbert

≤ Ck h1−1/p′ |f −Qkf |Lp′ (Tc)≤ Ck hk+1−1/p′ |f |W k,p′ (Tc)

.

�Remark 5.26.

1. In these quadrature approximations (5.170) the quadrature points Pj,Tcare usu-

ally chosen independently of the interpolation points for the FEs.2. The Pj,Tc

∈ Tc, Tc ∈ T hc in (5.170) are often determined as Pj,Tc

= FTc(Pj), Pj ∈

K. Then the computations are performed along the lines of Examples 4.11 and4.51. So let all the Tc ∈ T h

c be affinely equivalent to the reference element K,see Definition 4.6, hence, FTc

(K) = Tc, FTc= ATc

x + bTc, ATc

nonsingular. ThenqhTc

(f) =∑

Pj,Tc∈Tcwj,Tc

f(Pj,Tc) so we obtain the wj,Tc

:= det(ATc)wj , Pj,Tc

:=FTc

(Pj), from the quadrature formula on K, qh(f) =∑

Pj∈K wjf(Pj). For Tc atthe boundary slight modifications are necessary.

We apply the above qhTc

to the Tc ∈ T hc , to yield a quadrature formula qh(f) for

Ω or Ωhc . Note that for conforming and nonconforming FEs, satisfying the boundary

conditions, we have Ω = Ωhc and, hence,

∫Ωh

cf(x)dx =

∫Ωf(x)dx, otherwise we use the

extension fh : Ω ∪ Ωhc → R, cf. (5.178), (5.5), cf. (5.170) for qh

Tc(fh), in

qh(fh) :=∑

Tc∈T hc

qhTc

(fh) ≈∫

Ωhc

fh(x)dx ≈∫

Ω

f(x)dx. (5.172)


Often we additionally find, sometimes with C = 1, if all weights are positive,∣∣qhTc

(f)∣∣ ≤ C meas (Tc)‖f‖L∞(Tc) ∀Tc ∈ T h

c , (5.173)

admitting modified relations. We estimate the total quadrature error in (5.172) withProposition 5.25 and for p′ = 1 as37

|qh(fh)−∫

Ωhc

f(x)dx| ≤ Ckhk∑

Tc∈T hc

|fh|W k,1(Tc) = Ckhk|fh|W k,1(T h

c ). (5.174)

Some of the following estimates are applied to fh = uhvh, uh ∈W k,p, vh ∈W k,p′.

Proposition 5.27. For uh ∈W k,p(T h

c

), vh ∈W k,p′ (T h

c

), 1/p + 1/p′ = 1 we get

|qh(uhvh)−∫

Ωhc

uhvh(x)dx| ≤ Ckhk‖uh‖W k,p(T h

c )‖vh‖W k,p′ (T hc ). (5.175)

Proof. The standard and discrete Holder inequalities (1.45), (1.44) yield with p′ = 1in (5.171) and again with Ck indicating different constants, depending upon k,

|qh(uhvh)−∫

Ω or Ωhc

uhvh(x)dx|

≤ Ckhk∑

Tc∈T hc

|uhvh|W k,1(Tc)

≤ Ckhk∑

Tc∈T hc

∑α,β≥0

∑|β+α|=k

‖∂βuh∂α−βvh‖L1(Tc)

≤ Ckhk∑

Tc∈T hc

∑α,β≥0

∑|β+α|=k

‖∂βuh‖Lp(Tc)‖∂α−βvh‖Lp′ (Tc)

≤ Ckhk

⎛⎝ ∑Tc∈T h

c

∑|β|=k

‖∂βuh‖pLp(Tc)

⎞⎠1/p⎛⎝ ∑Tc∈T h

c

∑|α|=k

‖∂αvh‖p′

Lp′ (Tc)

⎞⎠1/p′

≤ Chk‖uh‖W k,p(T hc )‖vh‖W k,p′ (T h

c ). �

5.4.3 Quadrature for second order linear problems

Here we consider weak forms of second order problems, hence we choose U = V. Thebasic condition, compare the Sobolev Embedding Theorem 1.26 is a

bounded Ω ⊂ Rn with Lipschitz boundary , n ≥ 1 and f, aij ∈ C(Ω). (5.176)

u, u0, v ∈ C1(Ω), e.g., W k,p(Ω) ⊂ Cj(Ω), 1 ≤ p <∞, k > j + n/p for Ω,

see the modifications below for less smooth u, u0, v �∈ C1(Ω).

37 For some cases p′ = 1 is the optimal choice: It gives, in combination with (1.44), hp‖vh‖H1(T hc )

with largest p, e.g. p′ = ∞ would yield at most hp−1‖vh‖H1(T hc ).


The piecewise polynomials uh, vh are continuously differentiably extendable to eachTc. For a Pj,Tc1 ∈ Tc1 ∩ Tc2 the values uh(Pj,Tc1) and uh(Pj,Tc2) might differ. But it isno problem to exactly evaluate the uh, vh, ∂iuh, ∂jvh in the points Pj and to exactlyintegrate their products and the ah(·, ·). Usually the 〈f, vh〉 have to be approximated

by an 〈f, vh〉h. So there are good reasons to study both cases. Sometimes other formsof ah(·, ·) : Uh × Vh → R are defined by combining the exact uh, vh, ∂iuh, ∂jvh withappropriate means of the aij , f . We do not discuss those methods here, cf. Grossmann,Roos and Stynes [374–376].

The spaces here and their approximating spaces are, cf. (4.35), (4.36)

U = V ⊂ H1(Ω),Uhb ,Vh

b ⊂{L∞ (T h

c

): ∀Tc ∈ T h

c : uh|Tc∈ P},Pd−1 ⊂ P ⊂ Pd+τ ,

dimUhb = dimVh

b ,usually τ = −1, and norms ‖ · ‖Uh := ‖ · ‖Vh := ‖ · ‖H1(T hc ).

(5.177)

These Uhb ,Vh

b indicate the FE spaces, satisfying the boundary conditions eitherexactly or approximately, corresponding to the chosen FEM. If we use Uh,Vh weomit boundary conditions, often for the bilinear forms or operators. SometimesU ⊂W 1,p(Ω),V ⊂W 1,p′

(Ω) or U ⊂Wm,p(Ω),V ⊂Wm,p′(Ω),m > 1, are appropriate.

This can be handled similarly to below via the Sobolev Embedding Theorem 1.26.In addition to the quadrature versions fh, ah(·, ·), Ah we need the approximate

projectors Q′h, cf. Propositions 4.49, 4.50. All versions are obtained by applying

the quadrature formulas (5.172) to different combinations, f, fvh, aijuhvh, . . . As

in Section 5.2, we denote, e.g. the original and the sometimes necessary extensions,note (5.5) omitting Ec,

extensions of f, a(·, ·), A,G, to Ω ∪ Ωhc as fh, ah(·, ·), Ah, Gh, a.s.o., so (5.178)

ah(uh, vh) := 〈Ahuh, vh〉 :=

∑Tc∈T h

c

∫Tc

⎛⎝ n∑i,j=0

aij∂i uh∂j vh

⎞⎠ dx ∀uh ∈ U ∪ Uh,

ah(uh, vh) := 〈Ahuh, vh〉h :=

∑Tc∈T h

c

qhTc

⎛⎝ n∑i,j=0

aij∂i uh∂j vh

⎞⎠ , and (5.179)

〈fh, vh〉 :=

∑Tc∈T h

c

∫Tc

(n∑

i=0

fi∂ivh

)dx ∀vh ∈ V ∪ Vh, fj ∈ Hk

(T h

c

)orf ∈ H−1

k

(T h

c

),

〈fh, vh〉h :=

∑Tc∈T h

c

qhTc〈fh vh〉 =

∑Tc∈T h

c

∑Pj,Tc∈Tc

wj,Tc

(n∑

i=0

fi∂ivh

)(Pj,Tc

)

Q′h : V → Vh′

, 〈Q′hf − f, vh〉h = ((Q′hf − fh)vh)h = 0 ∀vh ∈ Vh

b . (5.180)

Note that for applying the quadrature formulas, we have to impose the additionalconditions f or f = (fj)n

j=0, aij ∈ Hk(Ω), or f ∈ H−1k (Ω), k large enough for the


UbG V ′

btested by V

Ph Φh

Uhb Vh

bVhb′

Q′h

Gh appropx. tested by

Figure 5.5 Quadrature approximate FEMs.

required function evaluations, cf. (5.184), (5.185). Similarly to (5.5) we get

ah(uh, vh) := 〈Ahuh, vh〉 := 〈Ahuh, vh〉

∣∣∣Uhb ×Vh

b= ah(uh, vh)

∣∣∣Uh

b ×Vhb

,

a(u, v) = ah(u, v)|Ub×Vb= 〈Au, v〉, and (5.181)

ah(uh, vh) := 〈Ahuh, vh〉h := ah(uh, vh)∣∣∣Uh

b ×Vhb, 〈fh, vh〉h := 〈fh, v

h〉h∣∣∣Vh

,

for u, v ∈ Hk(Ω) : ah(u, v) := 〈Ahu, v〉h := ah(u, v)|U×V := 〈Ahu, v〉h|U×V .

The approximate solutions uh0 are defined for these approximations uh

0 ∈ Uhb⟨

Ahuh0 , v

h⟩

= ah(uh

0 , vh)

= 〈fh, vh〉h or = 〈Ahuh0 , v

h〉h = ah(uh

0 , vh)∀vh ∈ Vh

b .

(5.182)

Reformulating with the exact and quadrature projectors yields

Ah = Q′hAh|Uh

b, Ahuh

0 = Q′hf, and Ah = Q

′hAh|Uhb, Ahuh

0 = Q′hf. (5.183)

Again a slight modification of the general discretization theory in Chapter 3, seeFigure 5.5, cf. Figures 4.27, 5.2, yields convergence for a consistent and stable methodin Phu0 with uniformly bounded Qh, since Ph = Ih or IhEc is again uniformlybounded, note Summary 4.52.

We will show in Theorems 5.28, 5.30 that ah(·, ·), 〈fh, ·〉h

in (5.182) are stable andconsistent in Phu, simultaneously with ah(·, ·), 〈fh, ·〉h and the above Ph and the newQ

′h, QhΠ are uniformly bounded for smooth enough data, hence if

k > max{d + τ − 1,

n

2

}for 〈fh, ·〉

h, f =

(fj ∈ Hk

(Ω ∪ T h

c

))nj=0

∈ H−1k (. . .) (5.184)

k > 2(d + τ − 1) for ah(·, ·), aij ∈W k,∞ (Ω ∪ Ωhc

), τ ≥ −1, usually τ = −1. (5.185)

Then, by the Sobolev Embedding Theorem 1.26, the quadrature formulas in (5.179)–(5.180) are well defined. We assume (5.184), (5.185), but do not show (5.185) to benecessary for a stable Ah; analogously to (4.137) we define ‖f‖H−1

k (T hc ).

Hence k is often pretty large. Solutions u0 and general functions u will usually�∈ Hk+1(Ω). So we combine in (5.190) the Ecu with Ihu. If u is smooth enough,the quadrature formula can be applied to u directly. This is not a necessary condi-tion, but is satisfied, e.g. if u ∈ Hs(Ω) with s > n/2, and the quadrature formularequires only function values, cf. Theorem 1.26. FEMs for fully nonlinear prob-lems, their linearization and isoparametic FEs do need the extension operator Ec.


Quadrature approximate FEMs for nonconforming FEs are delayed to Section 5.5Subsection 5.5.9 we omit the Ec here, except for fully nonlinear equations.

Theorem 5.28. Convergence of the fh and Q′h, consistent ah(uh, vh):

1. Assume (5.176), and for the quasiuniform subdivision T h the Condition 4.16.Choose a quadrature formula as in (5.172), (5.174), satisfying (5.184) and

(5.185) for 〈fh, ·〉h

and ah(·, ·), with fj ∈ Hk(Ω) or f ∈ H−1k (Ω), aij ∈W k,∞(Ω),

and let h > 0 be small enough.2. Then the following consistency results hold, with u ∈ Hd−p(Ω), p,≥ 0, in (5.190),

notice the different norms in (5.188), (5.189),

|〈fh, vh〉 − 〈fh, vh〉h| ≤ Chk−(d+τ−1)‖f‖H−1

k (T hc ) · ‖vh‖Vh , (5.186)

‖Q′hf − Q′hf‖Vh′ ≤ Chk−(d+τ−1)‖f‖H−1

k (T hc ), lim

h→0‖Q′hf‖Vh′ = ‖f‖V′ ,

(5.187)

|ah(uh, vh)− ah(uh, vh)|‖vh‖Vh

≤ Chk−2(d+τ−1)Maij‖uh‖Vh , for vh �= 0 (5.188)

≤ Chk−(d+τ−1) ·Maij‖uh‖Hd+τ (T h);Maij

:=n

maxi,j=0

‖aij‖W k,∞(Ωhc ). (5.189)

|ah(u, vh)− ah(Ihu, vh)|‖vh‖Vh

≤ C(hd−1−p + hk−(d+τ−1)Maij

)‖u‖Hd−p(Ω). (5.190)

Remark 5.29. The condition of a quasiuniform subdivision, T h, in this and thefollowing theorems, can be slightly relaxed with the results in Subsection 4.2.5. Thisrequires the restriction to the most important cases for FEMs, n = 2, 3, and requir-ing Assumptions 4.21 and 4.26. Then in the sum in (5.191) the terms have tobe estimated for each T ∈ T h separately. The results in Theorems 4.30–4.33 allowappropriate estimates for continuous and L2 FEs. We leave the details to the interestedreader.

Proof. We start using (5.172),(5.174) estimating for f ∈ Hk(Ω)

|〈fh, vh〉 − 〈fh, vh〉h| =

∣∣∣∣∣∣∑

Tc∈T hc

n∑j=0

∫Tc

fj∂jvhdx− qh

Tc(fj∂

jvh)

∣∣∣∣∣∣≤ Ckh

k∑

Tc∈T hc

n∑j=0

|fj∂jvh|W k,1(Tc)

= Ckhk

n∑j=0

|fj∂jvh|W k,1(T h

c ). (5.191)


We estimate these derivatives and use (5.184)

|fj∂jvh|W k,1(Tc)

=∑|α|=k

‖(fj∂jvh)(α)‖L1(Tc)

≤∑|α|=k

∑0≤�≤α

C�α‖f (�)j (∂jvh)(α−�)‖L1(Tc) by (5.312)

≤ C∑|α|=k

∑0≤�≤α

‖f (�)j ‖L2(Tc)‖(∂jvh)(α−�)‖L2(Tc) by (1.44) (5.192)

≤ C

⎛⎝∑|α|=k

∑0≤�≤α

‖f (�)j ‖2L2(Tc)

⎞⎠1/2⎛⎝∑|α|=k

∑0≤�≤α

‖(∂jvh)(α−�)‖2L2(Tc)

⎞⎠1/2

= C‖fj‖Hk(Tc)‖∂jvh‖Hd+τ (Tc) = C‖fj‖Hk(Tc)‖vh‖Hd+τ−(1−δj0)(Tc)

.

Combining (5.191), (1.44) with the usual δij yields, cf. (4.137), and [141], 8.x.15,

|〈fh, vh〉 − 〈fh, v

h〉h| = |〈fh, vh〉 − 〈fh, vh〉h|

(1.44)≤ Ckh

k (5.193)∑Tc∈T h

c

n∑j=0

‖fj‖Hk(Tc)‖vh‖Hd+τ−(1−δj0)(Tc)

≤ Ckhk−(d+τ−1)‖f‖H−1

k (T hc )‖vh‖H1(T h

c ).

thus (5.186) by the inverse estimate (4.47) for the quasiuniform T hc . Relation (5.187)

is an immediate consequence of (5.186).Similarly we have to estimate the quadrature errors for the weak bilinear form,

replacing (5.184) by (5.185). For modifying (5.192), (5.193), we use (5.185) and theCauchy–Schwarz inequality, (1.48). Then

|(aij∂iuh∂jvh)|W k,1(Tc)

≤ C‖aij‖W k,∞(T hc )‖∂iuh‖Hk(Tc)‖∂jvh‖Hk(Tc) (5.194)

≤ C‖aij‖W k,∞(T hc )‖∂iuh‖Hd+τ−(1−δi0)(Tc)

‖∂jvh‖Hd+τ−(1−δj0)(Tc)

.

uh, vh are always ∈ Hk(T hc ). This implies with the discrete Holder inequality∣∣∣∣∣∣

∑Tc∈T h

c

∫Tc

aij∂iuh∂jvhdx− qh

Tc(aij∂

iuh∂jvh)

∣∣∣∣∣∣ (5.195)

≤ Ckhk‖aij‖W k,∞(Ωh

c )

∑Tc∈T h

c

‖∂iuh‖Hd+τ−(1−δi0)(Tc)‖∂jvh‖

Hd+τ−(1−δj0)(Tc)

≤ Ckhk‖aij‖W k,∞(Tc)‖∂iuh‖Hd+τ−(1−δi0)(T h

c )‖∂jvh‖Hd+τ−(1−δj0)(T h

c )

with ‖u‖Hk+1(Ω) for ‖∂iuh‖Hd+τ−(1−δi0)(T hc ) if uh is replaced by u.


The analogous summation as in (5.193) and the inverse estimate (4.47) again yield(5.188), (5.190), since by (5.177), (5.180). (5.185), (5.195), we find

|ah(uh, vh)− ah(uh, vh)|

=

∣∣∣∣∣∣∑

Tc∈T hc

n∑i,j=0

(∫Tc

aij∂i uh∂j vhdx− qh

Tc. . .

)∣∣∣∣∣∣≤ Chk n

maxi,j=0

‖aij‖W k,∞(Ωhc )

⎛⎝ ∑Tc∈T h

c

n∑i,j=0

‖∂iuh‖Hd+τ−(1−δi0)(Tc)‖∂jvh‖

Hd+τ−(1−δj0)(Tc)

⎞⎠≤ Chk n

maxi,j=0

‖aij‖W k,∞(Ωhc ) ·(‖∇uh‖Hd+τ−1(T h

c )‖∇vh‖Hd+τ−1(T hc )

+ ‖∇uh‖Hd+τ−1(T hc )‖vh‖Hd+τ (T h

c ) + ‖uh‖Hd+τ (T hc )‖∇vh‖Hd+τ−1(T h

c )

+ ‖uh‖Hd+τ (T hc )‖vh‖Hd+τ (T h

c )

); (5.196)

note that ∇vh indicates ∂jvh ∈ Pd+τ−1, j = 1, . . . , n. We employ (4.47) for uh, vh, e.g.‖∇vh‖L2(T h

c ) ≤ ‖vh‖H1(T hc ) and verify (5.188) by (5.185).

If we replace uh by a smooth enough u, such that the quadrature formula is definedfor u, we end up with (5.190). If only u ∈ Hd−p(Ω), p,≥ 0, we use Ihu. Then ah(u−Ihu, vh) causes the hd+τ−p term by the interpolation error, the quadrature error,ah(Ihu, vh)− ah(Ihu, vh) remains unchanged by (5.189). �

Theorems 5.30 and 5.31 imply stability and a convergent quadrature approximateFEM, see Definition 4.46.

Theorem 5.30. Stability of Ah: Under the conditions of Theorem 5.28 the ah(·, ·)and ah(·, ·) are simultaneously bounded on Uh × Vh. Let ah(·, ·) be Uh

b -coercive or, e.g.for Uh

b �= Vhb , satisfy a discrete inf–sup condition and (4.145) for Uh and Vh, hence

Ah is stable and let k > 2(d + τ − 1). Then, for small enough h, Ah and ah(·, ·) havethe same property.

Proof. We only consider the case of a discrete inf–sup condition for ah(·, ·). This implieswith Theorem 5.28

ε′‖uh‖Uh ≤ inf0�=vh∈Vh

b

|ah(uh, vh)|/‖vh‖Uh

≤ inf0�=vh∈Vh

b

|ah(uh, vh) + (ah(uh, vh)− ah(uh, vh))|/‖vh‖Uh

≤ inf0�=vh∈Vh

b

|ah(uh, vh)|/‖vh‖Uh (5.197)

+ sup0�=vh∈Vh

b

|(ah(uh, vh)− ah(uh, vh))|/‖vh‖Uh

≤ inf0�=vh∈Vh

b

|ah(uh, vh)|/‖vh‖Uh + Ch‖uh‖Uh ,


so finally

(ε′ − Ch)‖uh‖Uh ≤ inf0�=vh∈Vh

b

|ah(uh, vh)|/‖vh‖Uh . (5.198)

Analogously the inequalities for the inf0�=uh∈Uhb

are handled. This shows the discretecoercivity or discrete inf–sup condition for ah(·, ·) for sufficiently small h > 0. �

Theorem 5.31. Stability and convergence for different quadrature approximations:

1. Let ah(·, ·) induce a stable Ah and assume the conditions of Theorem 5.28.Then, for small enough h, simultaneously ah(·, ·) induces a stable Ah. If fur-

thermore (5.184) is correct, then ah(uh0 , v

h) = 〈f, vh〉h∀vh ∈ Vhb , defines a stable

and convergent discretization for Au0 = f . For the other approximations, men-tioned above, this result remains correct if the ah, Ah are replaced by ah, ah,Ah, Ah.

2. For conforming FEMs and u0 ∈ Hs(Ω), Maij=

nmaxi,j=0

‖aij‖W k,∞(Ω), we obtain

∥∥u0 − uh0

∥∥Uh ≤ Chmin{s−1,k−(d+τ−1)} (5.199)(

(Maij+ 1)‖u0‖Hmin{s,k+1}(Ω) + ‖f‖H−1

k (Ω)

).

3. If only approximations for 〈f, vh〉h and none for ah(uh, vh) are used in (5.182),then the term (1 + Maij

)‖u0‖Hmin{s,k+1}(Ω) in (5.199) is dropped.

Proof. This theorem is an immediate consequence of (5.187), (5.190). The original(conforming) FEM is based upon an appropriate combination of spaces, projectorsQ

′h, linear and bilinear forms. Then (5.187) shows that Q′h ∈ L(H−1

k (T hc ) ⊂ V ′,Vh′

)and with (5.190) we obtain for conforming FEMs and u0 ∈ Hk+1(Ω)

ah(uh

0 , vh)− a(u0, v

h) = 〈fh, vh〉h − 〈f, vh〉

=⇒ ah(uh

0 − u0, vh)

= a(u0, vh)− ah(u0, v

h) + 〈fh, vh〉h − 〈f, vh〉

=⇒∥∥uh

0 − u0

∥∥Uh ≤ Chk‖u0‖Hk+1(Ω)

+Chk−(d+τ−1)(Maij· ‖u0‖Hk+1(Ω) + ‖f‖H−1

k (Ω)).

For u0 ∈ Hs(Ω), s < k + 1, we combine the interpolating FE Ihu0 with bounded‖Ihu0‖Hs(T h

c ) < C‖u0‖Hs(Ω) <∞ and ‖u0 − Ihu0‖H1(T hc ) < Chs−1‖u0‖Hs(Ω),

cf. (4.42), or (4.105), with Theorem 4.17 and 4.61, the discrete Brezzi–Babuska


condition (4.169),

ε∥∥Ihu0 − uh

0

∥∥U ‖v

h‖cV h ≤∣∣ah(uh

0 − Ihu0, vh)∣∣

= |(ah(uh

0 , vh)− ah(Ihu0, v

h))

+ (ah(Ihu0, vh)

− ah(Ihu0, vh)) + (a(u0, v

h)− a(u0, vh))|

≤∣∣〈fh, vh〉h − 〈f, vh〉

∣∣+ ∣∣ah(Ihu0, vh)− ah(Ihu0, v

h)∣∣+ ∣∣ah(Ihu0, v

h)− a(u0, vh)∣∣

≤ C(hk−(d+τ−1)(‖f‖H−1

k (T hc ) + Maij

· ‖Ihu0‖Hd+τ (T h)) + ‖u0 − Ihu0‖U)· ‖vh‖Vh

≤ C(hk−(d+τ−1)(‖f‖H−1

k (T hc ) + Maij

‖u0‖Hs(Ω)) + hs−1‖u0‖Hs(Ω)

)· ‖vh‖Vh

With ‖u0 − uh0‖U ≤ ‖u0 − Ihu0‖U + ‖Ihu0 − uh

0‖U we have proved the claim. �

5.4.4 Quadrature for second order fully nonlinear equations

We do not formulate the results for second order quasilinear problems, since the generalcase in Subsection 5.4.5 is nearly the same. Second order fully nonlinear equationsrequire U �= V. Since the boundary conditions remain unchanged, we start with thequadrature differential operator. The approximate solution has to satisfy cf. (5.3)

uh0 ∈ Uh

b : (Gw(·, uh

0 , Duh0 , D

2uh0

), vh)

h=∑

Tc∈T h

qhTc

(G(uh

0

), vh)

= 0∀vh ∈ Vhb

(5.200)

thus defining Gh = QhcG|Uh . We need the spaces, approximate projectors Qh, the

nonlinear operators and the generalizations, see (5.77)–(5.79), and their linearization,cf. (5.73) with the aij , and the strong bilinear forms, ah

s (uh, vh),

U ⊂ H2(Ω),V ⊂ L2(Ω),Uhb = Sh ⊂ S1

d

(T h

c

),Vh

b = Sh ∩ C10

(Ωh

c

), (5.201)

Uc ⊂ H2(Ωh

c

), . . . , ‖ · ‖Uh := ‖ · ‖H2(T h

c ), ‖ · ‖Vh := ‖ · ‖L2(T hc ), the linearization

(As,huh, vh) = as,h(uh, vh) =

∑Tc∈T h

c

∫Tc

⎛⎝ n∑i,j=0

(−1)j>0∂j(aij∂

i uh) vh

⎞⎠ dx with

(Ah

suh, vh

):= ah

s (uh, vh) =∑

Tc∈T hc

qhTc

⎛⎝ n∑i,j=0

(−1)j>0∂j(aij∂

i uh) vh

⎞⎠.

In Theorems 5.32 and 5.33 we will show the consistency and stability of ourapproximate FEM (5.200). The stability is equivalent to showing under which con-ditions ah

s (·, ·) inherits the boundedness and discrete inf–sup condition from as(·, ·)on Uh

b ,Vhb . The proofs of these results are very similar to those in Subsection 5.4.3.


Note the consequences of the different norms in (5.202) and testing in the L2 sensein (5.200) in contrast to the weak form in (5.182), ‖ · ‖Vh := ‖ · ‖L2(T h

c ) comparedto ‖ · ‖Vh := ‖ · ‖H1(T h

c ) in the last subsection. So the exponents in the consistencyestimates, e.g. k − d in (5.202) compared to k − (d + τ − 1) in (5.186), differ by 1.Furthermore, the �− 1 in Pn

�−1 ⊂ Sh ⊂ S1d , is relevant:

Theorem 5.32. Consistency for different quadrature approximations of fully non-linear problems: Assume (5.176), and for the quasiuniform subdivision T h the Con-dition 4.16. Choose a quadrature formula as in (5.172), (5.174), satisfying (5.184)and (5.185). Let the fully nonlinear extension G : Hk+2

(Ω ∪ Ωh

c

)→ Hk

(Ω ∪ Ωh

c

),

be differentiable, G′h(u0) be bounded with aij ∈W k+1,∞ (Ω ∪ Ωh

c

), see (5.73), f, fj ∈

Hk(Ω ∪ Ωh

c

), and let a u ∈ H�(Ω), yield Ecu ∈ H�

(Ω ∪ Ωh

c

), e.g. for u = u0. Choose

the approximating spaces Uhb ,Vh

b ⊂ Sh, Pn�−1 ⊂ Sh ⊂ S1

d , d ≥ 2n + 1, as in (5.201),with k > 2(d− 1), hence k > d, and let h > 0 be small enough. Finally, let uh beextended into uh ∈ Hk+2

(Ω ∪ Ωh

c

).

Then the following consistency results hold with Maij=

nmaxi,j=0

‖aij‖W k+1,∞(Ω)

|〈f, vh〉 − 〈fh, vh〉h| ≤ Ckhk−d‖f‖Hk(T h

c )‖vh‖Vh , k > d, (5.202)

‖Qhf − Qhf‖Vh′ ≤ Ckhk−d‖f‖H−1

k (Ω), limh→0

‖Qhf‖Vh′ = ‖f‖H−1k (Ω), (5.203)

|(G(uh), vh)− (G(uh), vh)h| ≤ Chk−d‖G(uh)‖Hk(T hc )‖vh‖Vh , and∥∥Qh

cG(uh)− QhcG(uh)

∥∥Vh′ ≤ Chk−d‖G(uh)‖Hk(T h

c ), or for Ecu, Ihu : (5.204)∥∥Qh

cG(Ecu)− QhcG(Ihu)

∥∥Vh′ ≤ Chk−d‖G(uh)‖Hk(T h

c ),

and for G′h :

∣∣ahs (uh, vh)− ah

s (uh, vh)∣∣ ≤ Chk−2(d−1)Maij

‖uh‖Uh‖vh‖Vh (5.205)

|ah(Ecu, vh)− ah(Ihu, vh)| ≤ Chmin{k−d,�−2}Maij

‖u‖Hmin{k+2,�}(Ω)‖vh‖Vh .

These inequalities only estimate the quadrature errors for the differential and not forboundary operators. For the complete consistency errors the terms in Theorem 5.4and L× the interpolation errors have to be added, cf. Theorem 5.33. L denotes theLipschitz constant for G.

Proof. The above conditions d ≥ 2n + 1 > 2, n ≥ 2, for S1d

(T h

c

)⊂ C1(Ω) and k >

2(d− 1), imply k > 2(d− 1) ≥ 2n+1 > n/2, n ≥ 2, k > d. Hence, by the Sobolevembedding theorem the W k,∞(Ω),Hk(Ω) are compactly embedded into C(Ω). Thusall the quadrature formulas for as and (G(uh), vh)), see e.g. (5.200), are well defined.If u or u0 should not be smooth enough, we use the interpolating Ihu0. Since theuh, vh ∈ S1

d

(T h

c

)⊂ Hk

(T h

c

)∩ C

(T h

c

), the estimates in (5.174) can be applied to

(5.201)–(5.200) under the above conditions to different choices of f or fh. Without


U F VΠ=V×VDtested by VΠ=V×Vb

Ph Φh QhΠ

Ub Fh VhΠ′ =Vh ′ ×Vh

bapprox. tested by Vh

Π =Vhb ×Vh

b

Figure 5.6 Nonconforming quadrature FEMs for fully nonlinear problems.

repeating all the details we get

|〈fh, vh〉 − 〈fh, vh〉h| =

∣∣∣∣∣∣∑

Tc∈T hc

n∑j=0

∫Tc

fj∂jvhdx− qh

Tc(fj∂

jvh)

∣∣∣∣∣∣≤ Ckh

k∑

Tc∈T hc

n∑j=0

|fj∂jvh|W k,1(Tc) (5.206)

≤ Ckhk−d‖f‖H−1

k (T hc )‖vh‖Vh .

As in the proof of Theorem 5.28, the last inequality follows by the inverse estimates(4.57) for the piecewise polynomials vh of degree ≤ d with d > 2, the discrete Holderinequality and ‖fhv

h‖L1(T hc ) ≤ ‖vh‖Vh ‖fh‖L2(Ωh

c ). Again (5.203) is an obvious conse-quence of (5.202). We get for k > d

|(G(uh), vh)− (G(uh), vh)h| ≤ Chk‖G(uh)‖Hk(T hc )‖vh‖Hd(T h

c ) (5.207)

≤ Ckhk−d‖G(uh)‖Hk(T h

c )‖vh‖Vh .

The other estimates follow by similar modifications compared to above. �

Now we consider the fully nonlinear problem, including the approximate boundaryconditions. In particular, we update the original product projectors and the FEM, see(5.62),(5.63), into the quadrature version, based upon Qh

c and the unchanged boundaryprojector Qh

b and Fh =(G,Bh

D

), cf. Figure 5.6.

Qhc,Π :=

(Qh

c , Qhb

): Vc × Vc,D → Vh′

Π , uh0 ∈ Uh

b : Fh(uh

0

):= Qh

c,ΠFh|Uh

(uh

0

)= 0.

Theorem 5.33. Convergence for different quadrature approximations of fully nonlin-ear problems: Under the conditions of Theorem 5.32 (k > 2(d− 1), d ≥ 2n + 1,Pn

�−1 ⊂Sh, � > 2) with G ∈ CL(Br(Phu0)) and boundedly invertible F ′(u0), this quadratureapproximate FEM Fh : Uh

b → Vh′Π is stable and consistent in Phu0, and has a unique

solution uh0 for Fh(uh

0 ) = 0. It converges, for u0 ∈ H�(Ω) with Ecu0 ∈ H�(Ω ∪ Ωhc ),

p = 2 and > 2 for polyhedral Ωh and curved Ωhc , and ‖uh‖Uh = ‖uh‖H2(T h

c ), cf.(5.67)(5.110), (5.205), Conjectures 4.40, 4.41,∥∥Ecu0 − uh

0

∥∥Uh ≤ Chmin{�−2,p,k−d}‖u0‖Hmin{�,p+2,k−d+2}(Ω), (5.208)

for the nonconforming, and with p, p + 2 deleted in the exponents in (5.208) for theconforming FEM.


Remark 5.34. The combination of the inequalities for k and d requires large valuesfor k, e.g. k > 8 for n = 2 and k > 16 for n = 3. If the stability of Gh is provedvia inverse estimates, as in (5.205), this is unavoidable. Otherwise, the quadratureformulas could be applied to subtriangles of the Tc ∈ T h

c and k could be reduced.

Proof. We show that the as(·, ·) and ahs (·, ·) are simultaneously bounded and, for

small enough h, satisfy the discrete inf–sup condition on Uhb × Vh

b with equiboundedlyinvertible Ah : Uh

b → Vh′. This implies the stability of the corresponding (Gh(u0))′ :

Uhb → Vh′

, and hence of Gh : Uhb → Vh′

in Phu0. By Lemma 5.10 we can assume thediscrete inf–sup condition for as(·, ·) and obtain

ε′‖uh‖Uh ≤ inf0�=vh∈Vh

b

|as(uh, vh)|/‖vh‖Vh ∀uh ∈ Uhb

= inf0�=vh∈Vh

b

∣∣ahs (uh, vh) + (as(uh, vh)− ah

s (uh, vh))∣∣ /‖vh‖Vh

≤ inf0�=vh∈Vh

b

∣∣ahs (uh, vh)

∣∣ /‖vh‖Vh (5.209)

+ sup0�=vh∈Vh

b

∣∣(as(uh, vh)− ahs (uh, vh)

)∣∣ /‖vh‖Vh

≤ inf0�=vh∈Vh

b

∣∣ahs (uh, vh)

∣∣ /‖vh‖Vh + Ch‖uh‖Uh ,

with (5.205) and k > 2(d− 1), so finally

(ε′ − Ch)‖uh‖Uh ≤ inf0�=vh∈Vh

b

∣∣ahs (uh, vh)

∣∣ /‖vh‖Vh . (5.210)

The inequalities for the inf0�=uh∈Uhb

are handled analogously. This shows the discreteinf–sup condition for ah

s (·, ·) for sufficiently small h > 0.The classical consistency error, see (5.205), is combined with the preceding (5.204)

yielding with k > d, u0 ∈ Hk+2(Ω) and Theorem 5.13∥∥FhPhu0 − QhΠFu0

∥∥Vh′

Π

=∥∥(Qh

cG(Phu0),−QhbBDu0

)∥∥Vh′

Π

≤∥∥((Qh

cG(Phu0)−QhcGPhu0

)+ Qh

cGPhu0,−QhbBDu0

)∥∥Vh′

Π

≤ Chk−d‖u0‖Hk+2(Ω) + CLhmin{�,k+2,p+2}−2‖u0‖Hk+2(Ω)

≤ C ′hmin{�,p+2,k−d+2}−2‖u0‖Hmin{�,k+2}(Ω).

and hence, the claimed convergence.


We finally discuss the implications of the two inequalities for k and d.

d ≥ 2n + 1 =⇒ S1d

(T h

c

)⊂ C1(Ω) and k ≥ 2(d− 1) + 1 =⇒

inf–sup condition for ahs (uh, vh), both =⇒ k ≥ 2n+1 + 1 > 0.

This indeed requires the large values for k mentioned above. �

5.4.5 Quadrature FEMs for equations and systems of order 2m

Starting with linear and quasilinear cases we generalize the above set of spaces andapproximations as in (5.177), with the notation in (4.129), (4.136), cf. Figure 5.5,

U = V ⊂Wm,p(Ω,Rq),Uhb ,Vh

b ⊂{L∞ (T h

c ,Rq)

: ∀Tc ∈ T hc : uh|Tc

∈ P, uh

vanishes exactly or approximately along ∂Ωhc

},Pd−1 ⊂ P ⊂ Pd+τ , and

‖ · ‖Uh := ‖ · ‖Vh := ‖ · ‖Hm(T hc ,Rq), with dim Uh

b = dimVhb and (5.211)

p = 2 and p ≥ 2 for linear and quasilinear cases, respectively.

cf. Theorem 5.38 3. for 1 < p <∞. The generalized weak linear FEM form and theprojectors Q

′h are

For Aαβ(x) ∈ Hk,∞(Ω,Rq×q), fα ∈ Hk(Ω,Rq×q) define (5.212)

∑Tc∈T h

c

qhTc

⎛⎝ ∑|α|,|β|≤m

(Aαβ∂

β�uh0 , ∂

α�vh)

q

⎞⎠ =: 〈Ah�uh0 , �v

h〉h and determine �uh0 ∈ Uh

b s.t.

〈Ah�uh0 , �v

h〉h = ah(�uh

0 , �vh)

= 〈�fh, �vh〉h =

∑Tc∈T h

c

qhTc

⎛⎝ ∑|α|≤m

(�fα, ∂α�vh)q

⎞⎠ ∀ �vh ∈ Vhb ,

and Q′h : V → Vh′

, 〈Q′h �f − �f,�vh〉h = ((Q′h �f − �f)�vh)h = 0 ∀�vh ∈ Vh

b .

The previous Theorem 5.28 is generalized to these new linear problems. As aconsequence of the modified spaces in (5.211) the exponent d + τ − 1 in Theorem5.28 has to be replaced by d + τ −m here. So we update (5.184) and (5.185) for order2m and systems in Rq, usually with τ = −1, as

k > max{d + τ −m,

n

2

}for 〈�fh, ·〉

h, �f = (�fα ∈ Hk(. . .))|α|≤m ∈ H−m

k

(Ω ∪ T h

c ,Rq)

k > 2(d + τ −m) for ah(·, ·), Aαβ ∈W k,∞ (Ω ∪ Ωhc ,R

q×q), |α|, |β| ≤ m. (5.213)

On this basis Theorems 5.30 and 5.31 remain verbatim and again guarantee conver-gence of the quadrature approximation.

Theorem 5.35. Consistency of different quadrature approximations of linear equa-tions and systems of order 2m:

1. Assume (5.176), (5.211), and for the quasiuniform subdivision T h the Condition4.16. Choose a quadrature formula as in (5.172), (5.174), satisfying (5.213) for


vector valued functions, 〈�f, ·〉h and ah(·, ·), with (extended) �f ∈ Hk(Ω ∪ T h

c ,Rq),

Aα,β ∈W k,∞ (Ω ∪ T hc ,Rq×q

), �u ∈ Hs(Ω,Rq), s ≥ d, in (5.217).

2. Then, for small enough h > 0, the following consistency results hold

|(�fh, �vh)− ( �fh, �v

h)h| ≤ Chk−(d+τ−m)‖�f‖H−mk (Ω,Rq) · ‖�vh‖Vh , (5.214)

‖Q′h �f − Q′h �f‖Vh′ ≤ Chk−(d+τ−m)‖�f‖H−m

k (Ω,Rq), limh→0

‖Q′h �f‖Vh′ = ‖�f‖V′ , (5.215)

|ah(�uh, �vh)− ah(�uh, �vh)|‖�vh‖Vh

≤ Chk−2(d+τ−m)MAα,β‖�uh‖Vh , for vh �= 0 and

(5.216)

≤ Chk−(d+τ−m)MAα,β‖�uh‖Hs(T h

c ,Rq)

with MAα,β:= max

|α|,|β|≤m‖Aα,β‖W k,∞(Ω,Rq×q),

and|ah(�u,�vh)− ah(Ih�u,�vh)|

‖�vh‖Vh

≤ C(hd−1 + hk−(d+τ−1)MAα,β

)‖�u‖Hs(Ω,Rq).

(5.217)

Theorem 5.36. Convergence for different quadrature approximations of linear equa-tions and systems of order 2m:

1. Let ah(·, ·) induce a stable Ah and assume (5.212) and the conditions in Theorem5.35. Then, simultaneously ah(·, ·) induces a stable Ah. If furthermore (5.213)

is correct, then ah(�uh0 ,

�vh) = 〈�f, �vh〉h∀ �vh ∈ Vhb , defines a stable and convergent

discretization for A�u0 = �f . For the other approximations, mentioned above, thisresult remains correct if the ah, Ah are replaced by good enough approximationsah, ah, Ah, Ah.

2. For conforming FEMs, the above MAα,β, and C∗ := C(1 + MAα,β

), �u0 ∈ Hs(Ω),we obtain∥∥�u0 − �uh

0

∥∥Uh ≤ C∗hmin{s−m,k−(d+τ−m)}(‖�u0‖Hmin{s,k+m}(Ω,Rq) + ‖�f‖Hk(Ω,Rq)).

3. If only approximations for 〈�f, �vh〉h and none for ah(�uh, �vh) are used, then∥∥�u0 − �uh0

∥∥Uh ≤ Chk−(d+τ−m)‖�f‖Hk(Ω,Rq) for k > d + τ −m.

We turn to quasilinear systems with a “quasibilinear” ah(�uh, �vh), where

Aα(·, �uh, . . . ,∇m�uh) : Wm,p(Ω,Rq) → Lp′(Ω,Rq), 1/p + 1/p′ = 1, (5.218)

ah(�uh

0 , �vh)

=∑

Tc∈T hc

⎛⎝ ∑|α|≤m

qhTc

⟨Aα

(·, �uh

0 , . . . ,∇m�uh0

), ∂α�vh

⟩q

⎞⎠ =: 〈Gh�uh0 , �v

h〉h.


The previous estimate (5.214) is not needed, but (5.215) is still needed here, and(5.216), (5.217) have to be reformulated for the quasilinear operator and its linearizedform, providing consistency and stability, hence convergence.

Theorem 5.37. Consistency of different quadrature approximations of quasilinearsystems of order 2m in W s,p(Ω), p ≥ 2:

1. Assume (5.176), the updated (5.211), and for the quasiuniform subdivision T h theCondition 4.16. Choose a quadrature formula as in (5.174), satisfying (5.213),(5.212), (5.218), a differentiable Gh, k > m + n/2, s ≥ d,m, and

1p

+1p′

= 1,∀�u ∈W s,p(Ω) h :=∂Aα

∂ϑβ(·, �u0, ..,∇m�u0) ∈W k,p/(p−2)(Ω), (5.219)

with ‖�u− �u0‖W m,p(Ω) < δ and h < h0.2. Then the following consistency results hold

|ah(�u,�vh)− ah(Ih�u,�vh)| = |〈Gh(�u), �vh〉 − 〈Gh(Ih�u), �vh〉h| ≤ C‖�vh‖Vh (5.220)(hd−m‖�u‖W s,p(Ω,Rq) + hk−(d+τ−m) · max

|α|≤m‖Aα(·, �u, . . . ,Dm�u)‖W k,p′ (Ω,Rq)

),

and for ‖�uh1 − Ih�u0‖Vh < δ :∣∣∣⟨(Gh)′(�uh

1

)�uh, �vh

⟩− 〈(Gh)′

(�uh

1

)�uh, �vh〉h

∣∣∣ ≤ Chk−2(d+τ−m) · (5.221)

max|α|,|β|≤m

{∥∥∥∥∂Aα

∂ϑβ

(·, �uh

1 , ..,∇m�uh1

)∥∥∥∥W k,p/(p−2)(T h

c )

}· ‖�uh‖Vh‖�vh‖Vh .

Theorem 5.38. Convergence of different quadrature approximations for quasilinearsystems of order 2m in W s,p(Ω), p ≥ 2:

1. Under the conditions of Theorem 5.37, and for a locally unique exact solution,�u0, of G(�u) = 0, with boundedly invertible G′(�u0), the FE equation Gh(�uh) =0, Gh = Q

′hGh|Uh in (5.218) has a unique solution �uh0 converging to �u0.

2. With MAα:= max

|α|≤m‖Aα(·, . . . , Dm�u0)‖W k,p′ (Ω,Rq) and for conforming FEMs we

get with �u0 ∈W s,p(Ω,Rq)∥∥�u0 − �uh0

∥∥Uh ≤ Chmin{s−m,k−(d+τ−m)}‖�u0‖Wmin{k,s},p(Ω,Rq)(1 + MAα

). (5.222)

3. With Theorem 4.67 and (4.194), (4.195) analogous results are possible for themonotone operator approach for quasilinear problems even for 1 < p <∞ anddiscrete Wm.p(T h) norms.


Proof. Relation (5.221) implies the simultaneous stability of Gh and Gh. For theconsistency of Gh with G in Ph�u0 we obtain by (5.174)∫

Tc

⟨Aα

(·, �uh

0 , . . . , Dm�uh

0

), ∂α�vh

⟩qdx− qh

Tc

( ⟨Aα

(·, �uh

0 , . . . , Dm�uh

0

), ∂α�vh

⟩q

)≤ Ckh

k∥∥Aα

(·, . . . , Dm�uh

0

)∥∥W k,p′ (Tc)

‖vh‖W k,p(Tc), (5.223)

hence, (5.220). For the stability we have to linearize our quasilinear operator, see(2.459), and get for �uh

1 , �uh, �vh ∈ Vh

b ⊂Wm,p(T h

c ,Rq), and with 〈·, ·〉 := 〈·, ·〉Vh′×Vh ,⟨

∂Gh

(�uh

1

)∂�uh

�uh, �vh

⟩=∑

Tc∈T hc

∫Tc

∑|α|,|β|≤m

⟨∂Aα

∂ϑβ

(x, ..,∇m�uh

1

)∂β�uh, ∂α�vh

⟩qdx.

(5.224)

With the above condition (5.219), we get∣∣∣∣∣⟨∂Aα

∂ϑβ

(·, ..,∇m�uh

1

)∂β�uh, ∂α�vh

⟩q

∣∣∣∣∣W k,1(Tc,R)

≤ C

∥∥∥∥∂Aα

∂ϑβ

∥∥∥∥W k,p/(p−2)(T h

c ,Rq×q)

‖∂β�uh‖W d+τ−|β|,p(Tc,Rq)‖∂α�vh‖W d+τ−|α|,p(Tc,Rq)

We modify (5.223) and estimate∣∣∣∣∣∫T h

c

⟨∂Aα

∂ϑβ∂β�uh, ∂α�vh

⟩q

dx− qh

⟨∂Aα

∂ϑβ∂β�uh∂α�vh

⟩q

∣∣∣∣∣ (5.225)

≤ Chk

∥∥∥∥∂Aα

∂ϑβ

∥∥∥∥W k,p/(p−2)(T h

c )

∑Tc∈T h

c

‖∂β�uh‖W d+τ−|β|,p(Tc)‖∂α�vh‖W d+τ−|α|,p(Tc)

≤ Ckhk

∥∥∥∥∂Aα

∂ϑβ

(·, ..,∇m�uh

1

)∥∥∥∥W k,p/(p−2)(T h

c )

‖∂β�uh‖W d+τ−|β|,p(T hc )‖∂α�vh‖W d+τ−|α|,p(T h

c ).

The inverse estimates show, with (5.212), the stability, hence the claim (5.222). �

Considering the many cases of semilinear equations as special cases of the precedingquasilinear problems with appropriately simplified conditions, the only missing classof elliptic equations are the nondivergent quasilinear and the fully nonlinear equationsand systems of orders 2 and 2m. So we recall the spaces, the extension operators, Ec,the approximate projectors Qh, the strong bilinear forms, nonlinear operators and thegeneralizations, see (5.77), (4.73), (5.78), (5.47), (5.128), cf. Figure 5.6. We replace in(5.200), (5.201), the 2m = 2 and q = 1 by general m and q and obtain, note U �= V,from here on

U ⊂ H2m(Ω,Rq),V ⊂ L2(Ω,Rq),Uhb = Sh = Sh

0 ⊕ Shb ⊂ S2m−1

d

(T h

c ,Rq), (5.226)

Vhb = Sh

0 = Sh ∩ Cm−10

(T h

c ,Rq),Vh

b = Shb , ‖·‖Uh := ‖·‖H2m(T h

c ,Rq), ‖·‖Vh := ‖·‖L2 .


The FE solution �uh0 has to satisfy the quadrature approximate differential operator

�uh0 ∈ Uh

b s.t.〈Gw(·, . . . , D2m�uh

0

), �vh〉hq =

∑Tc∈T h

c

qhTc

(G(�uh

0

)�vh)

= 0∀�vh ∈ Vhb .

(5.227)

For the stability we need the linearized form

ahs (�uh, �vh) =

∑Tc∈T h

c

qhTc

∑|γ|≤2m

⟨[∂ϑγGw(·, �ϑβ , |β| ≤ 2m)

] (�wh

0

)∂γ�uh, �vh

⟩q. (5.228)

So we replace the original approximation Gh = QhcG|Uh , by the quadrature approxi-

mation Gh = QhcG|Uh in (5.227). We start as in Subsection 5.4.4 with Gh, Qh, Qh

c :

Theorem 5.39. Consistency of different quadrature approximations for fully nonlin-ear systems of order 2m:

1. For m, q ≥ 1, k > 2(d−m), hence, k − d > 0, let

∀|γ| ≤ 2m : ∂ϑγ

(Gw(·, �ϑβ , |β| ≤ 2m)

) (�wh

0

)∈W k+|γ|,∞(Ω× RN2m),

cf (5.73), (5.17) (5.5), G : D(G) ∩Hk+2m(Ω) → Hk(Ω) and G : D(G) ∩Hk+2m

(Ω ∪ Ωh

c

)→ Hk

(Ω ∪ Ωh

c

). Choose the approximating Uh

b = Sh ⊃ Vhb =

Sh0 , d ≥ (2m− 1)2n + 1 as in (5.226). Let the quadrature formula satisfy (5.172),

(5.174), and let h > 0 be small enough. Again let, p = 2 for the conforming FEM,and for the nonconforming FEM 2 ≤ p ≤ 5, with p = 2 for polyhedral Ωh andp > 2 for better curved Ωh

c approximations for Ω ∈ C2m, cf. Conjectures 4.40and 4.41.

2. Then (5.202) holds and

‖Qhf − Qhf‖Vh′ ≤ Ckhk−d‖f‖H−m

k (Ω), limh→0

‖Qhf‖Vh′ = ‖f‖H−mk (Ω), (5.229)

|〈G(�uh), �vh〉Vc− 〈G(�uh), �vh〉h| ≤ Chk−d‖G(�uh)‖Hk

c (T hc )‖�vh‖Vh , or∥∥Qh

cG(�uh)− QhcG(�uh)

∥∥V ≤ Chk−d‖G(�uh)‖Hk

c (T hc ), and (5.230)∣∣ah

s (�uh, �vh)− ahs (�uh, �vh)

∣∣ ≤ Chk−2(d−m)‖�uh‖Uh‖�vh‖Vh

× max|γ|≤2m

‖∂ϑγ

(Gw(·, �ϑβ , |β| ≤ 2m)

) (�wh

0

)‖W k+|γ|,∞(T h

c ). (5.231)

3. Again, these inequalities only estimate the quadrature errors for the differentialand not for boundary operators. For the complete consistency errors the termsin Theorem 5.15 and L× the interpolation errors have to be added, cf. Theorem5.40. L denotes the Lipschitz constant for G.

Proof. Since �uh, �vh ∈ S2m−1d

(T h

c

)⊂ Hk

c

(T h

c

), the estimates in (5.174) again can be

applied to different choices of fc. All the above estimates follow as in the proof ofTheorem 5.32, by replacing d− 1 by d−m and updating the coefficients.


The above conditions d ≥ (2m− 1)2n + 1 > 2, n ≥ 2, for S2m−1d

(T h

c

)⊂ C2m−1(Ω)

and k > 2(d−m), imply k > 2(d−m) ≥ 2n+1 > n/2, n ≥ 2, k > d. Hence, by theSobolev embedding theorem the W k,∞(Ω),Hk(Ω) are compactly embedded into C(Ω).So all the quadrature formulas, even for a �u ∈W k,∞(Ω),Hk(Ω), are well defined. �

Now we add the (unchanged) boundary conditions, cf. (5.118), the

BhD�uh

0 :=

(∂j�uh

0

∂νj

∣∣∣∣∂Ωh

c

)m−1

j=0

= �0 ⇔(�vh

b , BhD�uh

0

)Vc,b

= 0 ∀�vhb ∈ Vh

b ⇔ Qhc,bB

hD�uh

0 = �0.

Replacing the previously introduced projectors Qhc,Π = (Qh

c , Qhc,b), by Qh

c,Π =(Qh

c , Qhc,b), we define the discrete quadrature operator and the FE solution �uh

0 ∈ Uhb as

Qhc,Π :=

(Qh

c , Qhb

): Vc × Vc,D → Vh′

Π , Fh(�uh

0

):= Qh

c,ΠFh|Uh

(�uh

0

)= 0 ∈ Vh′

Π .

(5.232)

Theorem 5.40. Convergence of different quadrature approximations for fully non-linear systems of order 2m:

1. Assume the conditions in Theorem 5.39, e.g. k > 2(d−m),Pn�−1 ⊂ Sh, d ≥

(2m− 1)2n + 1, a locally unique solution �u0 ∈ H�(Ω), � > 2m, and boundedlyinvertible F ′(�u0). Choose for the conforming FEMs for m = 1 and a convexpolyhedral domain, cf. (5.67), and for the nonconforming FEM for domains inCp, 2 ≤ p ≤ 5, the p = 2 for polyhedral approximation Ωh and p > 2 for bettercurved Ωh

c approximations for Ω ∈ C2, cf. Conjectures 4.40 and 4.41.2. Then the quadrature approximate FEM Fh : Uh

b → Vh′Π , Fh(�uh

0 ) = 0 in (5.232)has a unique solution �uh

0 , is stable and consistent in Ph�u0, hence convergent,such that, see (5.67),(5.110),∥∥Ec�u0 − �uh

0

∥∥Uh ≤ Chmin{�,p+2m,k−d+2m}−2m‖�u0‖∈Hmin{�,k+2m}(Ω). (5.233)

Proof. The proof of Theorem 5.32 remains essentially unchanged. The uh are turnedinto �uh, etc., and the last inequality, with k > 2(d−m), d > (2m− 1)2n, has to bemodified into∥∥FhPh�u0 − Qh

ΠF�u0

∥∥Vh′

Π≤ C ′hmin{�,p+2m,k−d+2m}−2m‖�u0‖Hmin{�,k+2m}(Ω).

We finally discuss the implications of the two inequalities for k and d. We considerm ≥ 1 and obtain

d ≥ (2m− 1)2n + 1 to guarantee S2m−1d

(T h

c

)⊂ C(2m−1)(Ω) and (5.234)

k ≥ 2(d−m) + 1,m ≥ 1 =⇒ inf–sup condition for ahs (�uh, vh), (5.235)

consequently k − d(5.235)

≥ d− 2m + 1(5.234)> 0. (5.236)

This indeed requires the large values for k mentioned above. �


5.4.6 Two useful propositions

At the end of this section we want to present two results which are important forintegrations and for applications in the final stability proof for nonconforming FEMs.Sometimes we have to rescale the integral over an interval [−1, 1] to [0, h] or a moregeneral reference domain K to T . Examples are (5.171), (5.172) and (5.173). We provethe following result on scaled Sobolev norms and apply it to the interpolation basis. LetP = span{φ1, φ2, . . . , φd} be a basis dual to N , hence Ni(φj) = δij and FT : K → Tthe affine bijection. Then, see Definition 4.8,

φTi = φi ◦ F−1

T = φi ◦ ϕ : T → R, i = 1, . . . , d,∀ T ∈ T h with the affine

ϕ := F−1T : T � t→ x := Bt/h + b ∈ K, B ∈ Rn×n, b ∈ Rn. (5.237)

is called an interpolation basis for Uhb or Vh

b . It is not necessarily a basis for Uhb or Vh

b ,but for a fixed T ∈ T h the φT

i , i = 1, . . . , d are a basis for PT .

Proposition 5.41. Transformation theorem: Let K ⊂ Rn be a reference domain,see Definition 4.1, 0 < h ≤ 1 and ϕ in (5.237) an affine scaling mapping. Chooseu ∈W s,p(K) and α ∈ Nn a multi-index with |α| ≤ s. Then(∫

T

(u ◦ ϕ))

= hn|det(B)|−1

(∫K

u

), and

(∫T

|∇α(u ◦ ϕ)|p) 1

p

= hn/p−|α|(∫

K

|∇αu|p) 1

p

, for B = I, else

(∫T

|∇α(u ◦ ϕ)|p) 1

p

≤ hn/p−|α|‖B‖|α||det(B)|−1/p

(∫K

|∇αu|p) 1

p

, (5.238)

specifically |u ◦ ϕ|W s,p(T ) ≤ C hn/p−s(max{1, ‖B‖})s|det(B)|−1/p|u|W s,p(K),

‖u ◦ ϕ‖W s,p(T ) ≤ C hn/p−s(max{1, ‖B‖})s|det(B)|−1/p‖u‖W s,p(K) and∥∥φTi

∥∥W s,p(T )

≤ C hn/p−s(max{1, ‖B‖})s|det(B)|−1/p‖φi‖W s,p(K),

with∥∥φT

i

∥∥H1(Ωh)

≤ C hn/2−1 dmaxj=1

{‖φj‖H2(K)},

so the∥∥φT

i

∥∥H1(T h)

are bounded for n ≥ 2. Defining a quadrature formula as in (5.172)for T , we start with a quadrature formula for K with the corresponding wj , Pj. Thenthe Pj,T := FT (Pj) and the wj,T := hn|det(B)|−1wj .

We omit the straightforward proof, but prove that for fh ∈ Vh′there exists an appro-

priate fh0 ∈ V ′ such that fh = Q

′hfh0 . By Theorem 2.35, (2.110f), (2.108) we know that

fh has the form (5.239) with bounded ‖fj‖Lp′ (T hc )=Lp′ (Ω), so f ∈W−1,p′

(Ω) = V ′.We need this property for the later proof of stability of nonconforming FEMs for


fh = Ahuh, for Q′h cf. (5.294).

〈fh, vh〉 :=∑

Tc∈T hc

∫Tc

(n∑

i=0

fi∂ivh

)dx ∀vh ∈ V ∪ Vh

b , fh ∈W−1,p′ (T h

c

), (5.239)

fj ∈ Lp′ (T hc

),with Q

′h : V ′ → Vh′, 〈Q′hf − f, vh〉 = 0 ∀vh ∈ Vh

b ∈W 1,p(Ω).

Proposition 5.42. Assume the local and nonlocal interpolation operators Ih inDefinition 4.8 (4.67) are well defined and allow the corresponding interpolation bases,the φT

i in (5.237) and let V = W 1,p(Ω). Then

∀fh in (5.239), ∃ fhe ∈W−1,p′

(Ω) ⊂ V ′, 1 ≤ p′ ≤ ∞, s.t. fh = Q′hfh

e , so⟨fh

e , vh⟩

= 〈fh, vh〉∀vh ∈ Vhb , lim

h→0

(‖fh‖Vh′ =

∥∥Q′hfhe

∥∥Vh′

)=∥∥fh

e

∥∥V′ . (5.240)

Proof. We aim for fhe ∈W−1,p′

(Ω) ⊂ V ′, 1 ≤ p, p′ ≤ ∞, 1/p + 1/p′ = 1 for fh in(5.239) with Vh ⊂W 1,p

(T h

c

). With the restrictions vh|Tc

and fhe |Tc

, we have

vh ∈ Vh ⊂W 1,p(T h

c

), fh ∈W−1,p′ (T h

c

)⇒ ∂ivh|Tc

∈ Lp(Tc), fi|Tc∈ Lp′

(Tc),

∀T ∈ T hc , i = 0, . . . , n. For this type of fh ∈W−1,p′ (T h

c

)⊂ Vh′

the fi|Tc∈ Lp′

(Tc)imediately extend to an fi,e ∈ Lp′

(Ω) s.t. fi,e|Tc∈ Lp′

(Tc). Then obviously theextended ⟨

fhe , v⟩

:=∫

Ω

(n∑

i=0

fi,e∂iv

)dx ∈ V ′

satifies (5.240). �

5.5 Consistency, stability and convergence for FEMs withvariational crimes

5.5.1 Introduction

Similarly to the conforming FEMs in Section 4.3 we restrict the discussion hereto U = V,Ub = Vb,Uh

b = Vhb . These we had exactly evaluated the weak bilinear and

higher nonlinear forms, e.g. a(·, ·) : Vh × Vh → R. So they are well defined and exactlyavailable. However, V = H1(Ω) for m = 1 requires Vh ⊂ C(Ω) (and V = H2(Ω) form = 2 even Vh ⊂ C1(Ω)), so often too restrictive conditions. We have to allowviolations of these conditions. The Vh

b �⊂ Vb, admit violated boundary conditions andVh

b �⊂ V violated continuity (and differentiability) across the boundary edges, e ofT ∈ T h. These are the so-called nonconforming FEs, see cases (1), (2) in Subsection5.5.2. Except partially in Subsection 5.5.7, we consider T h with curved triangles atthe boundary. So we again use the notation T h

c . As for quadrature approximations,the goal here is fitting the violated boundary and continuity conditions, such thatthe original convergence rate is not invalidated. This will motivate the appropriate

5.5. Consistency, stability and convergence for FEMs with variational crimes 369

choice of parameters for these methods. We will come back to this point in (5.325)ff. below. Before we discussed P2

d−1 ⊆ P ⊆ P2d+τ with τ ≥ −1. This allowed including,

as the most important case, the tensor product FEs Qnd , cf. (4.13) with τ = d− 2.

The following convergence results would remain correct for τ > −1, or even forτ = d− 2 only if unrealistic conditions were imposed. So we restrict the FEs, studiedhere, to

Vh ⊂{v : Ω ⊂ R → R2 : v|T ∈ P = P2

d−1 ∀ T ∈ T hc

}, Ω = ∪T∈T h

cT (5.241)

the piecewise polynomials of degree d− 1. For reasons discussed below, we restrict thediscussion in this section to R2 to, except for isoparametric FEs.

We unify the presentation again by using the same symbols u, uh, . . . ∈ H1(Ω), . . .instead of ∈ H1(Ω,Rq) and, as coefficients, aij instead of Aij , for real and vectorvalued functions as in (5.285).

Already in the last Sections 5.2–5.4 we studied different types of nonconformingFEMs. For fully nonlinear problems the available C1 FEs are not able to satisfy theboundary conditions exactly. In fact, we had to introduce an Ωh

c ≈ Ω for the new FEs.We will similarly discuss violated boundary conditions in Subsection 5.5.6 ff., andisoparametric FEMs in Subsection 5.5.10. In the Subsections before 5.5.10 we will,however, only study polynomial FEs, defined on a subsdivision of the original domainΩ or a curved approximation Ωc. For more complicated data the integrals in the FEequations have to be computed by quadrature or more general approximations as inthe previous Section 5.4.

The interplay between the weak and strong operators yields an essential tool forhandling nonconforming FEMs. Here we start with linear operators, estimating theviolation of Vh

b �⊂ Vb or Vhb �⊂ V. Later on we will extend this approach to nonlinear

problems as well. We systematically indicate by appropriate indexing, the weakand strong operators as A,Ah and As, A

hs , and the corresponding bilinear forms as

a(·, ·), ah(·, ·), ah(·, ·) and as(·, ·), as,h(·, ·), ahs (·, ·).

The following technique in Subsections 5.5.6 and 5.5.7 is a generalization of Brennerand Scott [141]. It extends their results for the Laplacian in R2 to general second orderlinear and quasilinear equations and systems. We systematically develop high accuracyemploying the high order Gauss-type quadrature formulas along the one-dimensionaledges. They exist no longer for the two-dimensional edges, e, of T ∈ T h ⊂ R3. Soeven for the few examples of two-dimensional cubature rules of higher order, thecorresponding FEMs lose much of their higher order convergence. So we omit themhere. The R2 results are closely related to the conditions for patch tests as discussedby Stummel, Shi et al. [583–585,587,611].

High accuracy FEMs with variational crimes are important for path-following ofparameter-dependent solutions of nonlinear problems and the study of turning andbifurcation points, in particular for avoiding spurious solutions. Furthermore, we willuse a similar approach for the spectral methods in [120].

Other possibilities are the DCGMs and the direct tensor product versions forrectangular subdivisions and collocation due to Bialecki and Fairweather [95], notdiscussed here.


We start with the Helmholz operator, Example 5.44 in Subsection 5.5.2, see Example4.45. This simple case motivates deriving the necessary extensions for bilinear formsand operators and are discuss the consequences of the nonconformity for the bilinearform. We elaborate the impact of variational crimes via the difference of the weakand strong bilinear forms and prove the basic Strang Lemma 5.52. In Subsection 5.5.3we generalize these results to second order linear and quasilinear problems as appro-priate general discretization concepts. These are again generalized Petrov–Galerkinmethods.

Subsection 5.5.4 shows the coercivity of the principal part as a first step towardsstability. It formulates, as a guideline for the further subsections, the variationalconsistency error for linear problems. The variational consistency errors are estimatedvia the quadrature errors of quadrature formulas applied to the edges of neighboringtriangles. So we introduce in Subsection 5.5.5 different types of high order Gaussquadrature formulas. Now we are ready to estimate, in Subsections 5.5.6 and 5.5.7,the classical and variational consistency for violated bondary conditions and continuityin linear and quasilinear problems. In Subsection 5.5.7 and for linear problems, theAubin–Nitsche trick for increasing the order of convregence by one with respect to theL2(Ω) norm is applied. By combining the results from the last three subsections, weprove stability for boundedly invertible linear operators with consistent discretizationin Subsection 5.5.8. We use here an anticrime transformation for a modification of theoriginal proof in Section 3.5. The next subsection, 5.5.9, proves convergence for theexact FEMs and their quadrature approximations for linear and quasilinear problems.For efficiently solving the nonlinear equations, we again apply the mesh independenceprinciple for our discrete problems. The last Subsection 5.5.10 considers isoparametericFEMs and strongly extends them to quasilinear problems.

In this section we assume that

Ω ⊂ R2 is a bounded domain, and the solution is u0 ∈ H2(Ω). (5.242)

For nonsmooth u0 �∈ H2(Ω) or nonconvex Ω, singular adapted FEs are added Vh. Thenthe strong regularity conditions can be reduced and still the usual high accuracy canbe guaranteed, for details cf., e.g. Blum and Rannacher [98].

5.5.2 Variational crimes for our standard example

Often, these nonconforming FEMs are only studied for cases, particularly interestingfor applications. This holds, e.g. for the results of Graham, Hackbusch and Sauter,[361–364] Their results, relevant for our approach, are summarized in Subsection 4.2.5.On this basis it is possible, generalizing the following results for nonconforming FEMson quasiuniform to more general triangulations.

Remark 5.43. Again, the condition of a quasiuniform subdivision, T h, in this and thefollowing theorems, can be relaxed, to more general triangulations cf. Subsection 4.2.5.Here we study only FEMs in R2, and require Assumptions 4.21 and 4.26. Again in themessy sums below,

∑T∈T h , the terms have to be estimated for each T ∈ T h separately.


The results in Theorems 4.30–4.33 allow appropriate estimates for continuous and L2

FEs. We leave the details to the interested reader.

Here we intend a discussion of general nonconforming FEMs, where we imposeCondition 4.16.

For presenting the essental ideas more clearly, we start studying, for our simpleExample 4.45, the impact of violated boundary conditions and continuity. Later on,we have to generalize linear and bilinear forms, operators and projectors, similarly toSections 5.2 and 5.4.

Necessary extensions towards variational crimes

Example 5.44. For Dirichlet boundary conditions, u ∈ H10 (Ω), f ∈ H−1(Ω), and the

Helmholtz operator we memorize, see (2.2) ff, (4.107), (4.109), (5.242), the boundedoperators and the corresponding bilinear forms and obtain by partial integration

A : V := H1(Ω) → H−1(Ω), As : H2(Ω) → L2(Ω) bounded, and for

u ∈ H2(Ω) : a(u, v) :=∫

Ω

〈∇u, ∇ v〉+ cuvdx =: 〈Au, v〉H−1(Ω)×H1(Ω) (5.243)

= as(u, v) := (Asu, v)L2(Ω) =∫

Ω

(−Δu + cu)vdx ∀v ∈ Vb := H10 (Ω).

The weak and FE solutions u0 ∈ Vb = H10 (Ω) and uh

0 ∈ Vhb ⊂ Vb are defined by

a(u0, v) = 〈f, v〉H−1(Ω)×H10 (Ω)∀v ∈ Vb (5.244)

a(uh

0 , vh)

=∫

Ω

(∇uh

0 , ∇ vh)2

+ c uh0 v

h dx = 〈f, vh〉... ∀ vh ∈ Vhb ⊂ Vb, (5.245)

with u0 ∈ H2(Ω) for f ∈ L2(Ω),Ω ∈ C2, and the inner product (·, ·)2 in R2. Indeed, byTheorems 2.43 and 2.45 u0, u

h0 uniquely exist for c ≥ 0. Obviously the integral is well

defined for conforming FEs Vhb ⊂ H1

0 (Ω). But here we want to study nonconformingfinite elements. �

Nonconforming finite elements

1. Finite elements violating the boundary conditions:In Section 5.2 we have discussed this problem of nonconforming FEs alreadyfor fully nonlinear problems. It is always possible to apply this technique tolinear and quasilinear operators and their nonlinear forms. However, here we aimfor the higher accuracy obtainable with polynomial FEs on curved domains inSubsection 4.2.7. For these FEs we recall (4.97), Ω = ∪T∈T h

cT and restrict the

discussion to homogeneous Dirichlet boundary conditions. So the FEs uh : Ω → Rsatisfy uh(Pj) = 0 only at certain points Pj ∈ ∂Ω or even only close to ∂Ω forcurved boundaries by (4.98). For homogeneous Dirichlet boundary conditionsBv = v|∂Ω = 0 , this implies

Vhb = {uh ∈ Vh ⊂ H1(Ω) = V : B(uh) ≈ 0},H1

0 (Ω) = Vb �⊃ Vhb , (5.246)


This implies that (5.243) is violated if u, v are replaced by uh, vh, yielding anestimate for the error due to violated boundary conditions in the correspondingFEM. For natural boundary conditions, only specific test functions vh ∈ Vb �⊃Vh

b ⊂ V will cause problems. But the original bilinear a(·, ·) and linear forms 〈f, ·〉are defined in Vh × Vh and Vh. We postpone isoparametric FEs to Subsection5.5.10.

2. Finite elements violating the continuity conditions:Let Vh

b �⊂ H1(Ω), e.g. we have discontinuous FEs, or more generally, cf. (5.300)

Vhb �⊂ V e.g. discontinuous across the edges e. (5.247)

This happens for nonconforming finite elements, e.g. the Crouzeix–Raviart FE,see Figure 4.10 [135,141]. So, the uh are continuous only in all the, say k, affinelyequivalent points P e

j ∈ e ⊂ T ∈ T hc on every e. Here k is so small that uh

l (Pj) =uh

r (Pj), j = 1, . . . , k, allows (uh|Tl)|e �≡ (uh|Tr

)|e. Usually, even for curved ∂Ω, thenumber of points Pj ∈ e ⊂ T on the edges interior to Ω and along ∂Ω is the same.Then violated continuity implies violated boundary conditions as well.

In these so-called nonconforming FEMs, (1), (2), we study polynomial FEs,

FE spaces with Vhb �⊂ Vb or Vh

b �⊂ V, (5.248)

uh|T , vh|T ∈ P ∀uh, vh ∈ Vh, T ∈ T hc with Pd−1 = P, of fixed degree,

with a sequence{Vh

b

}h∈H

indicated by h, 0 < h ∈ H, infh∈H h = 0.We combine these crimes with approximations for projectors, operators and linear,

bilinear and higher nonlinear forms, mainly by quadrature, cf Section 5.4. In thissubsubsection we mainly study the impact of nonconforming FEs onto necessarymodifications of the linear and bilinear forms.

For the following results we memorize, see Subsection 4.2.4: the vh ∈ Vh usually arediscontinuous across common edges e ⊂ T ∩ T1 of neighboring T, T1 ∈ T h

c , and hencevh �∈ H1(Ω), [174], pp. 207–208, or for short

vh �∈ C(Ω) ⇒ vh �∈ H1(Ω) (but C(Ω) �⊂ H1(Ω)). (5.249)

The original linear, bilinear and higher nonlinear forms, e.g. 〈f, ·〉 and a(·, ·), as(·, ·),have to be modified for the nonconforming FEs. This implies that the modified (5.243)is violated for uh, vh. We will estimate the errors due to nonconforming FEMs, causedby Vh

b �⊂ H10 (Ω) or �⊂ H1(Ω). As in Section 5.4 the 〈f, ·〉, a(·, ·), as(·, ·), A,As, . . . ,

have to be defined piecewise according to (5.251). We start the discussion, assum-ing Ω = Ωh = ∪T∈T h

cT . We summarize these cases, cf. (5.178), under the notation

vh ∈ Vh ⊂W 1,p(T h

c

), fh ∈W−1,p′ (T h

c

), . . . , with Lp(Ω) = Lp

(T h

c

). For the

important cases f ∈ V ′, fh ∈W−1,p′ (T hc

), including Ahu

h, we obtain the general form


for f ∈ V ′, and

Ω = Ωh = ∪T∈T hcT , 1/p + 1/p′ = 1, V = W 1,p(Ω), f ∈ V ′ ⊂W−1,p′ (T h

c

):

〈f, v〉W−1,q(Ω)×W 1,p(Ω) =∫

Ω

(f−1, ∇ v)2 + f0v dx, cf. Propos. 2.34, extended

〈fh, vh〉W−1,q(T h

c )×W 1,p(T hc ) :=

∑T∈T h

c

∫T

(f−1, ∇ vh)2 + f0vhdx (5.250)

for vh ∈ V + Vh, fh ∈ V ′ + Vh′, with f = fh|V , fh := fh|Vh .

Correspondingly to f , we define the following bounded extensions of the weak andstrong bilinear forms and the operators ah(uh, vh), Ahu

h, and as,h(uh, vh), As,h. Fora simplified presentation we use the Hilbert–Sobolev version with p = 2 for linearproblems:

ah(uh, vh) :=∑

T∈T hc

∫T

(∇uh ,∇ vh)2 + c uh vh dx =: 〈Ahuh, vh〉H−1(T h

c )×H1(T hc )

∀uh, vh ∈ V + Vh,with a(·, ·) = ah(·, ·)|V×V , ah(·, ·) := ah(·, ·)|Vh×Vh

(5.251)

and as,h(uh, vh) :=∑

T∈T hc

∫T

(−Δuh + c uh )vh dx =: (As,huh, vh)L2(T h

c ) (5.252)

∀uh ∈ H2(Ω) ∩ Vh,∀vh ∈ L2(Ω) + Vh

with as(·, ·) = as,h(·, ·)|H2(Ω)×L2(Ω), ahs (·, ·) = as,h(·, ·)|Vh×Vh .

Remark 5.45. We prevent a well-motivated misunderstanding: obviouslythe ah(u, vh), as,h(u, vh) are well defined for u ∈ Vb. Now, ah(u, vh) �=〈Au, vh〉H−1(T h

c )×H1(T hc ) ∀ vh ∈ Vh

b for the variational crimes with Vhb �⊂ Vb. This

inequality will be specified in Proposition 5.47. Note that 〈Au, vh〉H−1(T hc )×H1(T h

c )

always makes sense for u ∈ H1(Ω): the f := Au ∈ V ′ according to (5.243), is definedby testing ∀ v ∈ Vb. This f induces, by (5.250), an fh, now tested ∀ vh ∈ Vh

b ∪ Vb anda fh, now tested ∀ vh ∈ Vh

b . So for u ∈ H1(Ω) the Ahu is certainly well defined by

ah(u, vh) = 〈Ahu, vh〉H−1(T h

c )×H1(T hc ),but for variational crimes (5.253)

�= 〈Au, vh〉H−1(T hc )×H1(T h

c ) ∀ vh ∈ Vhb or Q

′hAu �= Q′hAhu,

for the projectors Q′h introduced in (5.294) below. This situation is essential for the

case of the later classical consistency errors.

Impact of variational crimes for the standard example

The relation between strong and weak bilinear forms is the essential tool for studyingthe impact of variational crimes in FEMs. They are reflected by the discontinuity


of the FEs crossing the interior edges e ⊂ T ∈ T hc and violated boundary conditions

along the boundary edges e ⊂ ∂T hc .

Generalizing (5.245), we determine the (weak) discrete solution as

uh0 ∈ Vh

b : ah(uh

0 , vh)

=⟨Ahu

h0 , v

h⟩

H−1(T hc )×H1(T h

c )= 〈fh, vh〉··· ∀ vh ∈ Vh

b ,

with Vhb �⊂ H1

0 (Ω) or �⊂ H1(Ω). (5.254)

cf. (5.248). For natural boundary conditions only Vhb , �⊂ H1(Ω) is problematic.

Similarly as in Section 5.2 we now systematically study the relation between weakand strong operators and forms, here for our standard example. We apply the firstGreen’s formula (2.23) to every T or T = Ω finding∫

T

−v Δu dx =∫

T

(∇u,∇ v)2dx−∫

∂T

v∂u

∂νds. (5.255)

This implies for the exact bilinear forms, see (5.243) for c �= 0 as well,

as(u, v) = a(u, v) ∀ u ∈ H2(T ), and either ∂uh/∂ν|∂T = 0 or v ∈ H10 (T ). (5.256)

Hence, (5.244), and a corresponding strong problem as(u0, v) = (f, v)∀ v ∈ Vb wouldhave identical solutions u0 ∈ H2(Ω), e.g. for f ∈ L2(Ω),Ω ∈ C2. This is satisfied foreither natural or Dirichlet boundary conditions.

For the extended bilinear forms ah(u, v) and as,h(u, v) in (5.251) and (5.252), nowfor c = 0, we apply the Green’s formula (5.255) for every T ∈ T h

c . With the surfaceintegrals

∫e· · · ds, e ∈ T and νe indicating the outer normal vector for T , perpendicular

to e, we get ∀u ∈ (V + Vh) ∩H2(T hc ), v ∈ V + Vh,

as,h(u, v) =∑

T∈T hc

∫T

−vΔudx =∑

T∈T hc

(∫T

(∇v,∇u)2dx−∑e∈T

∫e

v(∇u, νe)2ds)

= ah(u, v)−∑

T∈T hc

∑e∈T

∫e

v(∇u, νe)2ds

with BΔu : = (∇u, νe)2 = ∂u/∂νe. (5.257)

Obviously (5.257) reduces to (5.243) for u ∈ (V ∩H2(Ω)), v ∈ Vb :In (5.257) the surface integral

∫e

for every interior edge e ∈ T will be obtainedtwice with opposite normal directions for neighboring Tr , Tl ∈ T h

c and e ⊂ Tr ∩ Tl.The edges38 e ∈ ∂Ω will be summed up once.

We consider the transition from a triangle Tl to its neighboring Tr. Let ν = νl = νe

be the outward normal for e as the face of Tl. Then νr = −νl is the outward normalfor e as the face of Tr. We denote the restriction of v, analogously for ∇u, (∇u, νe)2,to Tl, Tr, and their extension to Tl, Tr as vl, vr, ∇ul, (∇u, νe)2. We introduce thestandard notation, see [665],

vl = v|Tl, vr = v|Tr

, [v] := vl|e − vr|e and {v} := (vl|e + vr|e)/2, (5.258)

38 We use the notation e ∈ ∂Ω, although either e ⊂ ∂Ω or even less for curved bouindaries, similarlye ∈ T h

c \ ∂Ω.


for the corresponding jumps and arithmetic means of v across an interior e, and

[v] := {v} := v|∂Ω along ∂Ω, and v arbitrary in R2\Ω.

For (5.257) we need, with e ∈ T hc \ ∂Ω or e ∈ ∂Ω, cf. [665], p. 153, [141], p. 206,∑

T∈T hc

∑e∈T

∫e

v∂u/∂νeds =∑

T∈T hc

∑e∈T

∫e

v(∇u, νe)2ds (5.259)

=∑

e∈T hc \∂Ω

∫e

(vl(∇ul, νl)2 + vr(∇ur, νr)2) ds +∑

e∈∂Ω

∫e

v(∇u, νe)2ds

=∑

e∈T hc \∂Ω

∫e

(vl(∇ul, νe)2 − vr(∇ur, νe)2) ds +∑

e∈∂Ω

∫e

v(∇u, νe)2ds

=∑

e∈T hc \∂Ω

∫e

([v]({∇u}, νe)2 + {v}([∇u], νe)2) ds +∑

e∈∂Ω

∫e

v(∇u, νe)2ds,

since ([v]{∇u}+ {v}[∇u]) (5.260)

=12(vl − vr)(∇ul +∇ur) +

12(∇ul −∇ur)(vl + vr) = (vl∇ul − vr∇ur).

This yields the following relation between the weak and strong bilinear forms

as,h(u, v) = ah(u, v)−∑

e∈T \∂Ω

∫e

([v]({∇u}, νe)2 + {v}([∇u], νe)2) ds

−∑

e∈∂Ω

∫e

v(∇u, νe)2ds ∀u ∈ (V + Vh) ∩H2(Ω), v ∈ V + Vh. (5.261)

We will mainly estimate the absolute value of the last messy sums for u, v ∈ Vhb ,

see below and Subsubsections 5.5.2 and 5.5.4. So we can choose any one of the twodifferent νT = νTl

or νT = νTr. The following relations for the as(u, v), a(u, v) and

ahs (u, v), ah(u, v) are basic to study the impact of variational crimes. This (5.261)

implies several cases: for u ∈ Vb ∩H2(Ω), v ∈ Vb we find again as(u, v) = a(u, v). Moreinteresting are the uh, vh ∈ Vh

b and we obtain from (5.261)

Proposition 5.46.

1. For our standard Example 5.44, we obtain for the bounded ah(uh, vh) andas,h(uh, vh) in (5.251) and (5.252), respectively, now with ν = νe,

as,h(uh, vh) = ah(uh, vh)−∑

e∈T \∂Ω

∫e

([vh]({∇uh}, ν)2 + {vh}([∇uh], ν)2

)ds

−∑

e∈∂Ω

∫e

vh(∇uh, ν)2ds ∀uh ∈ (V+ Vh) ∩H2(T hc ), vh ∈ V + Vh.

(5.262)


2. More generally, if the Vhb satisfy exact Neumann and Dirichlet boundary con-

ditions, respectively, then vh(∇uh, ν)2 = 0 ∀ e ∈ ∂Ω, uh, vh ∈ Vhb and the last

term∑

e∈∂Ω

∫evh(∇uh, ν)2ds in (5.262) vanishes.

Three special cases are important. Relation (5.262) simplifies by (1.67) as

ah(uh, vh) = ahs (uh, vh)−

∑e∈∂Ω

∫e

vh(∇uh, ν)2ds (5.263)

for vh ∈ Vhb ⊂ H1(Ω) and uh ∈ Vh

b ∩H2(Ω),

ah(uh, vh) = ahs (uh, vh)−

∑e∈T h

c \∂Ω

∫e

vh([∇uh], ν)2ds (5.264)

−∑

e∈∂Ω

∫e

vh(∇uh, ν)2ds for vh ∈ Vh ⊂ H1(Ω),

ah(uh, vh) = ahs (uh, vh) for vh ∈ Vh

b ⊂ H10 (Ω), uh ∈ Vh

b ⊂ H2(Jhc ). (5.265)

For the different types of variational crimes we combine (5.262)–(5.265), with thenonconformity Vh

b �⊂ Vb and Vhb �⊂ V.

Our final goal is an estimate for∥∥u0 − uh

0

∥∥Vh for the exact and discrete solutions u0

and uh0 , respectively. For general discretization methods, the classical and variational

consistency errors are relevant. If ah(·, ·) satisfies an inf–sup condition, this is achievedby estimates for ah

(u0 − uh

0 , vh)

or ah(u− uh, vh), cf. Subsection 5.5.4, based uponthe following proposition.

Proposition 5.47. For FEs with violated boundary conditions and/or continuitychoose an arbitrary u ∈ H2(Ω), fu := Au ∈ L2(Ω) and the corresponding discrete solu-tion Ahuh = Q

′hAu, e.g. exact and the discrete solutions u0 ∈ H2(Ω) and uh0 for

(5.244) and (5.254). Then we obtain, in particular for (u, uh) =(u0, u

h0

),

〈Ahu, vh〉H−1(T h

c )×H1(T hc ) − 〈Au, vh〉H−1(T h

c )×H1(T hc )

= 〈Ahu−Au, vh〉...

= ah(u− uh, vh) =∑

e∈T hc \∂Ω

∫e

[vh]∇u · νds +∑

e∈∂Ω

∫e

vh(∇u, ν)2ds. (5.266)

Remark 5.48. For variational crimes (5.263)–(5.265)and (5.266) imply ahs (uh,

vh) �= ah(uh, vh) and ah(u0 − uh0 , v

h) �= 0. So for small errors, hence a close rela-tion between the ah(·, ·) and ah

s (·, ·) : Vhb × Vh

b → R, the FEs have to be chosen suchthat along the common edges e the [vh] and on ∂Ω the vh disappear sufficientlyoften. Choosing these zeros among appropriate quadrature points, the correspondingquadrature rules applied to the edges e disappear and consequently the integrals, e.g.∫

evh([∇uh], ν)2ds, are very small. Extensions to R3 are possible using results as in

Cools [208–210]. Since these formulas have only lower order of converge, and since weconsider DCGMs in Chapter 7, we do not discuss these modifications here.


Remark 5.49. In Section 5.5.4 we will estimate the consistency errors introduced inDefinition 3.14. The standard form, for appropriate p, k, q, will be of the form

sup0�=vh∈Vh

b

{|ah(Phu− u, vh)|/‖vh‖Vh} = Cμhp‖u‖W kq (Ω). (5.267)

Now this u ∈W kq (Ω) is much smoother than the usual u ∈ V = H1(Ω) ⊃W k

q (Ω).Nevertheless we get vanishing consistency errors for u ∈ H1(Ω) as well. For a givenu ∈ V = H1(Ω) and any ε > 0 we have to show that

|ah(Phu− u, vh)|/‖vh‖Vh < ε ∀h < hε.

We choose us ∈W kq (Ω) such that ‖u− us‖V < C ′δ and determine δ such that the

previous < ε is satisfied. We combine (5.267), the boundedness of ah and (4.42) to get

|ah(Phu− u, vh)|≤ |ah(Phus − us, v

h)|+ |ah(Phu− Phus, vh)|+ |ah(us − u, vh)|

≤ |ah(Phus − us, vh)|+ C ′′‖Phu− Phus‖Vh‖vh‖Vh + C ′′‖us − u‖Vh‖vh‖Vh

≤(Cμhp‖us‖W k

q (Ω) + C ′′(CC ′ + 1)δ)‖vh‖Vh

≤ ε‖vh‖Vh for sufficiently small h, δ.

Due to the testing of the exact and discrete solutions u0, uh0 by v ∈ Vb, v

h ∈ Vhb

with Vhb �⊂ V,Vb, this argument is not sufficient for variational crimes.

Therefore, violated continuity and boundary conditions and approximations implymore or less complicated errors. The corresponding estimates for these nonconform-ing FEMs are highly technical. We will study the variational consistency errors inSubsections 5.5.4 ff. A first step in this direction is to update Proposition 5.47 forthe generalized elliptic operators considered in (2.135). We only have to replace the∇u0 · ν in (5.269) by Bau0. This yields

Proposition 5.50. For FEs with violated boundary conditions and/or continuity theexact solution u0 ∈ H2(Ω) and the discrete (weak) solutions uh

0 ∈ Vhb are related by

ah

(u0 − uh

0 , vh)

=∑

e∈T hc \∂Ω

∫e

[vh]Bau0ds +∑

e∈∂Ω

∫e

vhBau0ds ∀vh ∈ Vhb . (5.268)

Remark 5.51. For evaluating these ∂u/∂ν|∂Ω or Bau|∂Ω, and formulating the non-conforming FEMs, the situation is more complicated. An arbitrary u ∈ H1(Ω) doesnot allow a meaningful restriction to ∂Ω or to e to obtain ∂u/∂ν ∈ L2(∂Ω) and∂u/∂ν ∈ L2(e). We omit the reference to Bau|∂Ω. There are two ways out of thisdilemma. Either Theorems 1.37, 1.39 are used for replacing the above u ∈ H1(Ω) byu ∈ H3/2+ε(Ω) with any ε > 0. Then ∂u/∂ν|∂Ω ∈ Hε(∂Ω) ⊂ L2(∂Ω) or ∂u/∂ν|e ∈L2(e) is valid. Hence,

∫∂Ω

v∇udS, and∫

ev∇udS, are well defined, cf. (7.42). Or

u ∈ H1(Ω) is a function in the domain, e.g. of the Laplacian; then again∫

ev∇udS are

well defined. This problem of boundary traces has attracted some attention recently,cf. Taylor [618], Chapter 4, Proposition 4.5, and Jonsson and Wallin [425], Jerison,


and Kenig [419], Schwab [575], and Grisvard [373]. Readers not so familiar withthe H3/2+ε(Ω) results can in this chapter nearly everywhere replace H3/2+ε(Ω) byH2(Ω), with ∂u/∂ν|∂Ω ∈ L2(∂Ω) and ∂u/∂ν|e ∈ L2(e), thus requiring a bit more thannecessary.

Proof. For u ∈ H2(Ω) we modify (5.262) and use∫

e{vh}([∇u], ν)2ds = 0 along edges

e ∈ T hc finding

〈Ahu−Au, vh〉... = 〈Ahu, vh〉... − 〈Au, vh〉... with Ahuh = Q

′hAu

(5.251)= ah(u, vh) − ah(uh, vh)

u incorrectly – uh correctly tested by vh ∈ Vhb

= ah(u− uh, vh) (5.269)

(5.254)=

∑T∈T h

c

∫T

(∇u ∇ vh + c u vh )dx− ah(uh, vh)

(5.262)=

∑T∈T h

c

∫T

(−Δu + c u )vh dx

+∑

e∈T hc \∂Ω

∫e

({vh}([∇u], ν)2 + [vh]({∇u}, ν)2)ds

+∑

e∈∂Ω

∫e

vh(∇u, ν)2ds− ah(uh, vh)

u∈H2(Ω)= (f, vh)L2(Ω) − (f, vh)L2(Ω) +∑

e∈T hc \∂Ω

∫e

[vh]({∇u}, ν)2ds +∑

e∈∂Ω

∫e

vh(∇u, ν)2ds,


In Subsection 5.5.4 we will estimate the terms in the last sum for general problemsand thus estimate ah

(u0 − uh

0 , vh). That these estimates are crucial is shown in

the following Strang lemma, generalizing the Cea Lemma 4.48, see [135, 141]. Forgeneral cases we will proof convergence via the results of the general discretizationtheory in Chapter 3. Nevertheless, we include Strang’s proof for the convergencelimh→0

∥∥u0 − uh0

∥∥H1(Ω)

→ 0. It essentially applies to “coercive” bilinear forms.

Lemma 5.52. Strang lemma: Violated boundary and/or continuity conditions:Let Vh

b �⊂ Vb, and/or Vhb �⊂ V, hence the vh, uh ∈ Vh

b violate the Dirichlet boundaryconditions vh|∂Ω = 0 and the Neumann boundary conditions ∇uh|∂Ω = 0, and/or arediscontinuous. Let the bounded a(·, ·) and for more general cases let a(·, ·) satisfy aninf-sup condition and ah(·, ·) a uniform inf-sup condition. Let u0 ∈ Vb and uh

0 ∈ Vhb


be the exact and the unique approximate solution, defined by

a(u0, v) = 〈f, v〉V′×V ∀ v ∈ Vb, and ah(uh

0 , vh)

= 〈f, vh〉Vh′×Vh ∀ vh ∈ Vhb . (5.270)

Then, with C independent of h, and the broken Sobolev norms, ‖ · ‖Vh

∥∥u0 − uh0

∥∥Vh ≤ C

(inf

uh∈Vhb

‖u0 − uh‖Vh + sup0�=vh∈Vh

b

∣∣ah

(u0 − uh

0 , vh)∣∣

‖vh‖Vh

). (5.271)

Proof. For any uh ∈ Vhb ,∥∥u0 − uh

0

∥∥Vh ≤ ‖u0 − uh‖Vh +

∥∥uh − uh0

∥∥Vh (triangle inequality)

≤ ‖u0 − uh‖Vh +1ε

supvh∈Vh

b \{0}

∣∣ah

(uh − uh

0 , vh)∣∣

‖vh‖Vh

in (4.170)

= ‖u0 − uh‖Vh +1ε

supvh∈Vh

b \{0}

∣∣ah(uh − u0, vh) + ah

(u0 − uh

0 , vh)∣∣

‖vh‖Vh

≤ ‖u0 − uh‖Vh +1ε

supvh∈Vh

b \{0}

|ah(uh − u0, vh)|

‖vh‖Vh

(5.272)

+1ε

supvh∈Vh

b \{0}

∣∣ah

(u0 − uh

0 , vh)∣∣

‖vh‖Vh

(triangle inequality)

≤ ‖u0 − uh‖Vh +C

ε‖uh − u0‖Vh

(continuity in Vh

b × Vhb

)+

1ε

supvh∈Vh

b \{0}

∣∣ah

(u0 − uh

0 , vh)∣∣

‖vh‖Vh

(5.273)

=(

1 +C

ε

)‖u0 − uh‖Vh +

1ε

supvh∈Vh

b \{0}

∣∣ah

(u0 − uh

0 , vh)∣∣

‖vh‖Vh

Note that, by continuity,∣∣ah

(u0 − uh

0 , vh)∣∣

‖vh‖Vh

≤ C∥∥u0 − uh

0

∥∥Vh (5.274)

so that

∥∥u0 − uh0

∥∥Vh ≥

1C

supvh∈Vh

b \{0}

∣∣ah

(u0 − uh

0 , vh)∣∣

‖vh‖Vh

. (5.275)


Combining (5.272) and (5.275), we find

max

{1C

supvh∈Vh

b \{0}

∣∣ah

(u0 − uh

0 , vh)∣∣

‖vh‖Vh

, infuh∈Vh

b

‖u0 − uh‖Vh

}(5.276)

≤∥∥u0 − uh

0

∥∥Vh

≤(

1 +C

ε

)inf

uh∈Vhb

‖u0 − uh‖Vh +1ε

supvh∈Vh

b \{0}

∣∣ah

(u0 − uh

0 , vh)∣∣

‖vh‖Vh

.

�

Remark 5.53.

1. Inequality (5.276) indicates that the sum of approximation terms infuh∈Vhb

‖u0 − uh‖Vh , and nonconformity terms supvh∈Vhb \{0} |ah(u0 − uh

0 , vh)|/ ‖vh‖Vh

correctly reflects the size of the discretization error ‖u0 − uh0‖Vh . The same result

for the general cases is obtained by the general discretization results in Chapter3. Indeed by (3.33), the classical consistency error is estimated by the variationalconsistency and the interpolation error for the exact solution. For a ah(uh, vh),satisfying a uniform discrete inf-sup condition, or a stable Ah this again yields(5.271), cf.(3.33).

2. As a consequence of (5.266), the second term on the right-hand sideof (5.271) would be zero if Vh

b ⊆ Vb, compare (4.156) and Theorem4.54. Therefore, it measures the effect of nonconformity in Vh

b �⊆ Vb orVh

b �⊆ V.

5.5.3 FEMs with crimes for linear and quasilinear problems

In Subsection 5.5.2 we have studied the impact of FEMS with crimes for a speciallinear equation, the Helmholtz equation. Here we generalize these results to generallinear and quasilinear equations and systems of order 2 in R2. Since there arestrong similarities we do not repeat all details for (5.251)–(5.261). The correspond-ing Propositions 5.46–5.47 and Lemma 5.52 will be formulated for all these casessimultaneously. Extensions to Rn, n ≥ 3 can be worthwhile, whenever the availablecubature formulas for Rn−1, n ≥ 3, cf. Cools [208–210] are good enough for the specificproblem.

Compared to (5.243)–(5.245) with the original Δu and ∂u/∂ν we replace thisspecial differential and boundary operator by the general forms Au and Ba u inthe following (5.278), (5.279). An essential tool is again Green’s formula (2.9): forv, w ∈ H1(Ω), we obtain by partial integration and with the outer normal ν = (ν1, ν2)T

for ∂Ω ∫Ω

w∂v/∂xidx +∫

Ω

v∂w/∂xidx =∫

Ω


∂Ω

vwνids. (5.277)


We find the bounded bilinear forms and operators for Ω ⊂ R2

a(u, v) = 〈Au, v〉V′×V =∫

Ω

⎛⎝ 2∑i,j=0

aij∂iu ∂jv

⎞⎠ dx with aij ∈W 1−δj0,∞(Ω),

=∫

Ω

⎛⎝ 2∑i,j=0

(−1)j>0∂j (aij∂

iu)

⎞⎠ vdx +∫

∂Ω

(Ba u)vds (5.278)

=:∫

Ω

(As u)vdx +∫

∂Ω

(Ba u)vds ∀v ∈ V = H1(Ω), u ∈ V ∩H2(Ω),

with2∑

i,j=1

aijξiξj ≥ ε′|ξ|22 , ε′ > 0, for elliptic operators, and

Ba u =2∑

i,j=1

νjaij∂i u +

2∑j=1

νja0j u, and

Ba u = ∂u/∂ν for aij = δi,j ∀i, j = 0, 1, 2, e.g. Asu = −Δu + cu.

It is possible to transform the problem such that the above Bau is replaced by Ba u =∑2i,j=1 νjaij∂

i u, always satisfied for Ba u = ∂u/∂ν. Then sometimes one obtains ahigher order of consistency and convergence by the additional factor h.

Obviously, a(uh, vh) is defined ∀ uh, vh ∈ Vhb ⊂ V independent of the boundary

conditions. For FEs with crimes uh, vh ∈ Vhb �⊂ Vb,V, we again use the extended

〈fh, ·〉W−k,p′ (T hc )×W k,p(T h

c ), 1/p + 1/p′ = 1, ah(·, ·), and the broken Sobolev norms ‖ ·‖W k,p(T h

c ) and obtain, cf. (5.250)–(5.252), (5.248), (5.254),

ah(uh, vh) := 〈Ahuh, vh〉H−1(T h

c )×H1(T hc ) :=

∑T∈T h

c

⎛⎝∫T

2∑i,j=0

aij∂i uh∂j vh

⎞⎠ dx

=∑

T∈T hc

∫T

⎡⎣(As,h uh)vh :=2∑

i,j=0

(−1)j>0∂j(aij∂

i uh) vh

⎤⎦ dx (5.279)

+∑

e∈T hc \∂Ω

∫e

({vh}

[Bau

h]+ [vh]{Bau

h})ds +

∑e∈∂Ω

∫e

vhBauh ds

=: as,h(uh, vh) +∑

e∈T hc \∂Ω

∫e

({vh}

[Bau

h]+ [vh]{Bau

h})ds

+∑

e∈∂Ω

∫e

vhBauh ds ∀ uh ∈ (Vb ∩H2(Ω)) + Vh

b ⊂ H2(T h

c

),


vh ∈ Vb + Vhb ⊂ H1

(T h

c

), with

ah(·, ·) := ah(·, ·)|Vh×Vh , a(·, ·) = ah(·, ·)|V×V and

ahs (·, ·) := as,h(·, ·)|H2(T h

c )×L2(Ω), as(·, ·) = as,h(·, ·)|H2(Ω)×L2(Ω),

with ellipticity as in (4.138). The exact and FE solutions u0 and uh0 are defined by

u0 ∈ Vb : 〈Au0, v〉H−1(Ω)×H1(Ω) = a(u, v) = 〈f, v〉V′×V ∀ v ∈ Vb, (5.280)

uh0 ∈ Vh

b :⟨Ahu

h0 , v

h⟩


c )= ah

(uh

0 , vh)

= 〈fh, vh〉Vh′×Vh ∀ vh ∈ Vhb .

These relations (5.278)–(5.280) are easily extendable to elliptic linear sec-ond order systems or quasilinear equations and systems. We use the notationin (4.136) and ansatz and test functions �uh =

(uh

1 , · · · , uhq

), �vh ∈ Vh

b := Vhb (Rq),

now with Vhb �⊂ Vb = H1

0 (Ω,Rq), so violating the boundary conditions or belowthe continuity, Vh

b �⊂ H1(Ω,Rq). We start with linear systems, and, becauseof the high similarity with (5.278) ff., we only formulate the FE equations:again f, a(·, ·), fh, ah(·, ·) and A,Ah are bounded linear and bilinear forms andoperators.

〈Ah�uh, �vh〉H−1(T h

c )×H1(T hc ) = ah(�uh, �vh) :=

∑T∈T h

c

∫T

2∑k,l=0

(Akl∂l�uh, ∂k�vh)qdx

∀�uh, �vh ∈ Vb + Vhb , with Akl ∈ L∞(Ω,Rq×q), cf.(5.254) for Vh

b . (5.281)

The Vb-coercivity of the principal part follows from the uniform Legendre condition,see (2.343), (2.345), (4.138), (4.139), hence with W = L2(Ω,Rq), cf. Subsection 5.5.4,

∃ λ,Λ, s.t. 0 < λ < Λ <∞ : λ|�ϑ|2nq ≤2∑

k,l=1

(Akl(x)�ϑl, �ϑk)q ≤ Λ|�ϑ|2∀x ∈ Ω, (5.282)

=⇒ ap(�uh, �uh) > λ∗‖�uh‖2V and a(�uh, �uh) ≥ λ∗∗‖�uh‖2V − Cc‖�uh‖2W ∀�uh ∈ Vb + Vhb .

For relating ah(·, ·), Ah and as,h(·, ·), As,h, we require Akl ∈W 1−δk0,∞(Ω,Rq×q) :

ah(�uh, �vh) := 〈Ah�uh, �vh〉H−1(T h

c )×H1(T hc ) :=

∑T∈T h

c

∫T

2∑k,l=0

(Akl∂

l �uh, ∂k �vh)qdx

=

⎡⎣(As,h �uh, �vh)L2(Ω,Rq) :=∑

T∈T hc

∫T

2∑k,l=0

(−1)k>0(∂k (Akl∂l �uh), �vh)qdx

⎤⎦(5.283)

+∑

e∈T hc \∂Ω

∫e

(({�vh},

[Ba�u

h])q + ([�vh], {Ba�u

h})q

)ds +

∑e∈∂Ω

∫e

(�vh

l , Ba�uh)qds


=: as,h(�uh, �vh) +∑

e∈T hc \∂Ω

∫e

(({�vh},

[Ba�u

h])q + ([�vh], {Ba�u

h})q

)ds

+∑

e∈∂Ω

∫e

(�vh, Ba�uh)q ds ∀ �uh ∈ (Vb ∩H2(T h

c ,Rq)) + Vhb ⊂ H2

(T h

c ,Rq),

�vh ∈ Vb + Vhb ⊂ H1

(T h

c ,Rq)

with Ba �u :=2∑

k,l=1

νkAkl∂l �u +

2∑l=1

νkAk0 �u,

again with ah(·, ·) := ah(·, ·)|Vh×Vh , a(·, ·) = ah(·, ·)|V×V and

ahs (·, ·) := as,h(·, ·)|H2(T h

c )×L2(Ω), as(·, ·) = as,h(·, ·)|H2(Ω)×L2(Ω).

The exact and FE solutions �u0 and �uh0 are defined by

�u0 ∈ Vb : 〈A�u0, �v〉H−1(Ω)×H1(Ω) = a(�u,�v) = 〈�f,�v〉V′×V ∀ �v ∈ Vb, (5.284)

�uh0 ∈ Vh

b :⟨Ah�u

h0 , �v

h⟩


c )= ah

(�uh

0 , �vh)

= 〈�fh, �vh〉Vh′×Vh∀�vh ∈ Vhb .

We generalize these relation between ah(·, ·) and as,h(·, ·). to quasilinear equationsand systems. For avoiding a double formulation as for the linear case, we use thestandard notation (4.129), but the same symbols for all functions

u, u0 ∈ Vb = W 1,p0 (Ω,Rq), uh, uh

0 ∈ Vhb ⊂W 1,p

0 (T hc ,Rq), q ≥ 1, p ≥ 2, (5.285)

for equations and systems. Only (., .)q indicates the possibility for systems, q ≥ 1.Again we only formulate the FE equations, cf (4.173), (4.174) and use (5.277). Thenecessary conditions for the Nemickii operators Ai(x, uh,∇uh), e.g. for bounded formsand operators, ah(uh, vh) and Gh, are formulated in Sections 2.5.4, 2.5.6, 4.4 in (4.174)and 4.5 in (4.193). We additionally modify them, such that G is Lipschitz-continuous.With smooth enough

Ai : Ω× Rq × Rn×q → Rq, we consider Vhb as in (5.254), and

∀ uh ∈ (Vb ∩W 2,p(Ω,Rq)) + Vhb ⊂W 2,p

(T h

c ,Rq), vh ∈ Vb + Vh

b ⊂W 1,p(T h

c ,Rq).

Then we define the weak and strong forms, ah(uh, vh) and as,h(uh, vh), related by

ah(uh, vh) := 〈Ghuh, vh〉W−1,p′ (T h

c )×W 1,p(T hc ) (5.286)

:=∑

T∈T hc

∫T

2∑i=0

(Ai(x, uh,∇uh), ∂ivh)qdx =

=[as,h(uh, vh) := (Gs,h uh, vh)L2(Ω,Rq)

:=∑

T∈T hc

∫T

2∑i=0

(−1)i>0(∂iAi(x, uh,∇uh), vh)qdx]


+∑

e∈T hc \∂Ω

∫e

(({vh},

[BGu

h])q + ([vh], {BGu

h})q

)ds

+∑

e∈∂Ω

∫e

(vh, BGuh)q ds

and BG uh := BG (x, uh,∇uh) :=2∑

i=1

νiAi(x, uh,∇uh) with

ah(·, ·) := ah(·, ·)|Vh×Vh , a(·, ·) = ah(·, ·)|V×V and

ahs (·, ·) := as,h(·, ·)|H2(T h

c ,Rq)×L2(Ω,Rq), as(·, ·) = as,h(·, ·)|H2(Ω,Rq)×L2(Ω,Rq).

Again the exact and FE solutions u0 and uh0 are defined by

u0 ∈ Vb : a(u0, v) = 〈Gu0, v〉V′×V = 0 ∀ v ∈ Vb, (5.287)

uh0 ∈ Vh

b : ah(uh

0 , vh)

=⟨Ghu

h0 , v

h⟩

W−1,p′ (T hc ,Rq)×W 1,p(T h

c ,Rq)= 0 ∀ vh ∈ Vh

b ,

often for Ub = Vb,Uhb = Vh

b . We summarize the results for second order linear andquasilinear equations and systems and use the notation as in (5.285):

Proposition 5.54.

1. For either Dirichlet or natural boundary conditions and the bilinear or nonlinearforms ah(uh, vh), as,h(uh, vh) and the operators Ah, As,h, Gh, Gs,h in (5.279),(5.283) and (5.286), with the previously mentioned conditions for the Ai(·, ·, ·),are continuous. We find, cf. (5.256), with Bu := Bau, p = 2 for linear, and Bu :=BGu, 2 ≤ p <∞ for quasilinear problems with q ≥ 1

as(u, v) = a(u, v) ∀ u ∈W 2,p(Ω,Rq), v ∈W 1,p(Ω,Rq), and vBu|∂Ω = 0.(5.288)

Note that Vb = W 1,p0 ∩W 2,p(Ω,Rq) for Dirichlet boundary conditions.

2. Valid for any uh ∈ Vhb + (Vb ∩H2(Ω,Rq)) ⊂W 2,p

(T h

c ,Rq)

and vh ∈ Vhb + Vb ⊂

W 1,p(T h

c ,Rq)

we get, cf. (5.258), 2 ≤ p <∞, 1 ≤ q,

ah(uh, vh) = as,h(uh, vh)

+∑

e∈T hc \∂Ω

∫e

(({vh}, [Buh])q + ([vh], {Buh})q

)ds (5.289)

+∑

e∈∂Ω

∫e

(vh, (Buh))qds ∀ uh ∈ (Vb ∩H2(Ω,Rq)) + Vhb , v ∈ Vb + Vh

b .


For important special cases (5.289) simplifies, cf. (5.249), (5.263)–(5.265), as

ah(uh, vh) = ahs (uh, vh) for Vh

b ⊂W 1,p0 (Ω,Rq), Vh

b ⊂W 2,p(Ω,Rq), (5.290)

ah(uh, vh) = ahs (uh, vh) +

∑e∈∂Ω

∫e

vh(Buh)ds for only violated boundary cond.

and Vhb ⊂ C(Ω,Rq) ∈W 2,p(Ω,Rq), Vh

b ⊂ C1(Ω,Rq) ∈W 1,p(Ω,Rq), (5.291)

ah(uh, vh) = ahs (uh, vh) +

∑e∈T h

c \∂Ω

∫e

vh[Buh]ds (5.292)

+∑

e∈∂Ω

∫e

vh(Buh)ds for Vhb ⊂ C(Ω,Rq) ∈W 1,p(Ω,Rq).

Simplifying and shortening these formulas we use the notation

〈Ahu, vh〉H−1(T h

c ,Rq)×Vh := 〈Ahu, vh〉H−1(T h

c ,Rq)×H1(T hc ,Rq) for vh ∈ Vh, . . . (5.293)

We generalize the sequences of projectors Q′h in (4.117). For variational crimes we

have chosen for the f used in (5.244) the general form (5.250). Their obvious extensionsin (5.250) are defined for v ∈ V and simultaneously for vh ∈ Vh

b . Here we explicitelyreformulate Propositions 4.49, 4.50 for the Vb in (5.285). We only need the f, fh in(5.250). We define the generalized projectors Q

′h by testing as

Q′h ∈ L

(W−1,p′ (T h

c

),Vh

b

′): 〈Q′hfh − fh, v

h〉W−1,p′ (T hc )×Vh = 0∀vh ∈ Vh

b . (5.294)

Nearly analogous to (4.117) we find here with (5.296)

Q′hfh − fh ⊥ Vh

b ,

with

‖Q′hfh‖Vh′ = supvh∈Vh

b ,‖vh‖Vh=1

〈fh, vh〉W−1,p′ (T h

c )×Vh

= ‖fh‖Vh′ = ‖f‖V′(1 + o(h))

=⇒ limh→0

‖Q′hfh‖Vh′ = limh→0

‖fh‖V′ = ‖f‖V′ ,

fh := Q′hfh|Vh , lim

h→0‖Q′h‖L(V′,Vh′ ) = lim

h→0‖Q′h‖L(Vh′ ,Vh′ ) = 1. (5.295)

Proposition 5.55. For the extension fh ∈ V ′ + W−1,p′(T h

c ) in (5.250), of f ∈V ′ ⊂W−1,p′

(Ω) with W 1,p(T hc ) ⊃ Vh, fh = fh|Vh , and the projectors Q

′h ∈L(W−1,p′

(T hc ),Vh′

) in (5.294), the norms are asymptotically equal ∀ 1 ≤ p′ ≤ ∞, 1/p+1/p′ = 1,

‖fh‖Vh′ = ‖fh‖W−1,p′ (T hc )(1 + o(1)) = ‖f‖V′(1 + o(1)), fh := Q

′hfh|Vh ,

limh→0

‖Q′hfh‖Vh′ = ‖f‖V′ , limh→0

‖Q′h‖L(V′,Vh′ ) = limh→0

‖Q′h‖L(Vh′ ,Vh′ ) = 1. (5.296)

Particularly interesting are the 〈fh, v〉W−1,p′ (T hc )×Vh with fh = Ahu

h.


U G U ′ tested by←→ U

Ph Φh Q′h

Uh Gh Uh ′tested by←→ Uh with ⊄ Ub or ⊄ U

Figure 5.7 Nonconforming FEMs: Spaces and operators.

We summarize: with the projectors Q′h in (5.294), and the operators Ah and Gh,

extended to V + Vhb in (5.254), (5.279), (5.283), (5.286), we determine the discrete

(weak) solutions uh0 ∈ Vh

b from the following equations,⟨Ahu

h0 − fh, v

h⟩

H−1(T hc )×Vh = 0 ∀ vh ∈ Vh

b ⇔ Ahuh0 := Q

′hAhuh0 = fh = Q

′hfh,

and⟨Ghu

h0 , v

h⟩

W−1,p′ (T hc )×Vh = 0 ∀ vh ∈ Vh

b ⇔ Ghuh0 := Q

′hGhuh0 = 0. (5.297)

Reformulating, the discrete operators Ah and Gh have the form

ΦhA := Ah := Q′hAh|Vh

b= Q

′hAh, ΦhG := Gh := Q′hGh|Uh

b,

with the discrete solutions Ahuh0 − fh = 0, Ghuh

0 = 0. (5.298)

Again we sketch the different spaces and operators constituting the method in thediagram in Figure 5.7.

The necessary consistency errors for the nonconforming FEMs require, cf. Lemma5.52, estimates for the following ah(u− uh, vh).

Proposition 5.56.

1. For FEs with violated boundary conditions and/or continuity, choose an arbitraryu ∈ H2(Ω), fu := Au ∈ L2(Ω), e.g. the exact solution u0 ∈ H2(Ω), fu0 := Au0,with the discrete solution uh

0 , cf. (5.279), (5.283), (5.286). Then we obtain for uand the discrete solution uh of Ahuh = Q

′hAu

ah(u− uh, vh) = 〈Q′hAhu−Ahuh, vh〉H−1(T hc )×Vh = 〈Ahu−Au, vh〉... (5.299)

=∑

e∈T hc \∂Ω

∫e

([vh], {Bu})qds +∑

e∈∂Ω

∫e

(vh, (Bu))qds

∀ u ∈ H2(Ω,Rq), vh ∈ Vhb .

2. For quasilinear (5.286), the A,Ah,H−10

(T h

c ,Rq), have to be replaced by G,Gh,

W−1,p0

(T h

c ,Rq), p ≥ 2.

Proof. The proof of Proposition 5.50 remains correct. �

Remark 5.57.A solution u0 of Au0 = f with u0 ∈ V = H1(Ω) implies f ∈ H−1(Ω). Requiring u0 ∈

H2(Ω) implies f ∈ L2(Ω), so 〈f, v〉H−1(Ω)×H1(Ω) = (f, v)L2(Ω) =∫Ωfvdx.


5.5.4 Discrete coercivity and consistency

In Proposition 5.56 we have determined, not yet estimated, the variational errors forlinear and quasilinear equations and systems of order 2. As in (5.285), we use thesame symbols u, uh, . . . ∈ H1(Ω), . . . instead of ∈ H1(Ω,Rq) and, as coefficients, aij

instead of Aij , for real and vector valued functions. In fact, the same arguments arevalid for all the linear problems in the previous Subsection. For quasilinear problemsslight modifications are necessary, e.g. in Lemma 5.65.

The general convergence theory in Chapter 3 requires uniform discrete inf-supconditions, equivalently the stability, of the discrete linear and linearized operators andestimates for the variational discretization errors. These estimates will be discussedin Subsections 5.5.6, 5.5.7. Towards stability, we do need for a(·, ·), ah(·, ·) the Vh

b

coercivity for Dirichlet or natural boundary conditions. We prove this, as a firststep, for compactly perturbed principal parts, ap(·, ·) + ε′(·, ·)L2(Ω), of arbitrary ellipticbilinear (weak) forms. In Subsection 5.5.8 we will extend these results to the generalcase of a consistent compact perturbation A = B + C,Ah = Bh + Ch with boundedlyinvertible A, B and stable Bh; e.g. the original operator, A, induced by a(·, ·) is acompact perturbation of the operator B := Ap + ε′I induced by ap(·, ·) + ε′(·, ·)L2(Ω).Thus the results of Subsection 5.5.8 show the stability of Ah.

For the FEs with crimes in this section the standard approach requires Uhb = Vh

b .

For Ub = H10 (Ω) or = H1(Ω) ⊂ L2(Ω) = L2(Ω)′ ⊂ U ′

b = H−1(Ω)

we assume ∀ uh ∈ Uhb �= Vh

b ∃vh ∈ Vhb s.t ‖uh − vh‖Vh < β‖uh‖Vh (5.300)

or vice versa ∀ vh ∈ Vhb �= Uh

b ∃uh ∈ Uhb s.t ‖uh − vh‖Vh < β‖vh‖Vh ,

here β is an arbitrarily small constant and dim Uhb = dim Vh

b , with the

same norms in Uh,Vh, e.g. ‖uh‖Uh = ‖uh‖Vh = ‖uh‖H1(T hc ).

Remark 5.58. Beyond the usual case Uhb = Vh

b , this condition is satisfied, if Uhb

and Vhb are defined via the same interpolation or transition points along the interior

and boundary edges e ∈ T hc . Different degrees in Uh

b and Vhb are still allowed. The

nonlocal smooth approximations in Subsection 4.2.6 have this property as well, unlessthe application to fully nonlinear problems requires the splitting there.

A similar condition will be satisfied for the general case of Gelfand triples: letUb,U ,X be Hilbert spaces and let X = X ′ be identified such that Ub is dense inX and I : Ub ↪→ X , Iv := v ∈ X for all v ∈ Ub is continuous. Then (5.300) can begeneralized as Ub ⊂ X = X ′ ⊂ U ′

b and Ub ⊂ U is densely and continuously embedded.The convergence theory for mesh free methods in [120] will allow to generalize this

(5.300) for FEMs as well.

Obviously the linear and bilinear forms studied in the last subsection are uniformlycontinuous, such that there exists a constant, C, independent of h, with

‖v‖V = ‖v‖H1(Ω), ‖uh‖Vh = ‖uh‖H1(T hc ), . . .

|ah(uh, vh)| ≤ C‖uh‖Vh · ‖vh‖Vh ∀ uh, vh ∈ V + Vh, (5.301)

|〈fh, vh〉| ≤ C‖fh‖Vh′ ‖vh‖Vh ∀ vh ∈ V + Vh

b , fh ∈ V ′ + Vh′.


The following theorem applies to violated boundary conditions and continuity asdiscussed in Subsections 5.5.6, 5.5.7 and with the necessary machinery in Subsection5.5.10 and Section 5.4 as well. For the general case beyond ap(·, ·) + ε′(·, ·)L2(Ω) thestability proof is delayed to Subsection 5.5.8.

Theorem 5.59. Stability for modified principal parts:

1. Let this implies A, a(·, ·), Ah, ah(·, ·) and Dirichlet and/or natural bound-ary conditions be given as in (5.278)–(5.280), (5.284) and ε′ > 0. Letap(·, ·), ap,h(·, ·) be the principal parts. Then the ap,h(·, ·) + ε′(·, ·)L2(Ω) isbounded, say by M = C in (5.301), and Vb + Vh

b -coercive, hence a constant α > 0exists such that

ap,h(uh, uh) + ε′(uh, uh)L2(Ω) ≥ α(‖uh‖Vh)2 ∀uh ∈ Vb + Vhb . (5.302)

2. If for Uhb �= Vh

b , (5.300) is satisfied with βM/ε′ < 1, then the uniform discreteinf-sup conditions (2.52), (2.53) are valid,

sup0�=vh∈Vb+Vh

b

|ap,h(uh, vh) + ε′(uh, uh)L2(Ω)|/‖vh‖Vh ≥ ε‖uh‖Uh∀uh ∈ Ub + Uhb ,

3. In particular, for Dirichlet boundary conditions Ap and Ahp are boundedly and

equiboundedly invertible, hence Ahp is stable.

Note that the inf–sup condition for a(u, v) does not include the discrete inf–supcondition. This is possible under the conditions of compact perturbation results, e.g.in Theorem 5.78. Theorem 5.59 implies the unique existence of the exact solutionu0 and of the discrete solutions uh

0 for the special case of ap(·, ·) + ε′(·, ·)L2(Ω), if theconsistency results in Subsections 5.5.6 and 5.5.7 are valid. For isoparametric FEs forcurved boundaries, see Subsection 5.5.10, this result is available in Theorem 5.84.

Proof. The principal part of the above linear elliptic operators or bilinear forms,

ap(u, v) :=∫

Ω

⎛⎝ 2∑i,j=1

aij∂i u ∂j v

⎞⎠ dx

satisfies, cf. (4.131) ff.,

2∑i,j=1

aijξiξj + ε′η2 ≥ ε′(|ξ|22 + η2

), ε′ > 0,

with ξj , η ∈ R, and |ξ|2 the Euclidean norm in R2. We have chosen twice the sameε = ε′ as in (4.131). Now let ξj := ∂ju

h(x), η := uh(x) and obtain

2∑i,j=1

aij∂iuh∂ju

h(x) + ε′(uh(x))2 ≥ ε′(|∇uh(x)|22 + (uh(x))2

)and by integration


ap,h(uh, uh) + ε′(uh, uh)L2(Ω) =∑

T∈T hc

∫T

⎛⎝ 2∑i,j=1

aij∂iuh∂ju

h + ε′(uh)2

⎞⎠ dx

≥∑

T∈T hc

∫T

ε′(|∇uh|22 + (uh)2

)dx ≥ ε′(‖uh‖Vh)2

∀uh ∈ Vb + Vhb .

For the inf-sup conditions we need (5.300). Let Uhb ⊂/ Ub,Vh

b ⊂/ Vb,Uhb �= Vh

b . By(5.300) we know that for any uh ∈ Vh we may choose a uh ∈ Uh with ‖uh − vh‖Vh <β‖uh‖Vh . with the notation bh(uh, uh) := ah

p(uh, uh) + e′(uh, vh)L2(Ω) we find

‖bh(uh, vh)− bh(uh, uh)‖ = ‖bh(uh, uh − vh)‖ ≤M(‖uh‖Vh‖uh − vh‖Vh)

since bh(·, ·) is continuous. With ‖bh(uh, uh)‖ ≥ ε′‖uh‖2H1(Ω) this implies

‖bh(uh, vh)‖ ≥ ‖bh(uh, uh)‖ − ‖bh(uh, vh)− bh(uh, uh)‖≥ ε′(‖uh‖Vh)2 −M(‖uh‖Vh‖uh − vh‖Vh)

≥ ε′‖uh‖Vh

[‖uh‖Vh − (M/ε′)β‖uh‖Vh

]hence, going back to vh and for (M/ε′)β < 1 the inf-sup condition is proved. Forsystems, with (4.131) replaced by (4.138), the proof remains nearly unchanged. �

As the next step, we prepare estimating the variational consistency errors as inSubsections 5.5.6 and 5.5.7. In contrast to classical consistency errors, we always haveto assume here that the discrete solution uh

0 exists. So, we assume the existence of theexact and approximate solutions u0 and uh

0 of

∃u0 : a(u0, v) = (f, v) ∀v ∈ Vb or ∃uh0 : ah

(uh

0 , vh)

= (f, vh)h ∀vh ∈ Vhb . (5.303)

For nonconforming FEs we obtain as a special case of Proposition 5.56 the followingresults for the variational consistency error.

Variational consistency error for violated boundary conditions: (5.304)

ah

(u0 − uh

0 , vh)

=∫

∂Ω

vhBau0ds for u0 ∈ H2(Ω), ∀ vh ∈ Vhb ⊂ H1(Ω).

Variational consistency error for violated continuity, see (5.258): (5.305)

ah

(u0 − uh

0 , vh)

=∑

e∈T hc

∫e

[vh]Bau0ds for u0 ∈ H2(Ω),∀vh ∈ Vhb �⊂ H1(Ω).

Variational consistency error for general nonconforming FEs

ah

(u0 − uh

0 , vh)

=∑

e∈T hc

∫e

[vh]Bau0ds +∫

e∈∂Ω

vhBau0ds (5.306)

for u0 ∈ H2(Ω), ∀ vh ∈ Vhb �⊂ Vb,Vh

b �⊂ H1(Ω).


In Subsections 5.5.6 ff. we will aim for the classical consistency errors as well. Sowe generalize (5.304), (5.305), (5.306) from u0, u

h0 to general u and its corresponding

discrete counterpart uh, defined by Ahuh = Q′hAu, cf. (5.297), (5.298). Note that

the classical consistency error is independent of the existence of a (unique) discretesolution, in contrast to the variational consistency error.

We require, beyond Condition 4.16, specific approximating spaces:

Condition 5.60. Conditions for nonconforming FEs: Choose piecewise polynomialFEs locally in P = P2

d−1 in R2. Recall that Pd−1 = P1d−1 and Pd−1 = P2

d−1 denotethe uni-and bi-variate polynomials of degree d− 1, respectively Every T ∈ T h

c atthe boundary has at most one curved side, see Figures 4.21–4.23 above. Let ∂Ωbe piecewise smooth. Nonsmooth points of ∂Ω are used as vertices of subtriangles,T ∈ T h

c , compare Condition 5.64. Let the subdivision be quasiuniform, or generalizdnondegenerate, see Definition 4.12. This condition of quasiuniformity can be relaxedin R2 by applying the results in Subsection 4.2.5. Extensions to Rn, n ≥ 3, are possibleusing results as in Cools [208–210]. We only discuss R2.

5.5.5 High order quadrature on edges

These are an important tool for estimates of all types of nonconformity errors. Themain idea is very simple: the variational errors ah

(u0 − uh

0 , vh), listed in (5.304)–

(5.306), are small if the [vh] disappear often enough along interior edges e and vh

along the boundary ∂Ω. Then in R2 the Gauss-type approximations along edges eand the boundary ∂Ω vanish for the right-hand sides of (5.304)–(5.306). Conse-quently the corresponding right-hand side integrals, and hence the variational errorsah

(u0 − uh

0 , vh), are small. Similarly to the techniques in Section 5.4, this is shown

with inverse estimates, costing some powers of h. There quadrature is applied to thebilinear or higher nonlinear forms, compare (5.179) and (5.306), (5.330), on the T ∈ T h

c

in R2. Here we apply quadrature to boundary forms on edges of the triangles. Sincethe degree of freedom for a polynomial increases with the dimension n, cf. (4.9), thisconcept works well for Gauss-type formulas on edges and is still not too bad for highorder cubature on triangles, but seldom for Rn, n ≥ 4. We often have to distinguishinterpolation and quadrature errors and points.

Univariate Gauss–type quadrature formulas

For one-dimensional intervals or edges, we use three types of Gauss formulas, seeLemma 5.61 below. They are distinguished by the requirement that either none, oneor two boundary points of the interval are included in the corresponding quadraturegrid points. They are defined for Legendre and Chebyshev polynomials with respect tothe weight functions w ≡ 1 and w = (1− x2)

12 , respectively, on the interval [−1, 1]. We

introduce three types of univariate polynomials, Qρd′ ∈ Pd′

:= P1d′ , ρ = 0, 1, 2, based

on the Legendre and Chebyshev polynomials Pd′ ∈ Pd′and Td′ ∈ Pd′

, respectively,defining quadrature formulas of the type of


Gauss: ρ = 0 : Q0d′ := Pd′ or Q0

d′ := Td′ and for both casesGauss–Radau: ρ = 1 : Q1

d′ := Q0d′ + aQ0

d′−1,with a

such that Q1d′(−1) = 0,

Gauss–Lobatto: ρ = 2 : Q2d′ := Q0

d′ + aQ0d′−1 + bQ0

d′−2, with a, b

such that Q2d′(±1) = 0.

(5.307)

This implies d′ ≥ max{1, ρ}. The d′ quadrature points yρj := yj ∈ [−1, 1], j =

0, . . . , d′ − 1 for the quadrature formulas are defined as the roots of the polynomials

Qρd′(yρ

j

)= 0 for j ∈ {0, . . . , d′ − 1}, ρ = 0, 1, 2, max{1, ρ} ≤ d′, (5.308)

with yρ0 = −1 for ρ = 1, 2 and yρ

d′−1 = 1 for ρ = 2. The corresponding weights are thencomputed via the Lagrange elementary polynomials, pρ

j , as

pρj ∈ Pd′

: pρj (yρ

l ) = δj,l ∀ 0 ≤ j, l ≤ d′ − 1 as

wj : = wρj :=

∫ 1

−1

pρj (x)w(x)dx, ρ = 0, 1, 2, d′ ≥ max{1, ρ},

with the above weight function, w. The following proposition is proved in manytextbooks, e.g. [158].

Proposition 5.61.

1. Let −1 ≤ y0 < · · · < yd′−1 ≤ +1 be the above roots of the Qρd′ , for ρ = 0, 1, 2, and

wj := wρj the corresponding weights. Then the quadrature approximation qd′

w (u)is defined as

qd′

w (u) :=d′−1∑j=0

u(yρ

j

)wρ

j ≈∫ 1

−1

u(x)w(x)dx and qd′(u) := qd′

w≡1(u),

for u ∈ C[−1, 1], ρ = 0, 1, 2, max{1, ρ} ≤ d′ ∈ N, and satisfies

qd′

w (p) =d′−1∑j=0

p(yρ

j

)wρ

j =∫ 1

−1

p(x)w(x)dx, for all p ∈ P2d′−ρ−1. (5.309)

2. This implies for scalar products, (u, v)w, with respect to the weight function wthat

(u, v)w = (u, v)d′

w := qd′

w (u · v) =d′−1∑j=0

(u · v)(yρ

j

)wρ

j ∀u ∈ Pd′−ρ, v ∈ Pd′−1.

(5.310)

3. So (u, v)d′w = qd′

w (u · v) exactly reproduces scalar products for u, v ∈ Pd′−1 for ρ =0, 1. For ρ = 2 we have to require at least one of the u or v ∈ Pd′−2.

The next proposition is applied for proving Theorems 5.66, ff.


Proposition 5.62. For all d′ ∈ N with max{1, ρ} ≤ d′, hence 2d′ − ρ ≥ d′ ≥ 1,there exists C = Cd′ such that ∀0 < h < 1, 1 ≤ q ≤ ∞, f ∈W 2d′−ρ,q[0, h] ∩ C[0, h],∣∣∣∣∣∣

h∫0

f(x)dx− h/2d′−1∑j=0

wjf(hξj)

∣∣∣∣∣∣ ≤ Cd′ h2d′−ρ+(1−1/q)|f (2d′−ρ)|Lq(0,h). (5.311)

With ρ = 0, 1, 2, for Gauss, Gauss–Radau and Gauss–Lobatto quadrature, respectively,the points hξj := hξρ

j := h(yρ

j + 1)/2 and yρ

j are zeros of the Legendre polynomialsQρ

d′ of degree d′, see (5.308).

Proof. We use, for any T ⊂ R2, the Holder inequality, see (1.45),

∣∣∣∣∣∣∫T

f(x)g(x)dx

∣∣∣∣∣∣ ≤⎛⎝∣∣∣∣∣∣∫T

|f(x)|pdx

∣∣∣∣∣∣⎞⎠1/p⎛⎝∣∣∣∣∣∣

∫T

|g(x)|qdx

∣∣∣∣∣∣⎞⎠1/q

(5.312)

for all 1 ≤ p, q ≤ ∞ with 1/p + 1/q = 1 and f ∈ Lp(T ), g ∈ Lq(T ).Now we add and subtract the Taylor polynomial Q2d′−ρf for f in (5.311). Then (5.309)implies with f ∈W 2d′−ρ,q(0, h), and the Bramble–Hilbert–Lemma 4.15∣∣∣∣∣∣

h∫0

f(x)dx − h/2d′−1∑j=0

wjf(hξj)

∣∣∣∣∣∣≤

∣∣∣∣∣∣h∫

0

f(x)−Q2d′−ρf + Q2d′−ρfdx− h/2d′−1∑j=0

wjf(hξj)

∣∣∣∣∣∣≤

h∫0

1 · |f(x)−Q2d′−ρf |dx ≤ Cd′ h1−1/q|f −Q2d′−ρf |Lq(0,h)

≤ Cd′ h2d′−ρ+1−1/q|f (2d′−ρ)|Lq(0,h). �

5.5.6 Violated boundary conditions

In this Subsection we estimate the consisitency errors of continuous FEs violatingDirichlet boundary conditions. Natural boundary value problems without prescribedboundary conditions do not play any role in this context. We do need here theclassical and variational consistency error. With the discrete solution uh of Ahuh =Q

′hAhuh = Q

′hfu = Q′hAu they are introduced in Definition 3.14 as Q

′hAu−AhPhuand Q

′hAhu−Ahuh = Q′hAhu−Q

′hAhuh, respectively (Note that for conformingFEMs with Ah = A the variational consistency error, Ahuh −Q

′hAu vanishes!) By


Chapter 3, (3.33), they are, for a linear operator, A, related as


= Q′hAh (u− Phu)︸︷︷︸

interp.error

+AhPhu−Q′hAu︸︷︷︸

–class.cons.error

withAhuh = Q′hAu

(5.313)

For the first term, the variational consistency error, tested with vh,

〈Q′hAhu−Ahuh, vh〉Vh′×Vh = 〈Ahu−Ahuh, vh〉Vh′×Vh = ah(u− uh, vh),

we have an explicit form in (5.299). With the interpolation operator, Ph = Ih, themiddle term Q

′hAh(u− Phu) = Q′hAh(u− Ihu) can be estimated by the interpola-

tion error. So the classical consistency error can be estimated as

‖AhPhu−Q′hAu‖Vh′ ≤ ‖Q′hAhu−Ahuh‖Vh′ + ‖Q′hAh(u− Phu)‖Vh′ (5.314)

≤ (1 + o(h))‖A‖V′←↩V‖Phu− u‖V + ‖Q′hAhu−Ahuh‖Vh′ .

This (5.314) has to be slightly modified for quasilinear operators. Again, we introducefu := Gu and uh, the discrete solution of Ghuh = Q

′hfu = Q′hGu. From the possibly

many solutions uh for the nonlinear Gh we choose that with small ‖uh − u‖. We findwith Gh = Q

′hGh|Vhb

in (5.287),

Q′hGhu−Ghuh = Q

′h(Ghu−GhPhu) + (GhPhu−Q

′hGu). (5.315)

So we have to estimate these variational consistency errors, cf. (5.299),

ah(u− uh, vh) = 〈Ahu−Ahuh, vh〉Vh′×Vh ∀vh ∈ Vhb , with Ahuh = Q

′hAu and

for nonlinear G : 〈Ghu−Ghuh, vh〉Vh′×Vh ∀vh ∈ Vhb , for Ghuh = Q

′hGu. (5.316)

Explicit formulas for linear and quasilinear problems are listed in Proposition 5.56 andthe G,Gh modification of (5.299). We will update it for violated boundary conditions.

Proposition 5.63. Let Q′h and Ah, Gh be defined in (5.294)–(5.297), and choose an

arbitrary u ∈ Vb ∩H2(Ω) and the corresponding uh from Ahuh = Q′hAu or Ghuh =

Q′hGu. Then the violated boundary conditions are reflected in the variational consis-

tency error ∀ vh ∈ Vhb as, cf. Proposition 5.56,

〈Q′hAhu−Ahuh, vh〉Vh′×Vhor 〈Q′hGhu−Ghuh, vh〉Vh′×Vh =∫

∂Ω

vhBuds. (5.317)

We start with linear problems and Ba, for nonlinear problems with B = BG

cf. Theorem 5.68. According to (5.317), an estimate for the consistency error forvh ∈ Vh

b �⊂ Vb, hence vh|∂Ω �≡ 0, depends on how well vh approximates vh|∂Ω = 0and how this implies small |

∫∂Ω

vhBa u0ds|. The following steps involve quadratureerrors and inverse estimates. First, we use Proposition 5.61 estimating |

∫∂Ω

fds−∑Pj∈∂Ω wPj

f(Pj)| for f ∈W d′,∞(Ω), cf. (5.308) ff. Here, Pj := P ρj ∈ ∂Ω indicate

the quadrature points along one or all edges on ∂Ω, or on a subsection of ∂Ωor its approximation. The wj indicate the corresponding quadrature weights. For


FEMs we only consider Legendre polynomials with w ≡ 1. The Gauss-type pointsare used for nonconforming FEs to specify vh ∈ Vh

b . We choose for the boundary Ω aparametrization

∂Ω = {xb(s) : s ∈ [0, sb], s = arc length, sb = arc length of ∂Ω,

xb = linear on straight parts of ∂Ω}. (5.318)

We delay the isoparametric case to Subsection 5.5.10. There we will impose Dirichletboundary conditions in d′ points close to ∂Ω. Now we proceed as in Subsubsection4.2.7. For each edge e with endpoints Pi, Pe ∈ e ∩ ∂Ω such that Pi = xb(se), Pe =xb(se + he) we define a parametrization

e = {xe(s) : s ∈ [se, se + he]}; xe := xb|[se,se+he]. (5.319)

Any derivatives of this global and local parametrization in (5.318) and (5.319),respectively, are determined solely by ∂Ω and can be bounded independently of h. Withthe ξj in (5.311) we define the additional boundary nodes on the (curved) boundary,cf. (4.98),

e � P ρ,ej := P e

j := xe(se + heξj) ∈ e, j = 0, 1, . . . , d′ − 1

hξj := h(y2

j + 1)/2 ∀ e ∈ T h

c with |{e ∩ ∂Ω}| = 2 (5.320)

often with Gauss–Lobatto quadrature ρ = 2 and Pi = P e0 , Pe = P e

d′−1,

cf. Proposition 5.61. This allows defining the Vhb ⊂ Vh, cf. (5.241), as

Vhb :=

{v : Ω→ R : v ∈ Vh,∀e ∈ T h

c , v(P e

j

)= 0∀P e

j in (5.320)},Ω = ∪T∈T h

cT.

(5.321)

We apply (5.311) to the special case of f = (vhBau) ◦ xb with the above parame-trization xb for ∂Ω. We use (1.67) for p = ∞, and vhBau ∈ W k+1,∞(Ω), (5.278) andthe product and chain rule estimating

‖((vhBa u) ◦ xb)(k)‖L∞(∂Ω) ≤ Cxb‖u‖W k+1,∞(Ω)‖vh‖Wmin{k,d−1},∞(Ω)

∀u ∈W k+2∞ (Ω), vh ∈ Vh; (5.322)

this Cxbdepends on powers of order ≤ k of different derivatives x

(j)b , j = 0, . . . , k, for

the above parametrization. Finally, inverse estimates yield for (5.311) and Condition5.60, cf. e.g. Theorems 4.17, 4.19 and Proposition 4.18 with n = 2,

‖vh‖W j,p(Ω) ≤ C hl−j−(2/q−2/p)‖vh‖W l,q(Ω) for FEs and

0 ≤ l ≤ j ≤ d− 1, 1 ≤ q ≤ p ≤ ∞ ∀vh ∈ Vh. (5.323)

We get estimates for the interpolation errors with Condition 5.60 and v ∈W d,p(Ω):

‖v − Ihv‖W s,p(Ω) ≤ C hd−s|v|W d,p(Ω) and

‖v − Ihv‖W s,p(T ) ≤ C (diam T )d−s|v|W d,p(T ) for 0 ≤ s ≤ d. (5.324)


We have to consider the consequence of (5.320), (5.321) for imposing Dirichletboundary conditions in d′ points on the edges, e. In R2 we require

for ρ = 0, 1, 2 choose max{1, ρ} ≤ d′ hence 2d′ − ρ ≥ d′ ≥ 1. (5.325)

This would still allow many different orders of convergence in the following results andrather complicated case studies yielding only few really interesting results. For avoidingthese technicalities, we choose the other way round. We exclude all nonoptimal cases,and prove the results only for the optimal cases. Exotic cases violate d− 1 ≤ 2d′ − ρ.Furthermore, as already indicated in the introduction 5.5.1, we want to maintainthe order of convergence, d− 1, of the original FEM in its violated form as well.For d− 1 ≤ 2d′ − ρ a detailed discussion would yield as order of convergence, with2d′ − ρ− (d− 1) ≥ 0, the min{2d′ − ρ− (d− 1) + 1/2, d− 1} aimed to be ≥ d− 1.So we assume

d− 1 ≤ 2d′ − ρ and min{2d′ − ρ− (d− 1) + 1/2, d− 1} ≥ d− 1. (5.326)

We distinguish two cases. For polygonal boundaries the trivial boundary conditionscan be satisfied exactly. This is no longer correct in combination with discontinuousFEs, see Subsection 5.5.7. So we include polygonal boundaries here, too, mainlywith ρ = 0. For curved boundaries, Dirichlet boundary conditions usually cannot besatisfied exactly by polynomial FEs.

Polygonal ∂Ω violating boundary conditions, mainly for discontinuous FEs:

The piecewise linear xb, see (5.318), and vh|T ∈ Pd−1 yield (vh ◦ xb)|e ∈ Pd−1.Now d′ > d− 1 boundary points with vh(Pj) = vh(xb(se + heξj)) = 0, j = 0, . . . , d′ −1 imply (vh ◦ xb)e ≡ 0, hence the Dirichlet boundary conditions are satisfied exactly.So violated Dirichlet boundary conditions imply d′ ≤ d− 1. Condition 4.16, 6. assumesthe same number of functionals on every edge by requiring affinely equivalent(K,P,N ) and (T,PT ,NT ) ∀T ∈ T h

c . In this sense, max{1, ρ} ≤ d′ ≤ d− 1 is evenequivalent to violated continuity conditions, see Subsection 5.5.7. The second part of(5.326) yields, for ρ = 2, d′ ≥ d and, for ρ = 0, 1, d′ ≥ d− 1. With the above d′ ≤ d− 1,this excludes ρ = 2 and allows ρ = 0, 1 with d′ = d− 1 and max{1, ρ} = 1. So werequire for polygonal and allow curved ∂Ω also, so

for polygonal and general ∂Ω : ρ = 0, 1, and 1 ≤ d′ = d− 1. (5.327)

Curved ∂Ω with violated boundary conditions:

In this situation a d′ > d + τ and vh ◦ xb|e �∈ Pd+τ does not imply vh ◦ xb|e ≡ 0as above. For τ cf. Condition 4.16. Thus, violated Dirichlet boundary conditionsusually do not imply violated continuity. However, if we assume the same number ofinterpolation points on each e, including curved boundary edges as well, the abovearguments for a polygonal boundary apply again and yield (5.327), mainly withρ = 0. Another possibility, combining curved boundaries and violated continuity viaρ = 0, 1, 2 is discussed in the introduction of Subsection 5.5.7. Finally, (5.326) implies,for ρ = 2, d′ ≥ d. Since by (5.326) a d′ > d would not increase the convergence rate,


the only interesting case is the

standard case for curved ∂Ω : ρ = 2 ≤ d′ = d,Pd−1 = P. (5.328)

We summarize:

Condition 5.64. Conditions for boundary and boundary quadrature points: Let∂Ω be a piecewise smooth, e.g. a curved boundary and introduce the points Pj = P e

j

along ∂Ω as in Subsubsection 4.2.7, see (4.92) and (5.311), (5.318)–(5.320). Let thesegments e ⊂ ∂Ω, see (5.318),(5.319), be defined by Pi = Pj , Pe = Pj+1. Choose aquadrature approximation according to Propositions 5.61 and 5.62, along the aboveclosed (curved) segments e of the piecewise smooth ∂Ω ⊂ R2. Impose Dirichlet bound-ary conditions in d′ points P ρ,e

i ∈ ∂Ω, i = 0, . . . , d′ − 1 on each edge, e. We requireρ = 0, 1 ≤ d′ = d− 1 and ρ = 2 ≤ d′ = d only for curved ∂Ω, see Proposition 5.62 and(5.320),(5.321), (5.327), (5.328). In practical computations only ρ = 0, 2 are used.

The P ρ,ei , cf. (5.319), (5.320), are defined by the Gauss, Gauss–Radau and Gauss–

Lobatto points according to the chosen quadrature approximation ρ = 0, 1, 2 along thesegment e, parametrized by xe with respect to the arc length, see (5.319).

For the proof of Theorem 5.66 we need Lemma 5.65, discussing the estimatesfor violated boundary values along a single edge e. The following notation allowsa simultaneous formulation for the two cases (5.327), (5.328).

(ν0, ν0) :=

{(d− ρ, 2d− ρ− 1) for (5.327): ρ = 0, 1 ≤ d′ = d− 1,(d, 2d− 1) for (5.328): ρ = 2 ≤ d′ = d.

(5.329)

Lemma 5.65. Under Conditions 5.60 and 5.64 we choose a linear or a quasilineardifferential operator, A or G, and the corresponding boundary operator B = Ba or B =BG, satisfying (5.335), homogeneous Dirichlet boundary conditions as in (5.278) andan arbitrary u ∈W ν0+1,∞(Ω) ∩W ν0+1,∞(∂Ω), or u ∈ Hν0+2(Ω), cf. Remark 5.67.39

Furthermore, we define Vhb as in (5.321) and choose e ⊂ T , T ∈ T h

c . Then there existsa constant C = C(d, χ) such that∣∣∣∣∫

e

vhBu ds

∣∣∣∣ ≤ C hν0‖u‖W ν0,∞(T )|vh|H1(T ), ∀ vh ∈ Vhb �⊂ H1

0 (Ω). (5.330)

Proof. The (5.311), (5.322) and (5.323) with l = 1, j = d− 1, p =∞, q ≥ 2 are appliedto an edge e ⊂ T ∈ T h

c with length he and |{e ∩ ∂Ω}| ≥ 2. We use the parametrizationxe for e, see (5.319). Now we combine qd′

((vhBau) ◦ xe) = 0 ∀vh ∈ Vhb , Proposition

5.62 for q = ∞ with (5.321), (5.322), Theorem 4.19 with (5.323), and (1.67), requiring

39 The transition from line 1 to 2 in (5.331) requires more than u ∈ W ν0+1,∞(Ω), since Theorem

1.37 is wrong for p = ∞. This is not surprising since ∂Ω has measure 0 in R2 So u ∈ W ν0+1,∞(Ω)

does not imply any form of u ∈ W ν0+1,∞(∂Ω). So, motivated by Remark 5.67 and Theorem 1.26, we

impose u ∈ W ν0+1,∞(Ω) ∩ W ν0+1,∞(∂Ω) or u ∈ Hν0+2(Ω).


u ∈Wμ′+2,∞(Ω) with μ′ := 2d′ − ρ, for (5.331). We estimate for B = Ba∣∣∣∣∫e

vhBauds

∣∣∣∣ =∣∣∣∣∣∫ se+he

se

(vhBau) ◦ xeds

∣∣∣∣∣ ≤ Chμ′+1e ‖vhBau‖W μ′,∞(e)

≤ Chμ′+1e ‖u‖W μ′+1,∞(T )‖vh‖W d−1,∞(T ) with min{μ′, d− 1} = d− 1

≤ Chμ′+1h1−2/q−(d−1)‖u‖W μ′+1,∞(T )‖vh‖W 1q (T ) or (5.331)

≤ Chμ′+2−d‖u‖W μ′+1,∞(T ) · ‖vh‖H1(T ), for q = 2,

≤ Chμ′+2−d‖u‖W μ′+1,∞(T ) · |vh|H1(T ),

where we have replaced ‖vh‖H1(T ) by |vh|H1(T ) in (5.331). This is correct, sincevh(Pj) = 0 for at least one point Pj ∈ e. So we can use a modified Poincare–Friedrichinequality of the form

‖u‖L2(T ) ≤ h|u|H1(T )∀u ∈ H1(T ) with u(Pj) = 0 for at least one Pj ∈ e.

The additional changes for quasilinear problems with Lipschitz-continuous G areobvious by (5.286). �

Theorem 5.66. Consistency for FEMs violating boundary conditions and conver-gence under coercivity conditions:

1. Under Conditions 5.60 and 5.64 we choose A, a(·, ·), homogeneous Dirichletboundary conditions as in (5.278), ρ = 0, 2, and an arbitrary u ∈W 2d,∞(Ω) ∩W 2d,∞(∂Ω), or u ∈ H2d+1(Ω), cf. Remark 5.67. Furthermore, we define Vh

b asin (5.321) and choose the FEMs in (5.278)–(5.280), (5.294)–(5.298).

2. Then Ah = Q′hAh|Vh

bis consistent with A in u. Let uh , defined by Ahuh =

Q′hAu , exist. Then the variational and classical consistency errors vanish,40 cf.

(5.329) and Remark 5.67, such that, with an h-independent C = C(d,n,ρ,xb),

sup0�=vh∈Vh

b

|ah(u− uh, vh)|‖vh‖H1(T h

c )

≤ Chν0− 12 ‖u‖W 2d−1,∞(Jh

c ), and, by (5.329),

‖AhIhu−Q′hAu‖Vh′ ≤ Chd−1‖u‖W 2d−1,∞(Ω) for u ∈W 2d,∞(Ω). (5.332)

3. Let ah(·, ·) be Vhb -coercive, cf. (5.301) and Theorem 5.59. Then, for ρ = 0, 2, the

solutions uh exist uniquely and converge to u according to

‖uh − u‖H1(Ω) ≤ Chd−1‖u‖W 2d−1,∞(Ω). (5.333)

4. For ρ = 1 in (5.327) the convergence is only ≤ Chd−3/2‖u‖W 2d−2,∞(Ω), u ∈W 2d−1,∞.

40 In this and the next subsection we often use the pair u, uh with Ahuh = Q′hAu instead of the

exact solution u0, uh0 . This is motivated by the classical consistency error. It is defined not only for

u0 but for general u as well.


Remark 5.67.

1. Here we consider Gauss, Gauss–Radau, and Gauss–Lobatto points. For violatedcontinuity we restrict the discussion to Gauss points. Consequently, differenttechniques for the proofs are necessary.

2. An alternative result to (5.330), (5.332) is possible. Replace the ≤ Chμ′+1e

‖vhBau‖W μ′,∞(e) by ≤ Ch2(d−1)e |([vh]Bau)2(d−1))|L1(e) in the first line of

(5.331), and combine it with (1.68). Then the ‖u‖W 2d−1,∞(Ω) in (5.330), (5.332)can be replaced by ‖u‖H2d(Ω). The analogous result to (5.350) is incorrect forcurved boundaries, hence Theorems 4.60 does not hold either.

Proof. This collects the preceding results in Proposition 5.63, Lemma 5.65 estimating(5.332). Relation (5.317) shows that we have to estimate the sum over all edges e with|{e ∩ ∂Ω}| ≥ 2 along ∂ Ω,∣∣∣∣∫

∂ Ω

vhBauds

∣∣∣∣ ≤∑e

∣∣∣∣∫e

vhBauds

∣∣∣∣ with∑

e

:=∑

e:|{e ∩∂Ω}|≥2

≤ Chμ′+1−(d−1)−1/2‖u‖W μ′+1,∞(Ω)

(∑e

h1/2e |vh|H1(T )

)≤ Chμ′−(d−1)+1/2‖u‖W μ′+1,∞(Ω)|vh|H1(Ω),

since, by the Holder inequality and (4.36),

∑e

h1/2e |vh|H1(T ) ≤

(∑e

he

)1/2(∑e

|vh|2H1(T )

)1/2

≤(∑

e

he

)1/2

|vh|H1(Ω) and∑

e

he ≤ 2× (arc length of ∂Ω)

for small enough h. So we obtain (5.332).Relation (5.333) is obtained either by combining the stability result in Theorem

5.78 with the consistency estimate in (5.332) or by a combination with Lemma 5.52,see (5.271).

The last claim immediately follows, since, e.g. H2d+1(Ω) is dense in H1(Ω). So it isalways possible to choose a u ∈ H2d+1(Ω), such that ‖u− u‖ is small enough. �

For quasilinear problems the previous terms in (5.330)–(5.332),

u ∈ W 2d,∞(Ω), ‖u‖W 2d−1,∞(Ω), ‖u‖W d,∞(Ω) <∞,

u ∈ W 2d′−ρ+2,∞(T ), ‖u‖W 2d′−ρ+1,∞(T ) <∞, (5.334)


have to be replaced, with Bu = BGu in (5.286) and (5.330) in Lemma 5.65, by

Bu ∈ W 2d−1,∞(Ω), ‖Bu‖W 2d−2,∞(Ω), ‖Bu‖W d−1,∞(Ω) <∞,

Bu ∈ W 2d′−ρ+1,∞(T ), ‖Bu‖W 2d′−ρ,∞(T ) <∞. (5.335)

These estimates for the Bu in terms of u are possible with the results in Subsections2.5.4 and 2.5.6, based upon the second line of (5.334).

The linearized forms of these quasilinear problem are studied in Section 2.7, Subsec-tions 2.7.3 ff., e.g. Theorem 2.122 and have to require p ≥ 2 in (5.286). Furthermore,we already mentioned and require the necessary conditions for the Ai(x, uh,∇uh), e.g.for bounded forms and operators, ah(uh, vh) and Gh, formulated in Sections 2.5.4,2.5.6, 4.4 and (4.174), modified such that G is Lipschitz-continuous.

Theorem 5.68. Violated boundary conditions for quasilinear problems:

1. Let the FEM in (5.286) for the quasilinear problem in Section 4.4, (4.174), satisfyConditions 5.60 and 5.64, and define Vh

b as in (5.321). Furthermore let, foru ∈W ν0+1,∞(Ω) ∩W ν0+1,∞(∂Ω), or u ∈ Hν0+2(Ω), (5.334) imply (5.335), cf.Remark 5.67 and let the previous conditions be satisfied. Then all the consis-tency results in this subsection remain correct for these quasilinear problems aswell.

2. Let the bilinear form of the linearized quasilinear problem be bounded, Vhb -

coercive. Notice that we only consider W 1,p(Ω) for 2 ≤ p <∞ and the corre-sponding stability and convergence results as in Theorem 4.63. with respect to thediscrete H1(T h) norm. In this sense, the stability of the corresponding Gh andthe existence of uh

0 are secured. Then the solutions uh0 converge to u0 according

to (5.333).

Proof. The claims concerning consistency are an immediate consequence of Chap-ter 3, Theorem 3.21, and Lemma 5.65. It remains correct, if (5.334) implies(5.335).

Finally, the linearized principal part of the linearized quasilinear operator is H10 (Ω)-

coercive for 2 ≤ p <∞ or satisfies an inf-sup condition. If the quasilinear operator isbounded, implying its discretization to be equibounded, and consistent, cf. (5.286),we obtain the claim. �

5.5.7 Violated continuity

We consider nonconformity for piecewise polynomial Vh �⊂ V, including violated con-tinuity and boundary conditions. We have already seen that, for the case of polygonal∂Ω and the same number of functionals on every edge for FEs as in Definitions4.6, 4.8, violated boundary conditions are equivalent to violated continuity. Forpolygonal ∂Ω, violated boundary conditions are directly included in the followingdiscussion. In contrast to [141], we allow curved boundaries and discontinuous FEsby combining the different results and techniques here with those in Subsubsection


5.5.6 for the case ρ = 0. Isoparametric FEs in Subsubsection 5.5.10 require otherapproaches.

Other combinations would be possible as well: Curved boundaries with ρ = 2 anddiscontinuous FEs with ρ = 0 require ρ = 1 for discontinuous FEs on the edges ofthe triangulation with exactly one boundary point. In this case the affine equivalenceof the FEs, cf. Definition 4.6, would have to be replaced by three different types ofFEs on triangles with none, one and more than one point on the boundary ∂Ω. Thewhole theory would essentially remain correct. We leave its formulation with the manytechnical details to the interested reader.

So we only formulate the results with the same value of ρ = 0 for violated boundaryconditions and continuity. We restrict, as in Subsubsection 5.5.6, the presentation hereto the only practically relevant, standard case, cf. (5.327), required for the interior innerand the boundary edges,

standard discontinuous FEs with straight edges: 0 = ρ < d′ = d− 1. (5.336)

For FEs Vhb �⊂ V, we have already introduced the generalized 〈fh, ·〉, in (5.250), the

ah(·, ·) in (5.289) ff. and norms ‖ · ‖Vh , defined in (4.36). We study the impact ofdiscontinuous Vh

b �⊂ V onto the consistency estimates, cf. (5.279):

ah(uh, vh) =∑

T∈T hc

⎛⎝∫T

2∑i,j=0

aij∂i uh∂j vhdx

⎞⎠ ∀ u, v ∈ Vb + Vhb (5.337)

= as,h(uh, vh) +∑

e∈T hc \∂Ω

∫e

({vh}[Bauh] + [vh]{Bau

h})ds

+∑

e∈∂Ω

∫e

vh(Bauh)ds,

with uh ∈ Vb ∩H2(Ω) for as,h(uh, vh) and with Ba as in (5.278). Again we use theunified notation in (5.285). The exact and discrete solutions satisfy, cf. (5.280)

a(u0, v) = 〈f, v〉V′×V ∀v ∈ Vb, and ah(uh

0 , vh)

= 〈fh, vh〉Vh′×Vh ∀vh ∈ Vhb . (5.338)

In this two different approaches are possible for estimating the nonconformity errors.The μ-and σ-case are based on different proofs in Lemmas 5.65 and 5.70 The σ-caseyields slightly inferior consistency orders for less smooth solutions u0. The μ-caseallows slightly higher consistency orders but often needs much higher smoothness ofu0. For example, we get for the σ-case hd−1|u|Hd(Ω) instead of hd|u|H2d(Ω) for theμ-case.

Condition 5.69. Conditions for interpolation points on edges: We require Condition5.60 and modify Condition 5.64. In (5.336) we replace the Dirichlet conditions byinterpolation conditions on all straight inner and (curved) boundary edges, e ∈ T h

c .The uh, vh ∈ Vh, interpolate prescribed values vh(Pi) = yi, i = 0, . . . , d′ − 1,∀e ∈ T h

c


in the above Pi = P ρ=0,ei ∈ e, i = 0, . . . , d′ − 1, see (5.320). This implies continuity of

the FEs in the Pi, but not in e, even less in Ω.For problems with natural boundary conditions we do not have to impose boundary

conditions. For Dirichlet boundary conditions we additionally impose the originalCondition 5.64 along the boundary ∂Ω again with ρ = 0.

Accordingly we define for Ω = ∪T∈T hcT , see (5.321), possibly with curved e ∈ ∂Ω,

Vhb :=

{v : Ω → R : v|T ∈ Pd−1 ∀ T ∈ T h

c , [v](Pi) = 0,∀ e ∈ T hc (5.339)

v(Pi) = 0,∀ e ∈ ∂Ω ∀ i = 0, . . . , d′ − 1, Pi = P ρ=0,ei },

‖vh‖Vh := ‖vh‖H1(T hc ).

The second condition, v(P ρ=0,ei ) = 0, e ∈ ∂Ω, only applies to Dirichlet boundary con-

ditions; for natural boundary conditions it has to be omitted.Except these new Vh

b in (5.339), instead of (5.321) the whole formal procedure inthis and the preceding subsection is the same. Differences occure in the followingformulas, estimates and theorems due to the discontinuity in the many interior edgeshere, compared to only violated boundary conditions in the few boundary edges before.Correspondingly, we only recall the important concepts, Q

′h and discrete linear andquasilinear equations and operators are, cf. (5.294) ff.:

Q′h ∈ L

(W−1,p′ (T h

c

),Vh

b

′): 〈Q′hfh − fh, v

h〉W−1,p′ (T hc )×Vh = 0∀vh ∈ Vh

b , (5.340)

ah(uh

0 , vh)− 〈fh, vh〉Vh′×Vh =

⟨Ahu

h0 − fh, v

h⟩

H−1(T hc )×Vh = 0 ∀ vh ∈ Vh

b

and ah(uh

0 , vh)

=⟨Ghu

h0 , v

h⟩

W−1,p′ (T hc )×Vh = 0 ∀ vh ∈ Vh

b . (5.341)

Ah := Q′hAh|Vh

b, and Gh := Q

′hGh|Vhb⇒ Ahuh

0 − fh = 0, and Ghuh0 = 0. (5.342)

The variational discretization error for u ∈ H2(Ω) with the discrete solution uh ofthe linear Ahuh = Q

′hAu and the quasilinear generalization, see (5.305), (5.299), is

ah(u− uh, vh) =∑

e∈T hc

∫e

[vh]Buds +∑

e∈∂Ω

∫e

vhBuds ∀u ∈ H2(Ω). (5.343)

The classical consistency errors are, cf. (5.313), (5.339), (3.35),

‖AhIhu−Q′hAu‖V′ , ‖GhIhu−Q

′hGu‖V′ , (5.344)

and we need

supvh �=0

|ah(u− uh, vh)|‖vh‖Vh

= sup

| ∑e∈T h

c

∫e[vh]Buds +

∑e∈∂Ω

∫evhBuds|

‖vh‖Vh

.

The next lemma could be proved by the same technique as the previous Lemma5.65. However, we neglect the fact that along the straight edges e the parametrization


is affine. This allows modified estimates compared to Lemma 5.65 and a reducedsmoothness conditions for u. Since the interpolation error dominates the consistencyerror, cf. (5.348), we only discuss this updated form.

Lemma 5.70.

1. Under Conditions 5.60,5.69 and with the notation in (5.258), (5.318), we chooseA, a(·, ·), ah(·, ·), G, Gh as in (5.278) ff., the Vh

b as in (5.339), and u ∈ Hd(Ω).Let T1, T2 ∈ T h

c be two neighboring or T1 a boundary triangle with diam Ti ≤h, i = 1, 2 and a joint or a boundary straight edge and let

Ue := interior (T1 ∪ T2) for e ⊂ T1 ∩ T2 or Ue := T1, for |T1 ∩ ∂Ω| ≥ 2.

Then there exists a constant C = C(d, χ) such that, with |vh|H1h(Ue) :=

|vh|H1h(T1∪T2),

∣∣∣∣∫e

[vh − v]Bau ds

∣∣∣∣ ≤ C hd−1|u|Hd(Ue)‖vh − v‖H1h(Ue), ∀v ∈ H1(Ue) (5.345)

=∣∣∣∣∫

e

[vh]Bau ds

∣∣∣∣ ≤ C hd−1|u|Hd(Ue)|vh|H1h(Ue) for v = 0.

2. This remains correct for a straight boundary edge e ⊂ ∂Ω with [vh − v], . . . ,replaced by vh − v, . . . , but not for curved boundaries.

3. For quasilinear problems under the condition (5.335) for Theorem 5.68, theprevious estimates in (5.345) remain correct with Ba replaced by B = BG.

Proof. We apply a technique similar to Lemma 5.65 to our problem. For everystraight edge e ∈ T ∈ T h

c with length he, the parameterization xe is affine, see (5.319).This allows the following modification. We combine qd′

([vh]w) ◦ xe = 0 ∀vh ∈ Vhb ,

Propositions 5.61, 5.62 for q = 2 in (5.309), (5.311), with (5.339) to yield

∫ se+he

se

([vh]w) ◦ xeds =∫ se+he

se

([vh − v](w − p2)) ◦ xeds (5.346)

∀v ∈ H1(Ω) or ∀c1 ∈ R ∀ p2 ∈ Pd−2, [c] = 0 = [v], [vh] = [vh − v − c1],

for these c1, v. Note vhp2 ∈ P2d−3, hence∫ se+he

se([vh − v]p2) ◦ xeds = 0. We choose

in (4.34), Q1vh = c1, Qd−1Bau = p2. This allows estimating

5.5. Consistency, stability and convergence for FEMs with variational crimes 403∣∣∣∣∫e

[vh]Bauds

∣∣∣∣ = ∣∣∣∣∫e

[vh − v]Bauds

∣∣∣∣ (5.347)

=

∣∣∣∣∣∫ se+he

se

([vh − v − c1](Bau− p2)) ◦ xeds

∣∣∣∣∣≤ C‖xe‖L∞(e)‖[vh − v − c1]‖L2(e)‖(Bau− p2)‖L2(e) by (1.67)

≤ C(‖[vh − v − c1]‖L2(Ue)‖[vh − v − c1]‖H1

h(Ue)

)1/2

(‖(Bau− p2)‖L2(Ue)‖(Bau− p2)‖H1

h(Ue)

)1/2

≤ C(h‖vh − v‖2H1(Ue)

)1/2

(hd−1‖Bau‖Hd(Ue)hd−2‖u‖Hd(Ue))

1/2 with

(4.34),(4.41) and choosing in (5.346) the infimum for c1, p2,

≤ Chd−1|vh|H1h(Ue)|Bau|Hd−1(Ue) for v = 0 and

≤ Chd−1‖vh − v‖H1h(Ue)|Bau|Hd−1(Ue) ∀v ∈ H1(Ω),

hence, (5.345). The vh − v modification is an immediate consequence of (5.346) andthe use of ‖vh‖H1(T h

c ) instead of |vh|H1(T hc ).

With the above change of [vh − v], . . . , into vh − v, . . . , for a boundary triangleT2 ∩ ∂Ω �= ∅ this result (5.345) remains valid as well.

For quasilinear problems (5.347) remains correct for the nonlinear Bu = BGu. �

Theorem 5.71. Consistency for FEMs violating continuity conditions and conver-gence for coercive forms:

1. Let Ω and the FEs satisfy Conditions 5.60 and 5.69. With the notation in (5.258),(5.318), we choose A, a(·, ·), Ah, A

h, ah(·, ·),Vhb , Q

′h as in (5.278) ff., (5.339) ff.,and natural or Dirichlet boundary conditions.

2. Then Ah= Q′hAh|Vh

band Gh= Q

′hGh|Vhb

are consistent with A and G in u ∈ Hd

(Ω), respectively. The variational and classical consistency errors can beestimated for straight edges and polygonal ∂Ω, with an h-independent C=C(d,n,ρ,xb) as

sup0�=vh∈Vh

b

|ah(u− uh, vh)|‖vh‖Vh

≤ Chd−1‖u‖Hd(Ω), and (5.348)

‖AhIhu−Q′hAu‖Vh′ , ‖GhIhu−Q

′hGu‖Vh′ ≤ Chd−1‖u‖Hd(Ω).

3. Let ah(·, ·) be Vhb -coercive, e.g. the principal part, ah

p(·, ·). Then the solutions uh0

for Ahuh0 = Q

′hAu0 exist and converge to u0 according to∥∥uh0 − u0

∥∥H1(Ω)

≤ Chd−1‖u0‖Hd(Ω). (5.349)


4. For curved boundary edges in a curved ∂Ω, (5.349) has to be combined with ρ = 0in (5.332), and Condition 5.64 has to be imposed. Then (5.349) is replaced by∥∥uh

0 − u0

∥∥H1(Ω)

≤ Chd−1‖u0‖H2d(Ω). For the ρ = 0, 1, 2 combination indicated in

the introduction, we only have∥∥uh

0 − u0

∥∥H1(Ω)

≤ Chd−3/2‖u0‖H2d(Ω).

5. This remains correct for quasilinear problems under the conditions previous toTheorem 5.68, if the coercivity of ah(·, ·) is replaced by the coercivity of thebilinear form of the linearized operator G′(u0).

6. As in Lemma 5.70 the first estimate in (5.348) remains correct for vh − v,

sup|ah(u− uh, vh − v)|

‖vh − v‖Vh

≤ Chd−1‖u‖Hd(Ω), ∀ v ∈ H1(Ω). (5.350)

Proof. Similarly to Theorem 5.66 the consistency errors are immediate consequencesof Lemma 5.70. We sum over all e ∈ T h

c in Condition 5.60, where Ue is defined inLemma 5.70. We use |

∫e[vh]Bau ds| ≤ C hd−1|u|Hd(Ue)|vh|H1

h(Ue), see (5.306) and(5.345) estimating41

|ah(u− uh, vh)| =

∣∣∣∣∣∣∑

e∈T hc

∫e

[vh]Bau0ds

∣∣∣∣∣∣+∣∣∣∣∫

e∈∂Ω

vhBau0ds

∣∣∣∣≤ Chd−1

∑e∈T h

c ∪∂Ω

|vh|H1h(Ue)|u|Hd(Ue) (5.351)

≤ Chd−1∑

T∈T hc ∪∂Ω

|vh|H1(T )|u|Hd(T ) (by Holder)

≤ Chd−1|vh|Vh · |u|Hd(Ω).

The analogous ‖vh − v‖Vh extension of Lemma 5.70 is verified as there. �

Remark 5.72. Lemma 5.70 is strictly local with the local step size h and the locallyappropriate, e.g. ‖u‖Hd(Ue) valid for all u ∈ Hd(Ue). This can be used in two directions.Either locally large Bu and discretization errors are compensated by mesh refinementsor the so-called hp-methods are employed. For locally smooth solutions u0 and Bu0,high orders of convergence d− 1 and large he, for locally unpleasant u0 and Ba u0,small orders d− 1 and small he have to be combined. These are the so-called adap-tive FEMs, introduved in Chapter 6. The hp viewpoint is discussed for DCGMs inSection 7.14.

We generalize the Aubin–Nitsche Lemma 4.59 and the corresponding convergenceresult in Theorem 4.60 to the case of variational crimes, with straight edges e, see,e.g. [135]. Here, we need the extension ah(·, ·) :

(Vb + Vh

b

)×(Vb + Vh

b

)→ R instead

of Vhb × Vh

b → R. In addition to the original problem a(u0, v) = (f, v)∀v ∈ Vb we needthe solution, φg, of its dual problem,

φg ∈ Vb s.t. a(w, φg) = (g, w)∀w ∈ Vb. (5.352)

41 For 0 ≤ a1 ≤ a2, 0 ≤ b1 ≤ b2 we have (a1 + a2)(b1 + b2) ≤ a1b1 + 3a2b2 and every T ∈ T hc con-

tains only a few edges, e.g. three for triangulations.


Lemma 5.73. Generalized Aubin–Nitsche lemma: Let a(·, ·) : Vb × Vb → R and itscorresponding ah(·, ·) :

(Vb + Vh

b

)×(Vb + Vh

b

)→ R be continuous bilinear forms with

ah(·, ·)|Vb×Vb= a(·, ·), see (5.279). Assume that the original and its dual problem

satisfy the conditions for unique exact and FE solutions u0, uh0 and φg, φ

hg . For the

Banach space W ⊃ Vb with inner product and norm (g, w) and ‖g‖W , respectively,let the embedding Vb ↪→W be continuous. Then the error for the FE solution can beestimated in the W norm as∥∥u0 − uh

0

∥∥Wh ≤ sup

0�=g∈W(1/‖g‖W)

(C∥∥u0 − uh

0

∥∥Vh

∥∥φg − φhg

∥∥Vh (5.353)

+∣∣ah

(u0 − uh

0 , φg

)−(u0 − uh

0 , gh

)∣∣+∣∣ah

(u0, φg − φh

g

)−(fh, φg − φh

g

)∣∣ ).Proof. We employ the dual norm of an element w of a Banach space in (1.32) andcombine the solutions u0, u

h0 and φg, φ

hg of both problems to find, see [135],(

u0 − uh0 , gh

)= ah(u0, φg)− ah

(uh

0 , φhg

)= ah

(u0 − uh

0 , φg − φhg

)+ ah

(uh

0 , φg − φhg

)+ ah

(u0 − uh

0 , φhg

)= ah

(u0 − uh

0 , φg − φhg

)−[ah

(u0 − uh

0 , φg

)−(u0 − uh

0 , gh

)]−[ah

(u0, φg − φh

g

)−(fh, φg − φh

g

)].

For the equality in the second and fourth lines we use the linearity of the ah(·, ·) andsplit and replace the linear functionals in the [..] by the corresponding a(·, ·), ah(·, ·),respectively. Finally, use (1.32), and the continuity of ah(·, ·) to obtain (5.353). �

We combine Lemma 5.73 with Theorem 5.71, but not with Theorem 5.66:

Theorem 5.74. Aubin–Nitsche theorem for nonconforming FEMs: Under the condi-tions of Theorem 5.71 and Lemma 5.73 we allow standard FEs and FEs violating thecontinuity in (5.336) and assume u ∈ Hd(Ω). Then∥∥u0 − uh

0

∥∥L2(Ω)

≤ Chd‖u‖Hd(Ω). (5.354)

Proof. We apply Lemma 5.73 to our situation: we chooseW = L2(Ω) and Vb = H10 (Ω).

This is justified since, by ‖u‖L2(Ω) ≤ ‖u‖H1(Ω) ∀ u ∈ H10 (Ω), the H1

0 (Ω) is continuouslyembedded into L2(Ω).

We estimate the terms in (5.353). For the first term in the second line of (5.353) weget, with the unusual defining relations for φg φh

g and (5.347), (5.350),

[ah

(u0 − uh

0 , φg

)−(u0 − uh

0 , g)]

= [a(u0, φg)− (u0, g)]− [ah

(uh

0 , φg

)−(uh

0 , g)]

= −[ah

(uh

0 , φg

)− ah

(uh

0 , φhg

)]= −

[ah

(uh

0 , φg − φhg

)]= −

[ah

(u0 − uh

0 , φg − φhg

)].


Similarly we find for the second term in the second line of (5.353), with the definingrelations for u0 and uh

0 ,

[ah

(u0, φg − φh

g

)−(f, φg − φh

g

)]

= [a(u0, φg)− (f, φg)]−[ah

(u0, φ

hg

)−(f, φh

g

)]= −

[ah

(u0, φ

hg

)− ah

(uh

0 , φhg

)](5.355)

= −[ah

(u0 − uh

0 , φhg

)]= −

[ah

(u0 − uh

0 , φhg − φg

)],

again with (5.347), and (5.350), now applied to the dual problem (5.352). We combinethe (5.347) with the continuity of the ah(., .) and the estimates for∥∥u0 − uh

0

∥∥H1(T h

c )≤ Chd−1‖u0‖Hd(Ω),∥∥φg − φh

g

∥∥H1(T h

c )≤ Ch‖φg‖H2(Ω) ≤ Ch‖g‖L2(Ω).

This allows, with (5.355) and by the continuity of ah(., .), the estimates∣∣ah

(uh

0 , φg − φhg

)∣∣ = ∣∣ah

(uh

0 − u0, φg − φhg

)∣∣≤ C

∥∥u0 − uh0

∥∥Vh

∥∥φg − φhg

∥∥Vh

≤ Chd‖u0‖Hd(Ω)‖φg‖H2(Ω)

≤ Chd‖u0‖Hd(Ω)‖g‖L2(Ω).

In fact the second and the following lines in this inequality can be used for the termin the first line of (5.353) and by (5.355) the two last terms coincide. So estimatingthe three terms in (5.353) yields, with W = L2(Ω) and V = H1(Ω),∥∥u0 − uh

0

∥∥L2(Ω)

≤ sup0�=g∈L2(Ω)

(1/‖g‖L2(Ω))(C∥∥u0 − uh

0

∥∥Vh

∥∥φg − φhg

∥∥Vh

+∣∣ah

(uh

0 , φg − φhg

)∣∣+ ∣∣ah

(uh

0 − u0, φg − φhg

)∣∣ )≤ sup

0�=g∈L2(Ω)

(1/‖g‖L2(Ω))(Chd‖u0‖Hd(Ω)‖g‖L2(Ω)

),

the claim. �

5.5.8 Stability for nonconforming FEMs

We turn to proving the stability for FEMs with variational crimes. Chapter 3 showsthat we can confine the proof to the linear problems. By Theorem 3.23, stabilityis implied by the stability of the linearized Ah and some technical conditions. InSection 4.3 we could directly apply the results in Chapter 3. In Sections 5.2 and 5.4specific ideas allowed us to overcome the difficulties with variational crimes in the fullynonlinear and quadrature approximate FEMs. Here we will procede in a similar way.

We have already proved in Theorem 5.59, assuming (5.300), see Remark 5.58, thestability of the above nonconforming FEMs for the principal parts, Ap, or specific


compact perturbations, B, yielding a stable Bh. Here A was a linear operator for anyof our above elliptic equations or systems of order 2. Furthermore, we have proved inTheorems 5.66 and 5.71 the consistency for general linear operators and extended itto quasilinear operators in Theorems 5.68 and 5.71, 5.

We have seen in Sections 4.3 and 5.2 that the existence of a bounded inverse of alinear A is related to, but not sufficient to ensure that the discrete Ah is stable, see [128,567]. The next Theorem 5.78 guarantees the stability for nonconforming FEMs studiedin this section for all types of elliptic equations and systems of order 2. It shows thata linear invertible B with stable Bh inherits the stability to Ah, for a linear invertiblecompactly perturbed A, such that Ah is consistent with A. This consistency is alwaysvalid for conforming FEs and is assumed for general discretizations in Theorem 3.29.It is not too surprising that the perturbations by the nonconformity of the FEs haveto be small. Good convergence of the discrete solutions is only to be expected forconsistency of higher order, so only for higher order methods and smooth enough u0.

For the following considerations we recall the different consistency errors

Q′hAhu−Ahuh︸︷︷︸var.cons.err.


interp.err.


class.cons.err.

, withAhuh = Q′hAu.

(5.356)

Condition 5.75. Choose Vhb ⊂ V as in (5.321), or (5.339) and require Condition

5.60, (5.300), and Ph = Ih and Q′h as in Proposition 5.55, (5.294), (5.295). Choose

the original linear problems and the FEMs as listed in (5.278)–(5.280), (5.281)–(5.284), (5.297)–(5.298).

Lemma 5.76. Under Condition 5.75, let Ap : Vb → V ′ be a linear principal part,cf. (5.278) or (5.281). Then Apu = f ∈ H−1(Ω), and for small enough h0 > h ∈ H,

Ahpu

h = Q′hf as well, have unique solutions, and uh

0 converge to u0 as∥∥uh0 − u0

∥∥Vh ≤ ε ‖u‖V , for h < h0, hence lim

h→0

∥∥uh0 − u0

∥∥Vh = 0. (5.357)

For general A, u ∈ Vb, Ph = Ih, with Ahuh = Q′hAu, the preceding consistency errors

converge to 0, as

limh→0

‖Q′hAhu−Ahuh‖Vh′ = 0, limh→0

‖Q′hAu−AhPhu‖Vh′ = 0. (5.358)

Proof. By Proposition 5.54, all the bilinear forms ah(·, ·) are bounded and continuous.By Theorem 5.59, Ah

p is stable in Phu. In Theorems 5.66, 5.71, we have formulated thesmoothness of the u guaranteeing the optimal order of consistency. So we choose, e.g.u ∈ Hd(Ω), near u. Then Theorem 4.53 implies that for u and u we have convergence,hence, e.g. for Ah

p uh = Q

′hApu, cf. (5.357), (5.358),

‖u− uh‖Vh ≤ Chd−1‖u‖Hd(Ω) < ‖u‖Vε/3 for h < h0.

For estimating

‖u− uh‖Vh ≤ ‖u− u‖Vh + ‖u− uh‖Vh + ‖uh − uh‖Vh < ε,


we handle the first and the last term, ‖u− u‖Vh = ‖u− u‖V and ‖uh − uh‖Vh . Weemploy the definition of Ah

p , Ap cf. (5.279), and choose u such that with the followingbounded maximum,

max{

1,∥∥∥(Ah

p

)−1∥∥∥Vh

b ←↩V′hb

‖Q′h‖V′hb ←↩V′

b‖Ap‖V′

b←↩Vb

}‖(u− u)‖V < ‖u‖V ε/3,

hence∥∥Ahp(uh − uh)

∥∥V′h = ‖Q′hAp(u− u)‖V′h =⇒∥∥uh − uh

∥∥Vh ≤

∥∥∥(Ahp

)−1∥∥∥Vh

b ←↩V′hb

‖Q′h‖V′hb ←↩V′

b‖Ap‖V′

b←↩Vb‖(u− u)‖V ≤ ‖u‖V ε/3,

hence (5.357). For the general Ah this argument yields (5.358). �

We combine the form of the consistency errors in Proposition 5.56, the stability ofthe principal parts in Theorems 5.59, and the consistency, specified in Theorems 5.66and 5.71, to obtain the convergence of the discrete to the exact solution. This can beused defining an anticrime transformation from Vh

b to Vb, the essential tool for provingthe following stability result in Theorem 5.78, cf. Proposition 5.42.

Lemma 5.77. Anticrime transformation: Let A0 ∈ L (Vb,V ′b), e.g. A0 = Δ be bound-

edly invertible, Ah0 : Vh

b → Vh′be stable and Ah

0 be consistent with A0 for every u ∈ Vb.For a given uh ∈ Vh

b define fh := Ah0u

h ∈ H−1(T h

c

), cf. (5.250), (5.279). This fh ∈

H−1(T h

c

)defines, as proved in Proposition 5.42, by its components f−1, f0 via (5.250)

an fh ∈ H−1(Ω), such that ‖fh‖H−1(Ω) ≤∣∣|Ah

0

∣∣ |Vh′←↩Vhb‖uh‖Vh . Define Eh : Vh

b → Vb

by uh := Ehuh := A−10 fh ∈ Vb with ‖Ehuh‖H1(Ω) ≤ C1‖uh‖H1(T h

c ). Then we obtainconvergence as: ∀ε > 0 ∃h0 = h0

(‖uh‖H1(T h

c )

): ∀h < h0 :

‖Ehuh − uh‖H1(T hc ) < ε‖uh‖H1(T h

c ),hence, limh→0

‖Ehuh − uh‖H1(T hc ) = 0. (5.359)

For fh ∈ L2(Ω), we even get ‖Ehuh − uh‖Vh ≤ Ch‖fh‖L2(Ω) for h→ 0 implying con-vergence of order 1.

Theorem 5.78. Compactly perturbed invertible and consistent discretizationsinherit stability: Under the Condition 5.75, let A ∈ L (Vb,V ′

b) be boundedly invertibleor be one of the linear(ized) operators for equations or systems, studied in this section,see (5.278), or (5.281), their extensions (5.279), or (5.283), and the corresponding FEequations (5.280), or (5.284). Finally, let B ∈ L (Vb,V ′

b), Bh be stable and Ah, Bh,hence Ch, be consistent with A,B,C, for u ∈ Vb. Then, for small enough h,

A−1 ∈ L(V ′,Vb) =⇒ Ah(or its quadrature approx. Ah) is stable in Phu. (5.360)

Proof.

1. We employ the anticrime transformation, Eh : Vhb → H1

0 (Ω), introduced inLemma 5.77. We derive several relations and inequalities. The proof is similar tothat of Theorem 5.9. The differences, mainly from (5.369) ff., are caused by the


necessary PhEhuh. As discussed it suffices to show that A−1 ∈ L(V ′,Vb) impliesthe stability of Ah with

A,B,C ∈ L(Vb,V ′), C compact, A =: B + C, with B,Bh

boundedly invertiblestable, and Ah, Bh, Ch consistent with A,B,C

in u ∈ H1(Ω) or ∈ H2(Ω). (5.361)

In fact, for all nonconforming FEMs in this section, the consistency of the Bh, Ch

is as correct as that of Ah. We use the notation in Proposition 5.55, (5.294)–(5.298),

Ah, Bh, Ch :(Vb + Vh

b

)→ Vh′

, Q′h : (V ′ + Vh′

) → Vh′,

Ah = Q′hAh|Vh

b: Vh

b → Vh′, . . .

with ΦhA = Ah, . . . , cf. (5.298). The bounded invertibility and stability of Band Bh and consistency with B in u imply the unique existence of the followingsolutions u and uh, defined, for an arbitrary u ∈ Vb and v′ := Cu by

Bu = v′ and Bhuh = Q′hv′.

With equicontinuous Bh, and Q′h, and Φh(B · −v′) = Bh · −Q′hv′, let

T := B−1 and Th := (Bh)−1Q′h ∈ L

(V ′ + Vh′

,Vhb

). (5.362)

Then the convergence of the uh to u can be written as

∀u ∈ Vb : limh→0

||(T − Th)Cu||H1(T hc ) = 0. (5.363)

This implies, since C is compact and the T − Th are equibounded,

||(T − Th)C||H1(T hc )←↩Vb

→ 0, for h→ 0. (5.364)

The limit (5.363) remains correct if u ∈ Vb is replaced by any bounded uh =Ehu ∈ Vb, depending upon h, with ||uh||V = ‖Ehuh‖H1(Ω) ≤ C2‖uh‖H1(T h

c ).Indeed the consistency estimate, cf. Remark 5.49, and the stability of Bh canbe applied to any bounded uh ∈ Vb as well:

||(T − Th)CEhuh||H1(T hc )←↩Vh

b≤ C3||(T − Th)C||H1(T h

c )←↩Vb‖uh‖H1(T h

c ).

(5.365)

We combine uh ∈ Vhb with Eh. With the boundedly invertible A we estimate

‖uh‖Vh ≤ 2||Ehuh||V ≤ 2||A−1||Vb←↩V′ ||AEhuh||V′

= 2||A−1||Vb←↩V′ ||B(I + TC)Ehuh||V′ (5.366)

≤ 2||A−1||Vb←↩V′ ||B||V′←↩Vb||(I + TC)Ehuh||V ,

hence,

||(I + TC)Ehuh||V ≥ ‖uh‖Vh/(2||A−1||Vb←↩V′ ||B||V′←↩Vb). (5.367)


With the stability of Bh we get

‖(I + ThCh)uh‖Vh = ‖(Bh)−1Bh(I + ThCh)uh‖Vh

≤ ‖(Bh)−1‖Vhb ←↩Vh′ ‖Bh(I + ThCh)uh‖Vh′ .

This implies

‖Bh(I + ThCh)uh‖Vh′ ≥ ‖(I + ThCh)uh‖Vh / ‖(Bh)−1‖Vhb ←↩Vh′ . (5.368)

The linearity of FEM with variational crimes implies

Φh(A) = Ah = Φh(B + C) = Bh + Ch

= Bh(I + (Bh)−1Ch) = Bh(I + ThCh) : Vhb → Vh′

.

For the following estimate we refer to (5.359), and Theorem 4.17 and obtain with(5.357),

‖uh − PhEhuh‖H1(T hc ) ≤ ‖(Ehuh − PhEhuh)‖H1(T h

c ) (5.369)

+ ‖(uh − Ehuh)‖H1(T hc ) ≤ 2ε‖uh||Vh .

2. For the final stability estimates we return to the (Bh)−1Ahuh = (I + ThCh)uh

term. The identities Q′h|Vh′ = IVh′ , Th|Vh′ = (Bh)−1Q

′h|Vh′ = (Bh)−1, the equi-boundedness of Ch and Th, the consistency of Ch, and the indicated error terms,are combined with the triangle inequality estimating

‖(Bh)−1‖Vhb ←↩Vh′

b||Ahuh||Vh′ = ‖(Bh)−1‖Vh

b ←↩Vh′b||Bh(I + (Bh)−1Q

′hCh)uh||Vh′

≥ ‖(Bh)−1‖Vhb ←↩Vh′

b||Bh(I + ThCh)uh||Vh′ (5.370)

≥(||(I + TC)Ehuh||V − ||uh − Ehuh||Vh − ||ThChuh − TCEhuh||Vh

)≥(||(I + TC)Ehuh||V − ||uh − Ehuh||Vh − ||(Th − T )CEhuh||Vh

− ||Th(Chuh − CEhuh)||Vh

)≥(||(I + TC)Ehuh||V − ||uh − Ehuh||Vh − ||(Th − T )CEhuh||Vh

− ||Th(ChPhEhuh − CEhuh)||Vh − ||Th(Ch − ChPhEh)uh||Vh

)


≥(|| (I + TC)Ehuh︸︷︷︸

by (5.367)

||V − || (uh − Ehuh)︸︷︷︸FEM error (5.369)

||Vh − 2|| (Th − T )C︸︷︷︸by (5.364)

||Vhb ←↩Vb

||uh||Vh

− ||Th (ChPhEhuh − CEhuh)||Vh︸︷︷︸class.consistency of C,cf.Theorems 5.66, 5.71

− ||ThCh(

(uh − Ehuh)︸︷︷︸FEM error (5.369)

+ (Ehuh − PhEhuh)︸︷︷︸interpol.error (4.41)

)||Vh

).

We choose h small enough, such that all the preceding errors in (5.369), (5.364),(5.358), (4.41) are < ε||uh||Vh with

ε < 1/(4(3 + ||Th||Vh

b ←↩Vh′b

+ 2||ThCh||Vhb ←↩Vh

b)||A−1||Vb←↩V′ ||B||V′←↩Vb

).

This implies

||Ahuh||Vh′ ≥(||uh||Vh/

(4||A−1||Vb←↩V′ ||B||V′←↩Vb

))/||Bh−1||Vh

b ←↩Vh′

≥ K ′||uh||Vh , (5.371)

i.e. (Ah)h∈H is stable. �

Theorem 5.78 yields a criterion for the stability of discretizations of operators, whichare compact perturbations of coercive operators. An important class of compactlyperturbed coercive operators are the A ∈ L (Vb,V ′

b) satisfying a so-called Gardinginequality. Hence Theorem 3.32 can directly be generalized to our FEMs with varia-tional crimes.

5.5.9 Convergence, quadrature and solution of FEMs with crimes

Now we have proved all necessary results guaranteeing consistency, stability, andconvergence for the FEMs with variational crimes studied in this section. It wouldbe boring to repeate all the different complex conditions for the many different casesin this section. We prefer the formulation of a summarizing theorem. It collects thedifferent desired results.

Theorem 5.79. FEMs for elliptic problems of second order, violating homogeneousDirichlet boundary conditions and/or continuity:

1. These are stable under the conditions in Theorems 5.59 (coercivity), 5.78 (stabil-ity), consistent for violated Dirichlet conditions and/or continuity in Theorems5.66 and 5.71. Hence all these methods converge with the same estimates as theirconsistency, multiplied by a stability constant S.

2. Extensions to quasilinear problems in Theorems 5.68 and 5.71, 5. imply thecorresponding convergence results.

3. An improvement of the convergence rate for linear problems with violated conti-nuity and with respect to the L2(Ω) norm is formulated in Theorem 5.74.


The quadrature approximations are illustrated in Figure 5.5. Now Vhb are the FEs

violating boundary conditions and/or continuity. Formulating with the quadratureprojectors yields, cf. (5.183), with the Q

′h in (5.180), and Ah in Theorem 5.66 and in(5.342),

Ah = Q′hAh|Vh , Ahuh

0 = Q′hf. (5.372)

The stability for Ah = Q′hAh|Vh is proved in Theorem 5.30. So according to Theorems

3.21, 4.53 and the continuity of A, we only have to show the classical consistency ofAh with A in the exact solution u0 yielding convergence in Theorem 5.80.

The corresponding smoothness of Sobolev norms is collected in C = Cσ or C = Cμ.We require, for the coefficients aij and the inhomogeneity f ,

aij ∈ W k∞(Ω), i, j = 0, . . . , n, f ∈ Hk(Ω). (5.373)

We impose the condition k > max{2(m + τ − 1), 2} for a quadrature version of thesame order of convergence as the conforming FEM. u0 ∈ H μ(Ω) is motivated byTheorem 5.71 and Remark 5.72. We have proved in Theorem 5.30 that for k > 2(d− 2)the ah(·, ·) and ah(·, ·) are simultaneously bounded and define stable Ah and Ah.

Theorem 5.80. Quadrature for nonconforming FEs:

1. Assume n = 2, nonconforming FEs and quadrature formulas, exact for piecewisepolynomials of degree k − 1 with k > max{2(d− 2), 2}. Let (5.176),(5.373) forthe aij , f , and Conditions 5.60 and 5.69 be satisfied. Let ah(·, ·) induce a stableAh. Then

ah(uh

0 , vh)

= 〈f, vh〉h∀vh ∈ Vh, (5.374)

defines a stable and convergent discretization for Au0 = f . For the other approx-imations, mentioned above, this result remains correct if the ah, Ah are replacedby ah, ah, Ah, Ah.

2. For u0 ∈ Hd−p(Ω), p ≥ 0, we obtain (M := Maij:=

2maxi,j=0

‖aij‖W k,∞(Ωhc )) :∥∥u0 − uh

0

∥∥Vh ≤ Chk+2−d(‖f‖V′ + M‖Phu0‖Hd−1(T h

c )) + Chd−1−p‖u0‖Hd−p(Ω).

(5.375)

3. For violated boundary conditions on curved boundaries, hd−1‖u0‖Hd(Ω) has tobe replaced by hd−1‖u0‖W 2d−1,∞(Ω). For isoparametric FEs the Phu0 has to becombined with Ecu0.

4. If only approximations for 〈f, vh〉h and none for ah(uh, vh) are used in (5.374),then the term M‖u0‖Hmin{s,k+1}(Ω) in (5.375) is dropped.

Proof. Ah is stable by Theorems 5.59 and 5.30 for k > 2(d− 2) and the conditions inthe theorem. So it suffices to estimate the classical consistency errors

‖AhPhu0 − Q′hAu0‖Vh′ ≤ ‖AhPhu0 −AhPhu0‖Vh′

+‖AhPhu0 −Q′hAu0‖Vh′ + ‖Q′hf − Q

′hf‖Vh′ .


The terms on the right represent the quadrature errors for Au0 and Q′hf and the

nonconforming consistency errors of the exact FEM for Ah in u0. Estimates are provedin Theorems 5.66, 5.71, and 5.28, cf. Remark 5.67. We get

≤ Chk+2−d(‖f‖H−1k (T h

c ) + Maij‖Phu0‖Hd−1(T h

c )) + Chd−1d‖u0‖Hd(Ω). (5.376)

If only u0 ∈ Hd−p(Ω), p ≥ 0, the consistency error is reduced to Chd−1−pd‖u0‖Hd−p(Ω).Now the above k > 2(d− 2) implies k + 2− d ≥ d− 1. So the term Chd−1−pd‖u0‖Hd−p(Ω) dominates in (5.376) and we do not specify the other terms. �

Theorem 5.68 for violated boundary conditions and quasilinear problems in the lastsubsection remains nearly unchanged.

Theorem 5.81. Quadrature approximate FEMs with crimes for quasilinear prob-lems: For the quasilinear problem in Section 4.4, (4.174) let its FEM in (5.286) satisfythe conditions of Theorems 5.68 and 5.71. In particular we require the necessaryconditions for the Nemickii operators Ai(x, uh,∇uh), e.g. for bounded forms and oper-ators, ah(uh, vh) and Gh, as formulated in Sections 2.5.4, 2.5.6, 4.4 and (4.174). Weadditionally modify them, such that G is Lipschitz-continuous. Then for a boundedlyinvertible linearized operator the convergence and quadrature approximation results inthis subsection remain correct.

We discuss the solution of the nonlinear equations here for all the previous FEMswith crimes, including quadrature approximations. These FEMs with crimes are linear,cf. ( 4.154). Hence all the quasilinear equations and their quadrature approximationscan be efficiently solved with the methods in Section 3.7, combining continuation withNewton’s methods and the mesh independence priciple in Theorems 5.21, 3.40. By(5.375) and for u0 ∈ Hd−p(Ω,Rq), d− p > 1, we obtain convergence of order d− p−1 > 0. We impose the following Lipschitz-continuity for G′,

G′(·) ∈ CL(Br(u0) ∪Br(Phu0)), (5.377)

G|Vd−p: D(G) ∩

(Vd−p := Hd−p(Ω) ∪Hd−p

(Ωh

c

) )→ Hd−p−1(Ω) ∪Hd−p−1

(Ωh

c

).

Theorem 5.82. Newton’s method for the FE equations: In addition to the conditionsin Theorems 5.80 and 5.81 we assume (5.377). Start the Newton process for G andGh or Gh with u1 ∈ Br(u0) ∩ D(G) ∩ Vd−p and uh

1 := Phu1. Then, for small enough‖u0 − u1‖V and h, both methods converge quadratically and the sequences ui, u

hi satisfy,

independend of h, for i = 1, . . . ,∥∥uhi+1 − uh

i

∥∥Vh ≤ C

(∥∥uhi − uh

i−1

∥∥Vh

)2and

∥∥uhi − Phui

∥∥Vh ≤ Chd−p−1‖ui‖Vd−p

.

(5.378)

We have presented the proof and futher details for fully nonlinear problems inTheorems 5.21 and 4.64.


5.5.10 Isoparametric FEMs

In this subsection we generalize isoparametric FEMs from the standard second orderlinear elliptic equations to second order linear and quasilinear elliptic equationsand systems. Even extensions to order 2m,m ≥ 1, in Rn, would be possible, if theinterpolation theory in Subsection 4.2.7 could be approporiately generalized to Cm−1

reference elements. The essential tool is, as in the case of fully nonlinear problems, thegeneral discretization theory in Chapter 3 and Summary 4.52.

Motivated by Proposition 5.63 we only discuss Dirichlet boundary conditions. Forsimplicity we again formulate the computations for n = 2 and the Laplacian, andindicate the generalizations to linear equations and systems of order 2 and themodifications for the quasilinear case and n ≥ 2.

For the consistency estimates we recall the different tools introduced in Definition4.42, (4.99), (4.100), (4.101). We recall the definition of the original and approximatingpolygonal domains Ω and Ωh, the isoparametric FEs and the mappings in (4.102),(4.103), (4.100), (4.101). The isoparametric FEM uses Vh

b = Vhb .

Fh : Ωh → Ωhc := Fh(Ωh), Fc : Ωh → Ω, and Ec := Fh ◦ F−1

c : Ω → Ωhc ,

uh, vh : Ωhc = Fh(Ωh) → R, uh = (uh ◦ Ec) : Ω→ R,

Vhb = Vh

b , Vhb = Vh

b :={uh : uh ∈ Vh

b

}, and

f : Ωhc → R, f(x) := f((Ec)−1(x)) = (f ◦ (Ec)−1)(x) with (5.379)

t− Ec(t) = O(hd) =⇒ (Ec)′ − IdΩ = O(hd−1), ((Ec)−1)′ − IdΩhc=O(hd−1).

The isoparametric FEM with the above Vhb are formulated directly for the general,

including the quaislinear case. We use the notation aij∂i u ∂j v, redefining them

for systems and the matrices aij = Aij as aij∂i uh, ∂j vh = (Aij∂

i uh, ∂j vh)q. Fordefining the extensions ah(·, ·), . . . , Gh, as above, we

assume aij , f, Ai defined on Ω∪ := Ω ∪ Ωhc for Ai in the sense of (5.5). (5.380)

We denote the linear and quasilinear cases and the projectors as ah(·, ·), ah(·, ·),Ah, A

h, Gh, Gh, and Q

′h and Q′hc ., cf. (5.286), Proposition 5.55. For a unified notation

we use W 1,p(Ω∪) = H(Ω∪) with A = G, p = 2 for42 the linear case and D(G) =∩n

i=0D(Ai) ⊂ Vb. Then we consider

A : H1(Ω) = V → H−1(Ω) = V ′ or Vb → V ′b and G : D(G) ⊂ V → V ′ or ⊂ Vb → V ′

b

42 For simplicity often only f ∈ L2

(Ωh

c

). In this case the definition (f, vh)h :=

∫Ωh

cf(x)vh(x)dx

is used. J(Ec)′(t) denotes the absolute value of the determinant of the Jacobian of(Ec)′(t).


UbA, G

Ph Φh Q ′ch

Ubh

U ′ tested by←→ Ub

Ah, Gh Uh ′ tested by←→

Ubh ⊄ U

Figure 5.8 Nonconforming FEMs: Spaces and operators

Ah : H1(Ω∪) → H−1(Ω∪) : 〈Ahu, v〉H−1(Ω∪)×H1(Ω∪) := ∀u, v ∈ H1(Ω∪),

ah(u, v) : =∫

Ω∪

⎛⎝ n∑i,j=0

aij∂i u ∂j v

⎞⎠ (x)dx, and

ah(uh, vh) = 〈Ghuh, vh〉W−1,p′×W 1,p(Ω∪) =

∫Ω∪

n∑i=0

(Ai(x, uh,∇uh), ∂ivh)qdx

and ah(·, ·) := ah(·, ·)|W−1,p(Ωhc )×Vh

b, a(·, ·) = ah(·, ·)|W−1,p(Ω)×Vb

Q′h : H−1(Ω) → Vh′

: 〈Q′hf − f , vh〉H−1(Ωhc )×Vh

b∀vh ∈ Vh

b

=∫

Ωhc

(Q′hf − f)(x)vh(x)dx for f ∈ L2(Ω) (5.381)

=∫

Ω

(Q′hf − f)(t)vh(t)J(Ec)′(t)dt = 0 and

Q′hc : H−1

(Ωh

c

)→ Vh′

: 〈Q′hc f − f , vh〉H−1(Ωh

c )×Vhb

∀vh ∈ Vhb

=∫

Ωhc

(Q

′hc f − f

)(x)vh(x)dx = 0 for f ∈ L2

(Ωh

c

)and

〈fh, vh〉 := 〈f, vh〉h := 〈f , vh〉H−1(Ωhc )×H1

0 (Ωhc ) = 〈Q′hf, vh〉H−1(Ωh

c )×Vhc

and

Ah := Q′hc Ah|Vh

band Q

′hc Gh|Vh

b: Vh

b → Vh′, with the

above ah(uh, vh) = 〈Ahuh, vh〉Vh′×Vh and 〈Ghuh, vh〉Vh′×Vh .

Here the spaces Vhb �⊂ V are defined on Ωh

c , and thus cannot satisfy the boundaryconditions, cf. Figure 5.8.

Theorem 5.83. Summary for isoparametric FEMs for linear and quasilinear secondorder equations and systems, cf. the following proofs:

1. Let A ∈ L(Vb,V′

b) be a boundedly invertible linear elliptic differential operatorand f ∈ L2(Ω). Then the exact solution u0 of Au0 = f uniquely exists, withu0 ∈ H2(Ω) for convex Ω. For the quasilinear G assume a unique solutionu0 ∈ H2(Ω) with boundedly invertible G′(u0). Finally, let Ω ⊂ R2 be bounded andhave a piecewise smooth boundary.


2. Then the isoparametric FEMs with the equations Ahuh0 = fh and Ghuh

0 = 0, seeTheorems 5.84 ff. for the necessary conditions, are consistent and stable and(locally) unique discrete solutions uh

0 exist and converge to u0 ∈ Hd(Ω)∥∥u0 − uh0

∥∥H1(Ω)

≤ Chd−1(‖u0‖Hd(Ω) + ‖u0‖W 1

∞(Ω)

).

3. Quadrature approximations as in Theorems 5.80 and 5.81 yield correspondingconvergence.

4. For isoparametric FEs the above nonlinear Ec, see (5.379), determines thediscretization. However, since (Ec)′ − id = O(hd), see (4.101), the discrete Ah

and Gh for the isoparametric case is still k-times consistently differentiable withA and G in u0, respectively. This concept will play an essential role for the meshindependence principle and in Bohmer [120] for numerical methods in bifurcationand center manifolds.

In a first step we prove the discrete coercivity, cf. Theorems 5.59 and 4.61:

Theorem 5.84. Isoparametric coercivity: We choose the above Ω ⊂ R2, the isopara-metric FEs and the ah(·, ·), 〈f, ·〉h and Q

′h. If a(·, ·) is Vb-coercive, see (5.302), thenah(·, ·) is Vh

b -coercive.

Proof. We present it for the Laplace operator using the substitution rule in

ah(uh, vh) =∫

Ωhc

∇uh(x) · ∇vh(x)(dx) (5.382)

=∫

Ω

∇uh(Ec(t)) · ∇vh(Ec(t)) det(Ec)′(t)dt. (5.383)

The usual |det(Ec)′(t)| is replaced by det(Ec)′(t), since, with the identity id,

(Ec)′(t) = id +O(hd−1), hence, det(Ec)′(t) = 1 +O(hd−1) > 0 (5.384)

for sufficiently small h. For the above uh(t) = uh(Ec(t)) and uh(x) the derivatives∂ti ,∇t and ∂xi ,∇x are always understood with respect to the variable t for uh and xfor uh, respectively. Then we find with the unit vector ei

∂iuh(t) = ∂ti uh(t) = ∂((uh ◦ Ec)(t))/∂ti = ∇xuh(Ec(t))∂iEc(t)

= ∇xuh(x)(Ec)′(t)ei, with x = Ec(t) and

∇uh(t) = ∇tuh(t) = ∇x(uh(Ec(t))

)(Ec)′(t) = ∇xuh(x)(Ec)′(t) or

∂iuh(x) = ∂xiuh(x) = ∂((uh ◦ (Ec)−1)(x))/∂xi = ∇tuh(t)∂xi((Ec)−1(x)

)= ∇tuh(t)((Ec)−1)′(x)ei and (5.385)

∇uh(x) = ∇xuh(x) = ∇t((uh ◦ (Ec)−1(x))

)((Ec)−1)′(x)

= ∇tuh(t)((Ec)−1)′(x), with t = (Ec)−1(x).


Applied to the above (5.383) we find with (4.101)

ah(uh, vh) =∫

Ω

((Ec)′(t)T

)−1∇uh(t) · ∇vh(t)((Ec)′(t))−1 det(Ec)′(t)dt

= a(uh, vh) +O(hd−1)‖uh‖H1(Ω) · ‖vh‖H1(Ω). (5.386)

By (5.384), ‖vh‖H1(Ω) = ‖vh‖Vh(1 +O(hd−1)). So with (5.386) and Vhb ⊂ Vb, the Vb-

coercivity of a(uh, uh) implies the Vhb -coercivity of ah(uh, uh). Hence, there exists a

unique solution uh0 for ah

(uh

0 , vh)

= (f, vh)h ∀ vh ∈ Vhb , see (5.381).

For the general case in (5.381) we find similarly, for smooth enough aij

〈Ahuh, vh〉Vh′×Vhb

= ah(uh, vh) =∫

Ωhc

n∑i,j=0

aij(x)∂i uh∂jvh(x))

= a(uh, vh) +O(hd−1)‖uh‖H1(Ω)‖vh‖H1(Ω). (5.387)

An immediate consequence is

sup0�=vh∈Vh

b

|ah(uh, vh)− a(uh, vh)| / ‖vh‖H1(Ω) ≤ Chd−1‖u‖Hd(Ω), (5.388)

yielding the Vhb -coercivity of ah(uh, vh) for uh = vh. �

As a second step we consider the consistency and convergence, starting with linearproblems. With ah(·, ·), 〈f, ·〉h, Q′h, Q

′hc as introduced in (5.381), we solve

uh0 ∈ Vh

b :⟨Ahuh

0 , vh⟩Vh′×Vh

b

= ah(uh

0 , vh)

= 〈f, vh〉h ∀ vh ∈ Vhb . (5.389)

Theorem 5.85. Isoparametric FEMs for linear problems:

1. For a bounded, piecewise smooth Ω ⊂ Rn, our isoparametric FEs, a boundedlyinvertible A and for u ∈ Hd(Ω), d ≥ 1, let the discrete solution uh for (5.389)with f := Au exist. Then Ah is consistent with A in u, and the classical andvariational consistency errors for u vanish of order d− 1. They can be estimated,with C, independent of h, and for h sufficiently small, by

sup0�=vh∈Vh

b

|ah(uh, vh)− a(u, vh)|/‖vh‖H1(Ω) ≤ Chd−1‖u‖H1(Ω), (5.390)

‖AhIhu−Q′hAu‖Vh′ ≤ Chd−1‖u‖Hd(Ω), (5.391)

sup0�=vh∈Vh

b

|(f, vh)− (f , vh)|/‖vh‖H1(Ω)

≤ Chd−1‖f‖H−1(Ω) ≤ Chd−1‖u‖H1(Ω). (5.392)

2. For an invertible A, and a stable Ah, let u0 ∈W d∞(Ω) and uh

0 be the exact anddiscrete solutions of Au0 = f and (5.389), respectively. Then∥∥u0 − uh

0

∥∥H1(Ω)

≤ Chd−1(‖u0‖Hd(Ω) + ‖u0‖W 1

∞(Ω)

).


Proof. By (5.381) the Ih, Q′h, Q

′hc are equibounded linear operators with, e.g.

limh→0

‖Q′hf‖H−1(Ωhc ) = ‖f‖H−1(Ω).

Obviously (5.388) implies (5.390). The error term for f is estimated as:∣∣∣∣∣〈f, vh〉 −∫

Ωhc

fvhdx

∣∣∣∣∣ =∣∣∣∣∫

Ω

(f(t)vh(t)− f(t)vh(t)dt det((Ec)−1)′(t)

)dt

∣∣∣∣≤∣∣∣∣∫

Ω

f(t)(1− det((Ec)−1)′(t))vh(t)dt∣∣∣∣

≤ Chd−1‖f‖H−1(Ω)‖vh‖L2(Ω) (5.393)

≤ Chd−1‖u‖H1(Ω)‖vh‖L2(Ω) for f = Au.

The proof of (5.391) is delayed to the next theorem for the quasilinear case. �

The next theorem, some conditions, and its proof are very similar to Theorem 5.4,and e.g. (5.66), however with the � there replaced by d here.

Theorem 5.86. Isoparametric FEMs for quasilinear problems:

1. We require the conditions for Theorems 4.43, 4.44, 5.84, 5.85, a Lipschitz-continuous quasilinear operator G with a global constant L,

G,Ai ∈ CL(D(G)), i = 0, . . . n. (5.394)

2. Then for u ∈ Hd(Ω) ∩ D(G), d > 1, Gh is consistent with G in Ihu and

‖GhIhu−Q′hGu‖Vh′

Π=∥∥∥Q′h

c GhIhu−Q

′hGu∥∥∥Vh′

Π

≤ CLhd−1‖u‖Hd(Ω). (5.395)

3. If G(u) = 0 posesses a unique solution u0 ∈ Hd(Ω) with boundedly invertibleG′(u0), then the Gh

(uh

0

)= 0 are locally uniqely solvable as well and converge

according to ∥∥u0 − uh0

∥∥H1(Ω)

≤ CLhd−1‖u0‖Hd(Ω).

Proof. We estimate the left-hand side with (5.381), (4.105), and L in (5.394):∥∥∥Q′hc GhI

hu−Q′hGu

∥∥∥Vh′ −

∥∥∥Q′hc GhEcu−Q

′hGu∥∥∥Vh′ (5.396)

≤∥∥∥Q′h

c GhIhu−Q

′hc GhEcu

∥∥∥Vh′ ≤ CL‖Ihu− Ecu‖Vh ≤ CLhd−1‖u‖Hd(Ω).

We test Q′hc GhEcu−Q

′hGu with vh, and combine it with (5.379), (5.380), (5.381).For the estimate we need the set of triangles with at most one boundary point,

T hi :=

{T ∈ T h

c in Definition 4.42 1.}

: Ωb := Ω \(∪T∈T h

iT),

Ωhb := Ωh

c \(∪T∈T h

iT).


Then we obtain with Gh(Ecu)vh −Gu(vh ◦ Ec) ≡ 0 in Ωhi := Ωh

c ∩(∪T∈T h

iT)

= Ωi∣∣∣∣⟨Q′hc GhEcu−Q

′hGu, vh⟩Vh′×Vh

∣∣∣∣ =∣∣∣∣∣∫

Ωhc

GhEcuvhdx−

∫Ω

Guvhdx

∣∣∣∣∣ ∀vh ∈ Vhb

≤∣∣∣∣∣∫

Ωhb

Gh(Ecu)vhdx−∫

Ωb

Gu(vh ◦ Ec)dx

∣∣∣∣∣ .(5.397)

In Ωhb ∩ Ωb the integrands are Gh(Ecu)vh −Gu(vh ◦ Ec) = O(hd) by (4.101) and

(5.394). Furthermore, the remaining strips Ωhb \ Ωb and Ωb \ Ωh

b have a width O(hd)and the integrands are equibounded. So (5.394), (4.100), (4.101) imply (5.395).The convergence claim is, for a coercive G′(u0), a consequence of Theorem 5.84and (5.395). �

6

Adaptive finite element methods,by W. Dorfler

6.1 Introduction

The finite element method (FEM) has become a flexible tool to solve partial differentialequations numerically. It is especially useful in program packages where the usertypically adds his geometry. This is comfortable even for the nonexpert. However, suchuse can be dangerous if no controls exist. For example, in 1991, the offshore oil platformSleipner broke down and this failure could be traced back to an underestimation of theshear stresses in some part of the structure because of insufficient grid resolution [51],Ch. 1.2. Thus the remedy would be to set up an algorithm that provides sufficientrefinement in an efficient way and that provides a measure for the quality of thesolution. The task of this chapter is to present techniques to overcome these deficienciesby a posteriori error estimation and local mesh refinement.

The theory of a posteriori error estimation for partial differential equations came upabout 1980 [49,50]. Combined with methods of local mesh refinement (or coarsening)it is called the adaptive finite element method (AFEM). Its applicability has now beendemonstrated in many circumstances and can be viewed as a standard tool nowadays.

These techniques are now widespread, concerning the type of error estimation andmesh refinement techniques. In this short overview we will present the main ideas onlyand we will further restrict to the Poisson problem with Dirichlet boundary conditions(6.1), (6.2). This equation can be seen as a simplified model of linear elasticitywhich played a role in the example mentioned above (Section 6.1.1). We explainthat solutions of such equations may exhibit singular behavior and the consequencesfor the finite element method (Section 6.1.2). A heuristic argument shows that localmesh refinement can overcome this problem (Section 6.1.4). We develop the theoryof a posteriori error estimation with the residual error estimator (Section 6.2). Thisstates that the adaptive algorithm can be designed to converge as an iterative method(Section 6.2.6) and that it leads in fact to the optimal grid in a certain sense (Section6.2.7).

Only briefly will we mention some alternatives to the residual error estimator forestimating the energy norm in Section 6.2.8 and refer to the literature. In the finalSection 6.3 we report on the methodology of estimating errors in quantities of interest.

Finally we mention a few books and overview articles on all these methods such as,for example, [6, 51,77,499,655].


6.1.1 The model problem

Let Ω ⊂ R3 be the domain occupied by an elastic body. The continuum mechanicalmodel yields the equations

−∇·(1

2C(∇u + (∇u)d

))= f in Ω

for the deformation field u : Ω → R3 and the elasticity tensor C. For simplification westudy the equation

−∇·(a∇u) = f in Ω (6.1)

to obtain u : Ω → R for given a : Ω → R+ and f : Ω → R. To state the boundaryconditions we assume a disjoint decomposition of ∂Ω

∂Ω = ∂DΩ ∪ ∂NΩ

and we require the solution of (6.1) to satisfy

u∣∣∂DΩ

= gD, a∇u · n∣∣∂NΩ

= gN. (6.2)

6.1.2 Singular solutions

Let us consider the boundary value problem (6.1), (6.2). Our question is: are solutionsof this equation in C2(Ω) as the formulation would require? In general the answer isno, if ∂Ω is not smooth.

Corner singularities in 2DA point x at the boundary ∂Ω is called a corner point if the boundary arc is continuousand one-sided differentiable in x, but not differentiable. It can be shown that such asituation can be locally described by that of a solution in a suitable angular domainwith angle θ and tip at 0. Special solutions for the case a = 1, f = 0 are of the form

u(r, φ) = rγ sin(mπ

φ

θ

), r := |x|, φ = arg(x),

in the case of homogeneous Dirichlet boundary values for φ = 0 and φ = θ. Insertioninto −Δu = 0 yields (

−γ2 + m2π2

θ2

)rγ−2 sin

(mπ

φ

θ

)= 0

for all r ∈ (0,∞) and φ ∈ (0, θ). This equation for γ is solved by

γ2 = m2π2

θ2.

The smallest positive solution for γ is obtained for m = 1:

u(r, φ) = rπ/θ sin(πφ

θ

). (6.3)

422 6. Adaptive finite element methods

For γ < 0, the solution would not satisfy our least requirement that u has finite energy,i.e.∫Ω|∇u|2 exists. In the case of homogeneous Neumann conditions at {φ = 0} or

{φ = 0, π} we have to take

u(r, φ) = rγ sin(π

2φ

θ

)u(r, φ) = rγ cos

(πφ

θ

),

respectively. Note that we would get in the first case

u(r, φ) = rπ/(2θ) sin(π

2φ

θ

).

For the slit domain (θ = 2π), this yields the worst cases

u(r, φ) = r1/2 or u(r, φ) = r1/4.

We will study the regularity of these solutions in Sobolev function spaces. Recall thedefinitions in Section 1.4.3:

Wm,p(Ω) :={v : Ω → R : ||v||W m,p(Ω) :=

( ∑|α|≤m

∫Ω

|∇αv|p)1/p

<∞},

Hm(Ω) := Wm,2(Ω).

Moreover, recall that this definition can extended to m ∈ R, cf. Theorem 1.30 and [668].We may consider u as above on the sector Ω = {(r, φ) : r ∈ (0, 1), φ ∈ (0, θ)}. For uthat behaves like rγ near 0 we find that

||u||pW m,p(Ω) ≤ C

∫ 1

0

|rγ−m|pr dr = C

∫ 1

0

r(γ−m)p+1 dr!<∞

holds if (γ −m)p + 1 > −1 ⇔ γ > m− 2p . For u ∈ Hm(Ω) we need the bound m < 1 +

γ = 1 + π/θ, respectively 1 + π/(2θ). Solutions of Dirichlet boundary value problemsare thus at least in H3/2(Ω); for mixed Dirichlet and Neumann conditions at least inH5/4(Ω). If we consider the existence in W 2,p spaces, this allows arbitrary p ∈ [1,∞]for γ ≥ 2 but p < 2

2−γ for γ ∈ (0, 2). For γ < 1 we will have W 2,p regularity for somep ∈ [1, 2). Especially, u ∈W 2,1(Ω) holds in general. Only for γ ≥ 1 do we have H2

regularity.The general structure of singular solutions u of elliptic partial differential equations

is as follows. Given m ≥ 2, there exists nm ∈ N, smooth functions wj that are locallysupported near the singular point, e.g. 0, of ∂Ω, and singular functions ηj (oftenηj(x) = |x|αj for some αj ∈ R) such that

u(x) =nm∑j=1

wj(x)ηj(x) + w(x)

for w ∈Wm,2(Ω) [255,372,446,455].


Corner and edge singularities in 3DA well studied example of a corner point in three dimensions is the Fichera corner in0 for Ω := (−1, 1)3 \ [0, 1)3. The solution of Δu = 0 near 0 behaves like x �→ |x|γ forγ = 0.45 . . . . The problem of finding γ leads to a quadratic eigenvalue problem (λ2A +λB + C)u = 0 for λ ∈ R, u �= 0 [35]. Here A,B,C are suitable operators. Besides thecorners, there are additional singular solutions along the edges [212].

Problems with jumping coefficients

We consider Equation (6.1) for a that is piecewise constant in the four quadrants ofthe coordinate system with values a1, . . . , a4. This problem could appear for flow inporous media where large jumps in the permeability a may in fact occur. Again wetry an ansatz of the form u(r, φ) = rγs(φ) to get in each quadrant

a(s′′ − γ2s)rγ−2 != 0,

that is, the following eigenvalue problem for γ

s′′ = γ2s in(jπ

4, (j + 1)

π

4

)for j = 0, 1, 2, 3.

The solution must have a continuous flux (a∇u) · n along the normal n of theinterfaces, thus we can require periodicity of s and ajs

′(j π4 −

)= aj+1s

′(j π4 +

)for

j = 0, 1, 2, 3 [533]. In the case of a1 = a3 = 1 and a2 = a4 ≈ 100 one gets a valueγ = 0.1. Thus u(x) behaves like 10

√|x| near 0! Moreover, u ∈ H1.1(Ω) [444,498].

Boundary layers

We consider the case of, cf. [559]

−εΔu + b · ∇u + cu = f in Ω,

u = 0 on ∂Ω,

with constant coefficients ε, b, c, where b �= 0 or c �= 0 and 0 < ε� |b|+ |c|. For exam-ple, the typical behavior of solutions near outflow points of ∂Ω (i.e. where b · n > 0)is |∇lu| ∼ 1/εl.43 At other parts of the boundary, a weaker dependence on 1/ε willoccur. Although u might be smooth, it appears as a nonsmooth function unless thediscretization method resolves the scale ε, which will lead to prohibitive work onuniform meshes.

6.1.3 A priori error bounds

In the following, we will simplify (6.1) further by assuming a = 1. The weak form of

−∇·∇u0 = −Δu0 = f in Ω,

u = 0 on ∂Ω

43 a ∼ b :⇔ ∃c1, c2 : 0 < c1 < c2 such that c1a ≤ b ≤ c2.


is given by ∫Ω

∇u0 · ∇v =∫Ωfv for all v ∈ C∞

0 (Ω). (6.4)

As previously in this book, we denote solutions of specific problems, e.g. of (6.4)and (6.5), as u0 and uh

0 , respectively For given f ∈ L2(Ω), there exists a solutionu0 ∈ H1

0 (Ω) of this problem (cf. Theorem 2.2 and Theorem 2.43 in Chapter 2). Bydensity of C∞

0 (Ω) in H10 (Ω) we can equally well state “for all v ∈ H1

0 (Ω)” in (6.4).Let Vh be a finite dimensional subspace of V := H1

0 (Ω). Then the Galerkin approx-imation uh

0 ∈ Vh is defined by∫Ω

∇uh0 · ∇vh =

∫Ω

fvh for all vh ∈ Vh. (6.5)

It is well known that both problems are uniquely solvable and that (Cea’s lemma, cf.Lemma 4.48 in Section 4.3)

||∇(u0 − uh

0

)||L2(Ω) ≤ C inf

vh∈Vh||∇(u0 − vh)||L2(Ω). (6.6)

If we have a continuous mapping Ph : H10 (Ω) → Vh, we can further estimate the right-

hand side of (6.6) to get

||∇(u0 − uh

0

)||L2(Ω) ≤ C||∇

(u0 − Phu0

)||L2(Ω).

Let Vh be a finite element space that at least contains the linear finite elements. Thenthere exists a continuous interpolation operator Ph : H1

0 (Ω) −→ Vh with the property

||∇(v − Phv)||L2(Ω) ≤ Chmax||∇2v||L2(Ω)

for all v ∈ H2(Ω). Here, hmax is the maximal diameter of the mesh cells in theunderlying discretization. This is a special case of Theorem 4.17 in Section 4.2 ofChapter 4. However, u ∈ H2(Ω) is not always true as we have seen in Section 6.1.2.In fact, we get u ∈ H1+s(Ω) for s > 0 only.

Theorem 6.1. Interpolation operator: Let Vh be a finite element space that atleast contains the linear finite elements. Then there exists a continuous interpolationoperator Ph : H1

0 (Ω) −→ Vh with the property

||∇(v − Phv)||L2(Ω) ≤ Chsmax||∇v||Hs(Ω) for all v ∈ H1+s(Ω)

and suitable s ∈ (0, 1]. Here, hmax is the maximal diameter of the mesh cells in theunderlying discretization and C depends on properties of the mesh [141], Ch. 14.3.

This leads to the following error bound.

Theorem 6.2. A priori error bound: The error for the finite element solution of (6.5)on Vh as in Theorem 6.1 obeys the a priori error bound

||∇(u0 − uh

0

)||L2(Ω) ≤ Chs

max||∇u0||Hs(Ω)


for some s ∈ (0, 1]. Here, hmax is the maximal diameter of the mesh cells in theunderlying discretization and C depends on properties of the mesh (cf. Section 6.1.7).

The meaning of this result is that a lack of regularity (smaller s) leads to a lowconvergence rate for the error in terms of the maximal mesh size in the underlyingdiscretization. Or, in other words, for a fixed mesh, there is a significant loss in accuracyif one approximates less regular functions.

6.1.4 Necessity of nonuniform mesh refinement

As we have observed in Section 6.1.3, there are cases where solutions of (6.4) are inH1+s(Ω) for s ∈ (0, 1]. By Theorem 6.2 the H1 error behaves like hs on, say, a uniformmesh of mesh size h. In this case we have Nh = O(1/hn) unknowns for linear finiteelements in a bounded domain in Rn. In terms of Nh the H1 error thus behaves likehs ∼ (Nh)−s/n. We discuss this, as an example, in the case n = 2.

� Ω ⊂ R2 and s = 1 (which is the full regular case). Now the error behaves like(Nh)−1/2 and to achieve a prescribed tolerance TOL, one needs (Nh)−1/2 ∼ TOL,or, Nh ∼ TOL−2. Thus, if we start with an error of size 1 and if we want to reducethe error by a factor of 100, we have to increase Nh by a factor of 104.

� Ω ⊂ R2 and s = 1/2 (the case of the “slitted domain” Ω = B1(0) \ {x ∈ R2 : x1 <0, x2 = 0}). Now the error decreases like (Nh)−1/4 and to achieve a prescribed tol-erance TOL we need Nh ∼ TOL−4 unknowns. That means, in the same situationas above we now need about 108 unknowns!

6.1.5 Optimal meshes – A heuristic argument

(See [78].) Let Ω ⊂ Rn be bounded and let Ω be decomposed into simplices orrectangles. We assume that we are given a mesh-size function h : Ω −→ R+ with h(x)describing the local mesh size near x (i.e. x ∈ T ⇒ h(x) ≈ diam(T )). Let the functionσ denote a geometry dependent function,

σ : Ω −→ R+ with σ(xT ) =h(xT )n

vol(T ).

For linear finite elements we have approximately one unknown per simplex. Thereforethe total number of unknowns N (h) for the linear finite element space is approximatelygiven by

N (h) ≈∑T

1 =∑

T∈T h

vol(T )vol(T )

=∑

T∈T h

σ(xT )h(xT )n

vol(T ) ≈∫

Ω

σ(x)h(x)n

dx. (6.7)

Thus we define for a given mesh-size function h the number of unknowns as

N (h) :=∫

Ω

σ

hn.

Assume that we have a representation of the local error in the form

|∇(u0 − uh

0

)(x)| ≈ h(x)αE(x), (6.8)


where α is the convergence order and E does not depend on h. For linear finite elementsand u0 ∈ H2(Ω) we have for example α = 1, E(x) = |∇2u0(x)|. If the solution u0, weare trying to approximate, is in H2

loc(Ω), then the global error E(h) is

E(h) :=(∫

Ω

(hαE)2) 1

2.

We further assume that the total work to establish the approximate solution isproportional to N (h) (e.g. use a multigrid method cf. [141]). Our aim is to find a mesh-size function h that gets an error of TOL with lowest possible number of unknowns.In other words, we have to find a mesh-size function h such that

N (h) is minimal while E(h) = TOL.

This is a constraint optimization problem! To solve it we introduce the Lagrangefunction

L(h, λ) = N (h)− λ(TOL2−E(h)2

)(6.9)

=∫

Ω

σ

hn− λ(

TOL2−∫

Ω

h2αE2). (6.10)

The point (h, λ) we are looking for is a stationary point of L. Thus for all φ ∈ C∞(Ω)we get

0 = ∂1L(h, λ)[φ] =∫

Ω

{− nσ

hn+1+ 2αλh2α−1E2

}φ.

This requires the pointwise condition −nσ/hn+1 + 2αλh2α−1E2 = 0 and leads to

h2αE2 =n

2αλσ

hnin Ω.

To satisfy the constraint

TOL2 = E(h)2 =∫

Ω

h2αE2 =∫

Ω

n

2αλσ

hn=

n

2αλN (h)

λ should be taken as

λ =n

2αN (h)TOL2 .

Insertion into the equation for h yields

h2αE2 =TOL2

N (h)σ

hnor

1σhn+2αE2 =

TOL2

N (h). (6.11)

The left-hand side depends on x, but the right-hand side is independent of x. Hence ourresult is to distribute hn+2αE2/σ equally on the mesh. For any T in the discretization,let E2

T (h) :=∫

Th2αE2 be the local error on T and NT (h) :=

∫Tσ/hn be the local

number of degrees of freedom in T . Then we have by (6.11)

E2T (h) =

∫T

h2αE2 =TOL2

N (h)

∫T

σ

hn=

TOL2

N (h)NT (h).


In our setting, so far, NT is a constant depending on the finite element method.Thus our result is that we have to equidistribute the error over the decomposition

of Ω.From (6.11) we get the optimal mesh size h∗ and if

∫Ω|E|2n/(n+2α) <∞, we obtain

from this

N (h∗) ∼ TOL−n/α .

This recovers the result we would get for the finite element method for the full regularproblem (see Section 6.1.4)!

6.1.6 Optimal meshes for 2D corner singularities

We apply the previous result to the case of a typical corner singularity in two spacedimensions for −Δu = 0 as in Section 6.1.2. For simplicity, we assume that the cornerpoint is located at 0 and that Ω is near 0 a sector of the unit circle with angle θ = βπ,β ∈ (1, 2). Then the solution (6.3) behaves like |∇lu0(x)| ∼ r1/β−l for r = |x| ∈ (0, 1),l ≥ 0.

Let us assume a discretization with a uniform mesh size h(x) := h0 for all x ∈ Ω. InΩ ∩Bh(0) the error is given by

||∇(u0 − uh

0

)||2L2(Bh0 (0)) ≈ ||∇u0||2L2(Bh0 (0)) ∼

∫ h0

0

r2β −2r dr ∼ h

2β

0 ,

while in the remaining domain, by (6.8),

||∇(u0 − uh

0

)||2L2(Ω\Bh0 (0)) ∼ ||h0∇2u0||2L2(Ω\Bh0 (0)) ∼ h2

0

∫ 1

h0

r2β −3 dr ∼ h2

0h2β −2

0 ∼ h2β

0 .

Thus ||∇(u0 − uh

0

)||L2(Ω) ∼ h

1β

0 and the requirement

||∇(u0 − uh

0

)||L2(Ω) ∼ E(h0)

!= TOL

leads to the relation, cf. (6.7),

N(h0) ∼ h−20 =

(h

1β

0

)−2β

= TOL−2β .

We already noticed this relation for β = 2 in Section 6.1.4. But, what does theoptimal mesh-size distribution h look like, and what can we achieve with it? Letus assume that we established the optimal mesh-size distribution. We observed in(6.11) that hn+2α|∇2u0|2/σ = TOL2 /N (h) =: K. For a radial symmetric mesh-sizedistribution it follows from this that (recall n = 2, α = 1)

h(r)4r2β −4 ∼ K or h(r) ∼ K

14 r1− 1

2β . (6.12)


The constants in this estimate depend on σ, but we assume σ ∼ 1. So the error forsuch a choice of h will be

TOL2 =∫

Ω

h2|∇2u|2 ∼∫ 1

0

K12 r2− 1

β r2β −4 rdr = K

12

∫ 1

0

r1β −1 dr (6.13)

∼ K12 =

TOLN (h)

12, (6.14)

and so we finally achieve

N (h) ∼ TOL−2,

independent of β. We thus recover the complexity for the full regular case as stated inSection 6.1.4 (these statements are only valid in the sense that they show the sameasymptotic behavior for large N (h); the constants have not been considered).

Compared with the second example in Section 6.1.4, we notice that the number ofunknowns on the optimal mesh is a factor TOL2 less than on a uniform mesh, that isa factor 10−4 for TOL = 0.01 for this two-dimensional problem.

6.1.7 The finite element method–Notation and requirements

For the reader’s convenience we summarize some definitions for the conforming finiteelement method for Equation (6.4) for simplicial discretizations, cf. Section 4.2.

Let Ω ⊂ Rn, an open bounded set and polygonally bounded, be decomposed into aset of simplices T h, so that

Ω =⋃

T∈T h

T.

In this chapter, simplices are assumed to be closed sets, T ◦ ∩D◦ = ∅ for all T,D ∈ T h,T �= D and each vertex of T is also a vertex of a neighbor of T . T h is then called aregular discretization of Ω (or triangulation if n = 2). For T ∈ T h let

hT := diam(T )

and let h : Ω → R+ defined by h∣∣T

= hT be the mesh-size function. Furthermore letρT be the radius of the largest inscribed ball in T , σh the piecewise constant functionwith σT := hT /ρT ≥ 1, the shape factor or chunkiness parameter of T . Let N h be theset of all vertices, Eh the set of all edges. Boundary vertices and edges are N h,∂ andEh,∂ , respectively, and for the interior sets we define Eh,◦ := Eh \ Eh,∂ and N h,◦ :=N h \ N h,∂ . The space of piecewise polynomials of degree p ∈ N is

Pp(T h) :={v : Ω → R : v

∣∣T∈ Pp for all T ∈ T h

}.

The space of conforming finite elements of degree p is

Shp := Pp(T h) ∩ C0(Ω) ⊂ H1(Ω).


In the following we will work with V := H10 (Ω) and its subspace

Vh := Shp ∩H1

0 (Ω) ={v ∈ Sh

p : v∣∣∂Ω

= 0}. (6.15)

For each z ∈ N h,◦, there is a unique function φz ∈ Sh1 such that

φz(z′) = δzz′ for all z ∈ N h,◦, z′ ∈ N h.

Each vh ∈ Sh1 , the space of linear conforming finite elements, can be written as

vh =∑

z∈Nh,◦

αzφz.

Inserting such an ansatz into the equation∫Ω

∇uh0 · ∇vh =

∫Ω

fhvh for all vh ∈ Vh

yields a linear system of equations

A�u = �b

with Azz′ :=∫Ω∇φz′ · ∇φz, bz :=

∫Ωfhφz for z, z′ ∈ N h,◦ and the vector �u =

[αz]z∈Nh,◦ of the representation

uh0 =

∑z∈Nh,◦

αzφz.

In contrast to the interpolation estimates cited before in Theorem 6.1 the followingare stated for nonuniform meshes.

Theorem 6.3. Interpolation operator: There exists a continuous interpolationoperator

Ph : L2(Ω) → Shp

that has for all v ∈ Hm(Ω) with m = 0, 1, . . . , p + 1 and l = 0, 1, . . . ,min(p,m) thefollowing properties:

||∇lPhv||L2(Ω) ≤ C||∇lv||L2(Ω) (6.16)

and

||∇l(v − Phv)||L2(T ) ≤ Chm−lT ||∇mv||L2(ST ) for all T ∈ T h, (6.17)

||v − Phv||L2(E) ≤ Chl− 1

2E ||∇lv||L2(SE) for all E ∈ Eh, (6.18)

where ST or SE are the domains composed of all triangles that touch T ∈ T h orE ∈ Eh, respectively. If v ∈ H1

0 (Ω), then Phv ∈ H10 (Ω). The constants only depend on

m, p and σT ′ for T ′ ⊂ ST , respectively T ′ ⊂ SE .

Proof. See [141], Ch. 4.8 or Theorem 4.17 in Section 4.2 of Chapter 4 and 7.7,Corollary 7.8 in Section 7.7 of Chapter 7. �


6.2 The residual error estimator for the Poisson problem

6.2.1 Upper a posteriori bound

Definition 6.4. Jumps and residuals:

1. Let E ∈ Eh,◦. The edge jump [v]E of a piecewise continuous function v on T h

along the edge E is defined by

[v]E(x) := lims↘0

{v(x + snE)− v(x− snE)} for x ∈ E,

where nE is a fixed unit normal vector on E.2. Let uh

0 be the solution of the discrete equation (6.5) in Vh as in (6.15). Definethe element residual

RT := Δuh0 + fh on T ∈ T h

and the edge residual

RE := [∂nuh0 ]E on E ⊂ Eh,◦.

Note that this definition does not depend on the choice of nE.

Theorem 6.5. Upper a posteriori error estimate: Let u0, uh0 be the solutions of the

continuous equation (6.4) in V = H10 (Ω) and the discrete equation (6.5) in Vh as in

(6.15), respectively. Then there are sets of positive constants {cT }T∈T h , {cE}E∈Eh,◦

such that

||∇(u0 − uh

0

)||2L2(Ω) ≤

∑T∈T h

cTh2T ||RT ||2L2(T ) +

∑E∈Eh,◦

cEhE ||RE ||2L2(E)

+CP (Ω)||f − fh||2L2(Ω).

cT , cE depend on σh (the shape constant) and p (the polynomial degree of the finiteelement space). CP (Ω) is Poincare’s constant for Ω (cf. (1.57) and the residualsRT , RE have been defined in Definition 6.4.

Proof. Let u0, uh0 be the respective solutions and let v ∈ V, vh ∈ Vh be arbitrary.

Then by (6.4) and (6.5) and partial integration∫Ω

∇(u0 − uh

0

)· ∇v

=∫

Ω

{fv −∇uh

0 · ∇(v − vh)− fhvh}

=∫

Ω

{(f − fh)v + fh(v − vh)−∇uh

0 · ∇(v − vh)}

=∫

Ω

(f − fh)v +∑

T∈T h

{∫T

(fh + Δuh

0

)(v − vh)−

∫∂T

∂huh0 (v − vh)

}

=∫

Ω

(f − fh)v +∑

T∈T h

∫T

(fh + Δuh

0

)(v − vh)−

∑E∈Eh,◦

∫E

[∂huh0 ]E(v − vh). (6.19)

6.2. The residual error estimator for the Poisson problem 431

Equation (6.19) is called the error representation formula. Now we choose vh = Phvwith Ph as in Theorem 6.3 and use the interpolation estimates to get (note that cT , cE

will change from estimate to estimate)∣∣∣ ∫Ω

∇(u0 − uh

0

)· ∇v

∣∣∣ ≤ ∑T∈T h

cThT ||RT ||L2(T )||∇v||L2(ST )

+∑

E∈Eh,◦

cEh1/2E ||RE ||L2(E)||∇v||L2(SE) + ||f − fh||L2(Ω)||v||L2(Ω)

≤

⎛⎝ ∑T∈T h

c2Th2T ||RT ||2L2(T ) +

∑E∈Eh,◦

c2EhE ||RE ||2L2(E)

⎞⎠1/2

||∇v||L2(Ω)

+CP (Ω)||f − fh||L2(Ω)||∇v||L2(Ω).

Here we used the Cauchy–Schwarz inequality for sums and that ∪E∈Eh,◦SE and∪T∈T hST give a finite coverage of Ω with a constant that only depends on σh. Sincev is arbitrary, the bound will follow dividing by ||∇v||L2(Ω) and taking the supremumover all v ∈ V. �

Remark 6.6.

1. This estimate in Theorem 6.5 is in principle computable. However, the constantscT , cE have to be estimated. There are some attempts to get explicit or tightbounds, e.g. [164].

2. The estimate in Theorem 6.5 shows the expected dependence on h for smooth u0,since ⎛⎝ ∑

T∈T h

h2T ||RT ||2L2(T )

⎞⎠1/2

= O(hmax)

and ⎛⎝ ∑E∈Eh,◦

hE ||RE ||2L2(E)

⎞⎠1/2

= O(hmax)

because of [∂nuh0 ]E ≈ ∂2

nu0 hE and

hE ||[∂nu

h0

]E||2L2(E) ∼ h4

E

∣∣∂2nu0

∣∣2 ∼ h2T ||∇2u0||2L2(T )

for T with E ⊂ ∂T \ ∂Ω. Note that∑

E∈Eh,◦ h2T ∼

∑T∈T h |T | = |Ω|.

3. Note that the estimate in Theorem 6.5 does not require any higher regularity thanu0 ∈ V. Thus it covers also the cases when u0 �∈ H2(Ω), like in Section 6.1.2.


6.2.2 Lower a posteriori bound

Definition 6.7. Bubble functions: If φ1, φ2, φ3 are the linear basis functions at thevertices of a triangle T , then βT := φ1φ2φ3/h

3T defines a cubic function on T called an

element bubble. Note that βT can be extended by 0 to a function in V with support inT . For each edge E = T1 ∩ T2 �= ∅ (T1, T2 ∈ T h), say E = z1z2 for some z1, z2 ∈ N h,we define a piecewise quadratic polynomial θE := φz1φz2/h

2E on ωE := T1 ∪ T2 called

an edge bubble. Note that θE is continuous on E and it can be extended by 0 to afunction in V with support in ωE.

The following theorem gives a lower bound for the actual error of uh0 .

Theorem 6.8. Lower a posteriori error estimate: Let u0, uh0 be as in Theorem 6.5.

Then there is a constant c that depends only on σh and p, such that for each T ∈ T h

and for RT , RE and ωE as in Definitions 6.4 and 6.7, respectively,

c

(cEhE ||RE ||2L2(E) +

∑T⊂ωE

cTh2T ||RT ||2L2(E)

)

≤ ||∇(u0 − uh

0

)||2L2(ωT ) + ||h(f − fh)||2L2(ωT ).

The constants cT , cE are as in Theorem 6.5.

Proof. Let E ⊂ Eh,◦ and T be any of the adjacent triangles, and let v := βTRT . Thusv ∈ V and the error representation formula (6.19) specializes to∫

T

∇(u0 − uh

0

)· ∇(βTRT ) =

∫T

{βTR

2T + (f − fh)βTRT

}.

Since RT and βTRT are polynomials, there is a constant c, depending on σT and p,such that

||RT ||L2(T ) ≤ C||β1/2T RT ||L2(T )

and furthermore we have the inverse estimate

||∇(βTRT )||L2(T ) ≤ Ch−1T ||βTRT ||L2(T ) ≤ Ch−1

T ||RT ||L2(T ).

Hence we can derive the bound

||RT ||2L2(T ) ≤ C

∫T

βTR2T = C

∫T

{∇(u0 − uh

0

)· ∇(βTRT ) + (f − fh)βTRT }

≤ C(||∇(u0 − uh

0

)||L2(T )||∇(βTRT )||L2(T ) + ||f − fh||L2(T )||RT ||L2(T )

)≤ C

(h−1

T ||∇(u0 − uh

0

)||L2(T ) + ||f − fh||L2(T )

)||RT ||L2(T )

so that

hT ||RT ||L2(T ) ≤ C(||∇(u0 − uh

0

)||L2(T ) + ||h(f − fh)||L2(T )

).


For the edge term there is a similar trick: we extend RE onto R2 by a constantextension normal to E [654] and define as a test function v := θERE . Clearly, v ∈ Vand the error representation yields∫

ωE

∇(u0 − uh

0

)· ∇(θERE) =

∫E

θER2E +

∑T⊂ωE

∫T

RT θERE +∫

ωE

(f − fh)θERE .

Similar to the above, the inequality∫E

|RE |2 ≤ C

∫E

θER2E

≤ C

(||∇(u0 − uh

0

)||L2(ωE)||∇(REθE)||L2(ωE)

+

( ∑T⊂ωE

||RT ||2L2(T )

)1/2

||θERE ||L2(ωE) + ||f − fh||L2(ωE)||θERE ||L2(ωE)

)

≤ C

(h−1

E ||∇(u0 − uh

0

)||L2(ωE) +

( ∑T⊂ωE

||RT ||2L2(T )

)1/2

+||f − fh||L2(ωE)

)||RE ||L2(ωE)

leads, with ||RE ||L2(ωE) ≤ Ch1/2E ||RE ||L2(E) (recall the definition of RE on ωE), to

h1/2E

(∫E

|RE |2)1/2

≤ C

⎛⎝||∇ (u0 − uh0

)||L2(ωE) +

( ∑T⊂ωE

h2T ||RT ||2L2(T )

)1/2

+ ||h(f − fh)||L2(ωE)

⎞⎠ .

Now observe that the second term on the right has already been estimated in the firststep of the proof in terms of the other two terms. This yields the result. �

6.2.3 The a posteriori error estimate

We summarize the results of the last two sections in the following theorem.

Theorem 6.9. A posteriori error estimate: Let u0, uh0 be the solutions of the contin-

uous equation (6.4) in V = H10 (Ω) and the discrete equation (6.5) in Vh as in (6.15),

respectively. For each E ∈ Eh,◦ we define the local error indicator

η2E := cEhE ||RE ||2L2(E) +

∑T⊂ωE

cTh2T ||RT ||2L2(E) (6.20)

with constants cT , cE from Theorem 6.5, and residuals RT , RE from Definition 6.4.Then there are constants c, C, both depending only on σh and the polynomial degreep, such that


ceff

⎛⎝ ∑E∈Eh,◦

η2E

⎞⎠1/2

≤ ||∇(u0 − uh

0

)||L2(Ω) + ||h(f − fh)||L2(Ω)

and

||∇(u0 − uh

0

)||L2(Ω) ≤

⎛⎝ ∑E∈Eh,◦

η2E

⎞⎠1/2

+ CP (Ω)||f − fh||L2(Ω).

Proof. Use Theorem 6.5 and Theorem 6.8 (sum over E ∈ Eh,◦). �

Remark 6.10. A posteriori error estimate:

1. The upper bounds were first proved in [49, 50], the lower bound first appearedin [653].

2. Usually, fh is not given explicitly, but is implicitly defined by using quadraturerules for the integrals

∫Tfϕ, where ϕ is a basis function. In general, this

quadrature rule should be as cheap as possible but at least converge as fast asthe error or one degree higher (e.g. for linear finite elements choose a quadraturerule of order 1 or 2). This same approximation fh to f should also be taken intoaccount when we compute RT and ||f − fh||L2(T ).

3. The edge residuals RE can be avoided in one dimension since v − vh in the proofof Theorem 6.5 can be chosen to vanish in the vertices. The jump residuals RE

are unavoidable and even dominant in more than one dimension [165]. This is,for example, easy to see if p = 1 and f = 0 (for nonhomogeneous Dirichlet data).Note that globally Δuh

0 is not a function but a distribution. Its regular part is zero,its singular part is [∂nu

h0 ]E. For ϕ ∈ C∞

0 (ωE) and for some E ∈ Eh,◦ we obtain

−Δuh0 [ϕ] =

∫ωE

∇uh0 · ∇ϕ =

∫E

[∂nuh0 ]Eϕ−

∫T1∪T2

Δuh0ϕ =

∫E

REϕ.

4. The upper bound Theorem 6.5 is global while the lower bound Theorem 6.8 islocal. The upper bound cannot be local since a local error source will have globalinfluence. Only a part of the error can be judged by a local interpolation errorand local data error, while other parts of the error have its source somewhereelse. This part is also called pollution error.

5. Such residual error estimates can also be derived for nonlinear equations. Forthis see [655].

6.2.4 The adaptive finite element method

The macro discretizationWe need a first decomposition of Ω into simplices that resolves the geometric details.This is often a very difficult problem, especially in 3D. A major difficulty is to avoidsimplices T with large shape constant σT .


Initial discretizationThe macro-discretization may not resolve the data required by the problem. In order tojudge and improve data resolution we define a data error estimator (take ||f − fh||L2(Ω)

and perform mesh refinement until a certain coarse tolerance is met), e.g.

δT h :=

⎛⎝ ∑T∈T h

δ2T

⎞⎠1/2

with δ2T := ||f − fh||2L2(T ).

The adaptive iteration1. Solve for uh

0 (e.g. by a multigrid method). We assume in the following that uh0 is

the exact finite element solution.2. Estimate: Compute ηE as in (6.20) for all E ∈ Eh,◦. Stop the iteration if

ηT h :=

⎛⎝ ∑E∈Eh,◦

η2E

⎞⎠1/2

≤ TOL

is satisfied (or use TOL ||∇uh0 ||L2(Ω) on the right-hand side in the case of relative

errors).3. Mark a set of edges Fh ⊂ Eh,◦ for refinement. Possible strategies are:

� Maximum strategy. Let

E ∈ Fh ⇐⇒ ηE ≥ (1− θ)ηmax

with ηmax := maxE∈Eh,◦ ηE and some θ ∈ [0, 1]. Note that we get Fh ≈ ∅ forθ ≈ 0 and Fh ≈ Eh,◦ for θ ≈ 1.

� Fixed energy fraction strategy. For θ ∈ [0, 1] seek a minimal set Fh ⊂ Eh,◦ suchthat ∑

E∈Fh

η2E ≥ θ2η2

T h . (6.21)

Note that we get Fh ≈ ∅ for θ ≈ 0 and Fh ≈ Eh,◦ for θ ≈ 1.4. Mesh refinement. Aim: create a new regular mesh from T h and the set of marked

triangles

Ah := {T ∈ T h : T ⊂ ωE , E ∈ Fh}. (6.22)

� 1D problems. Interval bisection: bisect T ∈ Ah in its center.� 2D problems. A refinement of one triangle introduces a hanging node

(a vertex that is not shared with a neighboring triangle [655], Sect. 4.1) andenforces the refinement of neighboring triangles if we have to work on regularmeshes. On the other hand, our refining algorithm has to stay local. Arbitrarybisection of triangles may lead to very small angles over several refinementsteps and as a consequence the shape constant will grow. We will study thistopic in Section 6.2.5.


Figure 6.1 Left: Red refinement. Right: Blue refinement of the right triangle.

Remark : If we are bound to rectangular decompositions we cannot avoidworking with hanging nodes.

� 3D problems. The principal problems are the same as in 2D but solutions toit are more involved.

Now go back to 1.

6.2.5 Stable refinement methods for triangulations in R2

The aim of this section is to describe some methods that return a (refined) regularmesh for a given regular mesh T h and a set Ah of marked triangles to be refined. Sucha method is called geometrically stable if repeated application of the method leadsto a sequence of meshes such that the corresponding sequence of shape constants isuniformly bounded. The following methods will fulfill this requirement.

Red–green (–blue) refinement [655], Sect. 4.1In case T ∈ Ah we will use red refinement as shown in Figure 6.1, left. Note that allnew triangles have the same angles as their original triangle. For the neighbors that arenot in Ah, we enforce bisection to remove the hanging node, regardless of the size ofthe new angles. These so-called green refinements, Figure 6.1, right, will be coarsenedbefore such a triangle will be refined in a subsequent step. If the opposite angle thatis divided by bisection is very small, one might use bisection methods that avoid these(like the one in the next subsection). In this context, this refinement is called bluerefinement . Triangles refined in this way need not be removed in later steps.

Newest node bisection [67]We store the nodes P1, P2, P3 of triangle T in a counterclockwise sense: nodes [T, : ] =[P1,P2,P3]. If T is marked, we plan to bisect T from P1 towards the opposite edge asin Figure 6.2. The two new triangles must again have oriented nodes – entries withPnew in the first place. However, the neighboring triangle has to be added to Ah ifPnew is a hanging node. In a practical implementation one actually only refines simplesituations:

• two triangle share the marked edge;• the marked edge is a boundary edge;


P1P′1 Pnew

Pnew

Figure 6.2 Left: Bisection in the interior. Middle: Bisection near the boundary. Right:

Refinement with interior node property.

as seen in Figure 6.2. It can be shown that Algorithm 6.1, which is based on theseideas, stops after finitely many steps and results in a locally refined mesh. Therefinement method is geometrically stable if the macro-triangulation T0 has a “correct”distribution of marked edges. We only have to avoid all edges emanating from onevertex being marked edges.

A newest node bisection with m-times repeated bisection is as follows. The markingprocedure assigns to each T a number m(T ) ∈ N. Then Ah := {T ∈ T h : m(T ) > 0}.After each bisection, we decrease m(T ) until we obtain 0, see Algorithm 6.1.

Algorithm 6.1.Repeated newest node bisectionwhile any m(T ) > 0

for T with m(T ) > 0identify neighbor T ′ opposite P1

situation Figure 6.2 left/right?;Yes :

bisect T (and T ′);m(T ) := max{m(T ) − 1, 0};(respectively m(T ′) := max{m(T ′) − 1, 0});

No :m(T ′) := max{m(T ′), 1}

Longest node bisectionAlways refine towards the longest edge in T and only in one of the situations as inFigure 6.2. For that, work on a copy of the coordinate field with a random distortion(to assure that the longest edge in T is unique almost surely). The important questionis now whether this ends after finitely many steps in one of the mentioned situations?The answer is affirmative since following these choices the length of the largest edgeincreases and thus will end at an edge of locally maximal length or at the boundary.The algorithm is like Algorithm 6.1, but a special ordering in the nodes field is notnecessary.


6.2.6 Convergence of the adaptive finite element method

The following description is very special to the case of linear finite elements. Never-theless, the method works for arbitrary polynomial degree with some modifications,see Remark 6.15.

Definition 6.11. Interior node property: A refinement method has the interior nodeproperty, if all marked triangles of the coarse mesh are refined in such a way that alltheir edge centers are vertices of the refined mesh and that each such triangle containsin its interior a vertex of the refined mesh.

An example for such a method is the repeated newest node bisection with m(T ) = 3,Algorithm 6.11.

Theorem 6.12. Computable a posteriori estimates: Let T H be a triangulation, VH

the linear finite element space over T H and uH0 ∈ VH the finite element solution to

the discrete Poisson problem (6.5) in VH . Let T h be a refinement of T H , VH ⊂ Vh

the refined finite element space and uh0 the discrete solution in Vh. Assume that both

approximations fH and fh of f are piecewise constant. If FH ⊂ EH,◦ is any set ofrefined edges and if all triangles in AH as in (6.22) have been refined by a refinementmethod with the interior node property (see Figure 6.2), then

c′eff∑

E⊂FH

η2E

(uH

0

)≤ ||∇

(uh

0 − uH0

)||2L2(Ω) + ||H(fh − fH)||2L2(Ω)

with some constant c′eff depending only on σH , σh. Here, H is the mesh-size functionfor T H and ηE

(uH

0

)is the error indicator (6.20) with respect to uH

0 .

Proof. [499], Lemma 4.2. We can assume the following situation. Let E ∈ EH,◦ andT1, T2 ∈ T H with E = T1 ∩ T2. Both T1 and T2 have been refined and T h has verticesxE (the center of E) and xi in the interior of Ti for i ∈ {1, 2}. Let ϕE , ϕi be the (max-imally coarse) basis functions related to these vertices in Vh with ωi := supp(ϕi) ⊆ Ti

and ωE := supp(ϕE) ⊆ T1 ∪ T2. For all ϕ with ωϕ := supp(ϕ) ⊆ T1 ∪ T2∫ωϕ

∇(uh

0 − uH0

)· ∇ϕ =

∫ωϕ

fhϕ−∫

ωϕ∩E

[∂nu

H0

]Eϕ

=∫

ωϕ

(fh − fH)ϕ +∫

ωϕ

fHϕ−∫

ωϕ

[∂nu

H0

]Eϕ.

Let ϕ = ϕi for i ∈ {1, 2}. Since fH is constant and ϕi

∣∣E

= 0 we have

|fH |2∫

ωi

ϕi =∫

ωi

fHfHϕi =∫

ωi

∇(uh

0 − uH0

)· ∇(fHϕi)−

∫ωi

(fh − fH)fHϕi

≤ ||∇(uh

0 − uH0

)||L2(ωi)

||∇(fHϕi)||L2(ωi)+ ||fh − fH ||L2(ωi)

||fHϕi||L2(ωi).

Using inverse estimates on ϕi (see Theorem 4.19) and considering the nodal scalingϕi(xi) = 1 yields

Hi||fH ||2L2(ωi)≤ C

(H−1

i ||∇(uh

0 − uH0

)||L2(ωi)

+ ||fh − fH ||L2(ωi)

)||fH ||L2(ωi)


with a constant C that depends on σTi. Similar, we derive the bound for the constant

edge residual taking the test function ϕ = REϕE

||RE ||2L2(E) ≤ C

∫E

|RE |2ϕE = C

∫E

RE(REϕE)

= C

(∫ωE

∇(uh

0 − uH0

)· ∇(REϕE)−

∫ωE

fhREϕE

)≤ C

(||∇(uh

0 − uH0

)||L2(ωE)h

−1/2E ||RE ||L2(E)

+(||fH ||L2(ωE) + ||fh − fH ||L2(ωE)

)h

1/2E ||RE ||L2(ωE)

).

||fH ||L2(ωE) can be estimated with the first step of the proof and our stated resultfollows readily. �

Definition 6.13. Saturated data approximation: Let T H be a triangulation and T h

a refinement of T H , uH0 ∈ VH the solution of the discrete Poisson problem (6.5) in

VH . We say that the data approximation is saturated with a factor μ > 0, if

δHh := max{CP (Ω)||f − fH ||L2(Ω), CP (Ω)||f − fh||L2(Ω), ||H(fH − fh)||L2(Ω)

}≤ μηT H

(uH

0

). (6.23)

δHh is called the data approximation error. Note that this bound is quite naturalfor small h, since generally δHh will be of lower order compared with ηT H (uH

0 )(for example, if f is piecewise smooth, then ||f − fH ||L2(Ω) = O(h2

max). However, thefollowing results will depend on the quantitative bound (6.23) and not on a vage higherorder argument.

Theorem 6.14. Convergence of adaptive finite element method: Let T H ,VH , uH0 , uh

0 ,and fH , fh be as in Theorem 6.12. Assume that we define FH by the fixed energyfraction strategy (6.21) for some θ ∈ (0, 1) and that we refine T H to get T h as requiredin Theorem 6.12. Assume that

δHh ≤ μηT H

holds for a sufficiently small μ > 0, only depending on σH , σh and θ. Then there is aconstant c that again depends only on σH , σh, such that

||∇(u0 − uh

0

)||L2(Ω) ≤

√1− cθ2 ||∇

(u0 − uH

0

)||L2(Ω)

as long as ηT H > TOL. Thus, for a given tolerance TOL > 0 and a geometrically stablerefinement algorithm, this method reaches ηT H < TOL after finitely many steps.

Proof. We can achieve δHh ≤ μηT H by refining the mesh. We could even establishδHh ≤ μTOL for the given μ, so our requirement would hold as long as ηT H > TOL.Since the finite element approximations to f converge on refined meshes, this can beachieved in finitely many steps. (See [499], where δHh and ηT H are treated within oneiteration.)


We get for the lower bound from Theorem 6.12 with the definition of FH

c′effηFH ≤ ||∇(uh

0 − uH0

)||L2(Ω) + μηT H

which leads with the marking strategy (6.21) to

(c′effθ − μ)ηT H ≤ c′effηFH − μηT H ≤ ||∇(uh

0 − uH0

)||L2(Ω).

We use the approximate Galerkin orthogonality (since VH ⊂ Vh!)∫Ω

∇(u0 − uh

0

)· ∇(uh

0 − uH0

)=∫Ω(f − fh)

(uh

0 − uH0

)and conclude from this

2∣∣∣∣∫

Ω

∇(u0 − uh

0

)· ∇(uh

0 − uH0

)∣∣∣∣ ≥ −2||f − fh||L2(Ω)||uh0 − uH

0 ||L2(Ω)

≥ −2CP (Ω)||f − fh||L2(Ω)||∇(uh

0 − uH0

)||L2(Ω)

≥ −2μηT H ||∇(uh

0 − uH0

)||L2(Ω)

≥ −2μ2η2T H −

12||∇(uh

0 − uH0

)||2L2(Ω).

Using this inequality and Theorem 6.12, we get

||∇(u0 − uH

0

)||2L2(Ω) = ||∇

(u0 − uh

0 + uh0 − uH

0

)||2L2(Ω)

= ||∇(u0 − uh

0

)||2L2(Ω) + ||∇

(uh

0 − uH0

)||2L2(Ω) + 2

∫Ω

∇(u0 − uh

0

)· ∇(uh

0 − uH0

)≥ ||∇

(u0 − uh

0

)||2L2(Ω) +

12||∇(uh

0 − uH0 )||2L2(Ω) − 2μ2η2

T H

≥ ||∇(u0 − uh0 )||2L2(Ω) +

(12(c′effθ − μ)2 − 2μ2

)η2T H

≥ ||∇(u0 − uh0 )||2L2(Ω) + cθ2||∇(u0 − uH

0 )||2L2(Ω)

for a sufficiently small μ, depending on c′eff and θ, and this yields the assertion

||∇(u0 − uh0 )||L2(Ω) ≤

√1− cθ2 ||∇(u0 − uH

0 )||L2(Ω).

�

Remark 6.15. Convergence of adaptive finite element method:

1. The first convergence proof was in [54] (n = 1 with no rate). The method of The-orem 6.14 was developed in [300] and [498]. A more operator oriented approachwas [96].


2. For p > 1 the error indicator has the element residual RT

(uH

0

)= ΔuH

0 + fH .The first step towards convergence is to prove (compare Theorem 6.12)∑

E∈EH,◦

η2E

(uH

0

)≤ C

(||∇(uh

0 − uH0 )||2L2(Ω) + ||H(f − fH)||2L2(Ω) + ||h(f − fh)||2L2(Ω)

),

where one has to assume that marked edges and triangles in AH have been refinedin a suitable way. More generally, one has to prove the inequality

chT ||RT (uH0 )||L2(T ) ≤ sup

vh∈XT

∫TRT (uH

0 )vh

||vh||L2(T )

for some constant c depending on σh and p. Here XT is the surplus space on T :XT = Vh

∣∣T\VH

∣∣T. A similar inequality has to hold for RE [302].

3. The error decreases geometrically for fixed θ. If we miss the tolerance TOLslightly in one step, we have to perform an additional step and we will very likelyend up with an iterate that is more accurate than necessary. Note that the laststep determines the total work if the number of elements increases geometrically.However, we can use the error decay to lower θ near the tolerance level. Assumethat the error satisfies

E2k = (1− cθ2)E2

k−1.

From this we may eliminate the unknown constant c by

c = 1θ2

(1−(

Ek

Ek−1

)2)≈ 1

θ2

(1−(

ηk

ηk−1

)2),

where the error Ek is approximated by our guess ηk := ηTk. When the next error

is presumably below the tolerance, that is if

(1− cθ2)E2k < TOL2,

we use a safety factor 0.9 and define a new value θnew such that

(0.9TOL)2 =(1− cθ2

new

)E2

k ≈(1− cθ2

new

)η2

k.

This yields the explicit formula

θ2new = θ2 1− (0.9TOL)2/η2

k

1− η2k/η

2k−1

under the assumption that both nominator and denominator are positive [300].4. How do we compute FH? We may sort {ηE}E∈EH,◦ by size and then put E ∈ FH

beginning from the largest value until ηFH meets the requirement (6.21). Thecomplexity of this algorithm is NE log(NE) in the number NE of edges in T H .Alternatively, we may do only a rough sort and keep the linear complexity in NE ,Algorithm 6.2 [300].


Algorithm 6.2.Fixed energy fraction markingsum := 0; τ := 1; ν = 0.1; FH := ∅;while sum < θ2η2

T H

τ := τ − ν;for all E ∈ EH,◦

if E �∈ FH ;if η2

E > τ2η2max

FH := FH ∪ {E};sum := sum + η2

E

6.2.7 Optimality

Although the convergence of the algorithm is an important result, it does not guar-antee that we found the most efficient method. In this section we introduce optimalapproximation spaces and show that the proposed algorithm does indeed producemeshes of the optimal asymptotic complexity. This description follows [600].

If u0 ∈ H2(Ω) and if we choose a linear finite element method on a uniform mesh(hT ≡ h for all T ∈ T h), we know that

||∇(u0 − uh0 )||L2(Ω) ∼ h

is optimal with respect to the exponent in h. For Ω ⊂ R2 we have the relation

h ∼ N−1/2

with the number of elements (and thus number of unknowns) N . To achieve

||∇(u0 − uh0 )|| ∼ TOL,

we need N ∼ TOL−2 degrees of freedom and this is again optimal with respect to theexponent, cf. Section 6.1.4. If u0 ∈ H1+r(Ω) for some r ∈ (0, 1], then we know

||∇(u0 − uh0 )||L2(Ω) ∼ hr

and this yields that N ∼ TOL− 2r . We say that u0 ∈ Bs, for some s > 0, if the best

approximation error of u0 in an N -dimensional linear finite element space behavesasymptotically as N−s. Thus N = O(TOL− 1

s ) is the necessary dimensionality toachieve the approximation error TOL for u0. We now define the function space

Bs :=

{v ∈ H1

0 (Ω) : supε>0

{ε infT c : infvT c∈VT c ||∇(v−vT c )||

L2(Ω)≤ε{(�T c − �T c

0 )s} <∞}}

,

where �T denotes the number of triangles in a triangulation T . The infimum is takenover all conforming triangulations T c that can be achieved from the given initial meshT c

0 by newest node bisection, Section 6.2.5. VT c is the linear finite element space overT c. The space Bs is equipped with the norm

||v||Bs := ||∇v||L2(Ω) + [v]Bs .


By construction, v ∈ Bs guarantees that there is a refinement T c of T c0 with

�T c − �T c0 ≤ O(ε−

1s )

such that

infvT c∈VT c

||∇(v − vT c)||L2(Ω) ≤ ε.

As seen above (for r = 2s), a function in H1+2s(Ω) fulfills this requirement, henceH1+2s(Ω) ⊂ Bs, but Bs is definitely a larger set. This kind of space is related to Besovspaces for which a detailed analysis is well known [97]. Based on this definition we willformulate the following notion of optimality.

Definition 6.16. An adaptive method is of optimal complexity if it realizes a meshT c and a numerical approximation uT c of u0 such that44.

||∇(u0 − uT c)||L2(Ω) ≤ ε

with a number of O(ε−1s ) operations for every solution u0 ∈ Bs of (6.4).

The adaptive algorithm, for which we sketch the optimality proof, is the one weproposed before with fixed energy fraction marking (6.21) for some θ > 0 and arefinement as required in Theorem 6.12. It is important to control the amount ofthe actual refinement that is caused by our marking strategy. For this, the refinementstep is split into two steps, REFINE and MAKECONFORM.

Theorem 6.17. Complexity of MAKECONFORM: The refinement edges on T c0 can

be arranged so that the following holds true. Construct a sequence of meshes {T ck }k≥1

by% Refine marked triangles only, Tk+1 is nonconformingTk+1 = REFINE (T c

k , fk, uk);% Refine further to get a conforming meshT c

k+1 = MAKECONFORM (Tk+1);Here, fk is the representation of f and uk the discrete solution of (6.5) on T c

k . IfREFINE refines according to the requirements in Theorem 6.12 and if MAKECON-FORM uses the nodes of newest node bisection, then for n ∈ N

�T cn − �T c

0 ≤ C

n∑k=1

{�Tk − �T c

k−1

}, (6.24)

where C only depends on T c0 .

Proof. [96], Lemma 2.1, Theorem 2.4. �

Let T be a triangulation that is a refinement of T c0 and E◦T the set of interior edges

in T . For the following arguments we will require that f is piecewise constant on T c0 ,

hence we can take f = fT for any refinement T of T c0 . For vT ∈ VT and E ∈ E◦T we

44 To avoid over-indexing, we abbreviate our usual uh0,T c to uT c , and below uh

0,T , uh0,k to uT , uk.


have the error indicator, see (6.20),

ηE(T , f, vT )2 := cEhE ||[∂hvT ]E ||2L2(E) +∑

T∈T : T⊂ωE

cTh2T ||f ||2L2(T ).

The global error estimate is then given by

η(T , f, vT )2 :=∑

E∈E◦T

ηE(T , f, vT )2.

We recall what we know about a posteriori error estimation in our present settingfrom the previous chapters: if u0 and uT are continuous and discrete solutions, thenwe have the upper bound (note: f is assumed to be constant!)

||∇(u0 − uT )||2L2(Ω) ≤ c1

⎛⎝ ∑E∈E◦

T

ηE(T , f, uT )2

⎞⎠1/2

(6.25)

and the lower bound

c2η(T , f, uT ) ≤ ||∇(u0 − uT )||L2(Ω), (6.26)

see Theorem 6.9. Note that it is not assumed that T is a conforming triangulation.Here in this case one defines interpolation equations in the hanging nodes uT (xE) =1/2(uT (p1) + uT (p2)) if xE is the center of the edge p1p2 to get VT ⊂ H1(Ω). We nowstudy a refinement of the bounds (6.25), (6.26).

Lemma 6.18. Modified upper and lower bounds:

1. Let T c be given and T be a refinement. Let F be a set of edges E for which T ∈ Texists such that ωE ∩ T �= ∅. Then �F ≤ C

(�T − �T c

)and with c1 as in (6.25)

||∇(uT − uT c)||L2(Ω) ≤ c1

(∑E∈F

ηE(T c, f, uT c)2)1/2

. (6.27)

2. Let T c be given and F be a set of edges. For each edge E ∈ F refine the twoadjacent triangles by the refinement rule as required in Theorem 6.12. On theresulting triangulation T we have �T − �T c ≤ C�F for some universal constantC and with c2 as in (6.26)

c2

(∑E∈F

ηE(T c, f, wT c)2)1/2

≤ ||∇(uT − wT c)||L2(Ω) (6.28)

for arbitrary wT c ∈ VT c .

Proof. The proof of the first inequality is as in Theorem 6.5, we only have to notethat the chosen functions wT , wT c can be taken to be equal on triangles in T c ∩ T .The proof of the second inequality is as in Theorem 6.12. �

This relates the number of refined triangles to the regularity of u0.


Lemma 6.19. Complexity of REFINE: Let T c0 be a macro-triangulation as in Theo-

rem 6.17, f ∈ P0 (T c0 ), and u0 ∈ Bs be a solution of the continuous problem (6.4) for

some s > 0. Let T c be a refinement of T c0 , uT c the solution of the discrete problem

(6.5) on T c, and

T = REFINE (T c, f, uT c)

with some θ ∈ (0, c2/c1) in (6.21). Then

�T − �T c ≤ C||∇(u0 − uT c)||−1s

L2(Ω)[u0]1s

Bs .

The constant C is independent of s for s ≥ s0 > 0.

Proof. Fix θ ∈ (0, c2/c1), choose λ ∈ (0, 1) with (c2/c1)2(1− λ2) ≥ θ2 and let T beany refinement of T c (T is not conforming!) such that

||∇(u0 − uT )||L2(Ω) ≤ λ||∇(u0 − uT c)||L2(Ω).

Then, by Lemma 6.18, there is a set F ⊂ ET c and a constant C > 0 such that

�F ≤ C(�T − �T c)

and, with (6.27), Galerkin orthogonality, and (6.26),

c21∑E∈F

ηE(T c, f, uT c)2 ≥ ||∇(uT − uT c)||2L2(Ω)

= ||∇(u0 − uT c)||2L2(Ω) − ||∇(uT − u0)||2L2(Ω)

≥ (1− λ2)||∇(u0 − uT c)||2L2(Ω)

≥ c22(1− λ2)η(T c, f, uT c)2.

By the choice of λ it follows that∑E∈F

ηE(T c, f, uT c)2 ≥ θ2η(T c, f, uT c)2.

This set of edges F thus fulfills the inequality (6.21). But since the constructed set ofmarked edges F by REFINE is minimal with this property, we have

�T − �T c ≤ C�F ≤ C�F ≤ C(�T − �T c).

Now let T be the minimal refinement of T c0 by REFINE such that uT satisfies

||∇(u0 − uT )||L2(Ω) ≤ λ||∇(u0 − uT c)||L2(Ω)

and by this assumption

�T − �T c0 ≤

(λ||∇(u0 − uT c)||L2(Ω)

)− 1s [u0]

1s

Bs .


Now we can assume that T is the smallest common refinement of T c and T . Thus wehave

�T − �T c ≤ �T − �T c0

and we conclude

�T − �T c ≤ C(�T − �T c) ≤ C(�T − �T c

0

)≤ C||∇(u0 − uT c)||−

1s

L2(Ω)[u0]1s

Bs

with C depending on λ−1/s hence on θ and any lower bound s0 > 0 for s. �

Algorithm 6.3.Algorithm SOLVE, with f∈ P0(T c

0 ) assumed ![T c

k , uT ck] = SOLVE (T c

0 , f, ε)Solve for uT c

0; k := 0;

while c1η(T c

k , f, uT ck

)≥ ε

Tk+1 := REFINE(T c

k , f, uT ck

);

T ck+1 := MAKECONFORM (Tk+1);

Solve for uT ck+1

;

k = k + 1

We summarize the method in the routine SOLVE in Algorithm 6.3. The nexttheorem proves the optimality of this procedure according to Definition 6.16.

Theorem 6.20. Complexity of SOLVE: T c0 macro-triangulation, f ∈ P0 (T c

0 ), u0 ∈Bs solution of the continuous problem (6.4) for some s > 0 and uT c the solu-tion of the discrete problem (6.5) on T c. Then the adaptive algorithm (T c, uT c) =SOLVE (T c

0 , f, ε) (see Algorithm 6.3) terminates after finitely many steps with

||∇(u0 − uT c)||L2(Ω) ≤ ε

and

�T c − �T c0 ≤ Cε−

1s [u0]Bs .

Proof. The Galerkin orthogonality yields

||∇(u0 − uT c

k

)||2L2(Ω) = ||∇

(u0 − uT c

k+1

)||2L2(Ω) + ||∇

(uT c

k+1− uT c

k

)||2L2(Ω).

By the previous estimates (6.28) and (6.25)

||∇(uT c

k+1− uT c

k

)||2L2(Ω) ≥ (c2θ)2η

(T c

k , f, uT ck

)2≥(c2c1θ

)2

||∇(u0 − uT c

k

)||2L2(Ω).

and so with

δ :=

(1−(c2c1θ

)2)1/2

< 1


we arrive at

||∇(u0 − uT c

k+1

)||L2(Ω) ≤ δ||∇(u0 − uT c

k)||L2(Ω)

(this is the simplified version of Theorem 6.14). From this it follows that the algorithmSOLVE terminates after finitely many steps. It remains to show optimality. Let n bethe index with

c1η(T c

n−1, f, uT cn−1

)≥ ε

and

c1η(T c

n , f, uT cn

)< ε.

With Lemma 6.19 we get

�Tk+1 − �T ck ≤ C||∇(u0 − uT c

k)||−

1s

L2(Ω)[u0]1s

Bs

for k = 0, . . . , n− 1. From our results on newest node bisection, Lemma 6.19,

�T cn − �T c

0 ≤ C

n−1∑k=0

{�Tk+1 − �T ck } ≤ C

n−1∑k=0

||∇(u0 − uT ck)||−

1s

L2(Ω)[u0]1s

Bs

≤ Cn−1∑k=0

(δn−1−k)1s ||∇(u0 − uT c

n−1)||−

1s

L2(Ω)[u0]1s

Bs

≤ C||∇(u0 − uT cn−1

)||−1s

L2(Ω)[u0]1s

Bs ≤ Cη(T c

n−1, f, uT cn−1

)− 1s

[u0]1s

Bs

≤ Cε−1s [u0]

1s

Bs .�

This is the first part of the proof in [600]. It continues with the general case ofvariable f and uT c

kbeing replaced by an approximate solution of the discrete system.

Using interior loops, one establishes a right-hand side fT c and an approximate discretesolution wT c so that

||f − fT c ||H−1(Ω) + ||∇(uT c − wT c)||L2(Ω) ≤ ωη(T c, fT c , wT c)

(for ω sufficiently small, not depending on the mesh size and the data). If the discreteapproximate solution can be established with optimal complexity, then this adaptivealgorithm is of optimal complexity with regard to the regularity properties of u0 and f .

6.2.8 Other types of estimators

There are several possibilities to obtain alternative a posteriori error estimates inthe energy norm. The objective of these variants is to get formulas that are easy toimplement or that allow us to better quantify the constants (cT , cE in (6.20)). However,from the mathematical viewpoint it is decisive to have the upper and lower bounds as


in Theorem 6.9 that would allow us to prove convergence of the adaptive method andoptimality. Actually, these estimates mean that these error indicators are equivalent tothe residual error estimator [655], Sect. 1.3, and these extensions are straightforward(besides the technical details).

Subdomain residual methodOne (approximately) solves a local error equation∫

ωz

∇χz · ∇w =∫

ωz

{fhw −∇uh

0 · ∇w}

for all w ∈ H10 (ωz), (6.29)

where ωz is a patch of triangles around the node z and Vz is an enhanced finiteelement space with support in ωz. The local error estimators, given by ||∇χz||L2(ωz),are equivalent to the residual error estimator [6], Ch. 3.2, [655], Sect. 1.3.

Gradient recoveryThe idea is to replace ||∇(u0 − uh

0 )||L2(Ω) by ||G(uh0 )−∇uh

0 ||L2(Ω), where G(uh

0

)is a

smoothed gradient of uh0 . For example, if uh

0 is piecewise linear, ∇uh0 is piecewise

constant and G(uh

0

)is defined as a projection of ∇uh

0 into the space of linear finiteelements. The method is easy and works well in practise [6], Ch. 4.

Hierarchical error estimateThe idea is to define an enhanced discrete space (Vh)∗ and write the equation on(Vh)∗ as a system with respect to the decomposition (Vh)∗ = Vh ⊕Wh. Wh is calledthe surplus space. With some simplifications one can approximate the error on Vh byan equation on Wh that is easy to solve. For this method to work it is decisive thatthe data error is well resolved [6], Ch. 5.

Equilibrated residual methodIn order to work more locally than the previous method it is necessary to solvelocal Neumann problems. To obtain such bounds one needs to carefully choose thefluxes that define these local boundary problems. There are several approaches toconstructing such fluxes [6], Ch. 6.

6.2.9 hp finite element method

The method so far will work for mesh refinement (h method) for fixed polynomialdegree. If one is interested in variable polynomial degrees, one faces the problem ofdeciding whether to refine the mesh (for fixed p) or increase the polynomial degree forfixed h. The optimal choice of h and p will lead to exponential (instead of algebraic)decay of the error with the number of unknowns. For the error estimation one mayalso use the residual error estimator, however, the explicit dependence of the errorindicator from the polynomial degree is important [575]. A convergence proof for sucha method is only known in one space dimension [301]. Many numerical issues can befound in [280,281].

6.3. Estimation of quantities of interest 449

6.3 Estimation of quantities of interest

6.3.1 Quantities of interest

In the following considerations the functions mostly, but not always, are exact anddiscrete solutions. So we use the notation u and uh instead of the previous u0 and uh

0 .So far we have been interested in discrete solutions uh of −Δu = f and looked for

estimates for ||∇(u− uh)||2. In practical problems, however, one has often more specificquestions. For example, we could ask

u(x0) = ? , ∂1u(x0) = ? for some x0 ∈ Ω,∫Γ

∂nu = ? for some curve Γ ⊂ Ω, ||u− uh||Ω′ for Ω′ ⊂⊂ Ω.

A mesh to answer these specific questions might look very different from those weconsidered before. Therefore we will pose the generalized question

Q(u) = ? ,

where Q : V → R is a continuous operator. However, the examples Q(u) := u(x0),Q(u) := ∂1u(x0) or Q(u) :=

∫Γ∂nu do not satisfy this if u ∈ V = H1(Ω) is assumed!

Hence we have to regularize these operators. Note that solutions are usually moreregular than just in V, so that Q will have some meaning unless very special situationsoccur (e.g. ∂1u(x0) and x0 is a corner point). For example, we let

Qε(u) := 1|Bε(x0)|

∫Bε(x0)

u for x0 ∈ Ω and sufficiently small ε > 0.

Since in this case Q ≡ Qε is linear, we can write

Q(u)−Q(uh) = Q(u− uh) = Q(e).

Our aim is to define a computable quantity η(uh) for which one can show cη(uh) ≤|Q(e)| ≤ Cη(uh) for positive constants c, C.

6.3.2 Error estimates for point errors

Again we need a reasonable error representation. For this we use the duality argument ,to represent Q( . ) as a scalar product in V. We choose u∗ ∈ V to be the solution of∫

Ω

∇v · ∇u∗ = Q(v) = −∫

Bε(x0)

v ≈ v(x0) = δx0 [v] for all v ∈ V,

where δx0 is the Dirac measure in x0. This means that u∗ is an approximate Green’sfunction

u∗(x) ≈ G(x;x0),

that is, G( . ;x0) is the solution of −ΔG( . , x0) = δx0 , G( . , x0)∣∣∂Ω

= 0. Now we canlocalize the error using

e(x0) = δx0 [e] ≈ Q(e) =∫

Ω

∇e∗ · ∇u∗ =∫

Ω

∇e · ∇(u∗ − wh).


The last equality uses the definitions of u and uh. Now we can also solve the discretedual problem ∫

Ω

∇vh · ∇(uh)∗ = Q(vh) for all vh ∈ Vh

and by taking vh = (uh)∗ we obtain

Q(e) =∫

Ω

∇(u− uh) · ∇(u∗ − (uh)∗) =∫

Ω

∇e · ∇e∗. (6.30)

Thus, the dual error ∇(u∗ − (uh)∗) can be viewed as a weight for the error ∇(u− uh)of the primal problem. To proceed we transform the first integral in (6.30) as in thewell-known error representation (6.19) to get

Q(e) =∑

T∈T h

∫T

RT (u∗ − (uh)∗) +∑

E∈Eh,◦

∫E

RE(u∗ − (uh)∗),

with RT ≡ RT (uh) and RE ≡ RE(uh) as in Definition 6.4. Proceeding as in Theorem6.5 leads to

|Q(e)| ≤ C∑

T∈T h

⎛⎝h2||RT ||L2(T ) +∑

E⊂∂T\∂Ω

h3/2E ||RE ||L2(E)

⎞⎠ ζT

with ζT := max{h−2

T ||u∗ − (uh)∗||L2(T ), h−3/2T ||u∗ − (uh)∗||L2(∂T )

}and some positive

constant C that depends only on the shape regularity σh and the polynomial degreep. Unfortunately, we do not know u∗ and it is additional work to compute (uh)∗.

Heuristic approachIn R2, we know that u∗ has the form u∗(x) = 1

2π log(|x− x0|) + w(x) where w issmooth. Therefore we take

ζT ≈hT

|x− x0|2 + ε2.

Defining ηT := hT ||f ||2;T +∑

E⊂∂T\∂Ω h3/2E ||RE ||2;E , we obtain with ε = hT

|Q(e)| ≤ C∑

T∈T h

h2T

|xT − x0|2 + h2T

ηT ,

where xT denotes the center of gravity of T and C is as above.Thus we have derived a computable estimate for |Q(e)| in terms of the local

contribution and we can now proceed with the marking and refinement strategiesof Section 6.2.4.

Estimates using the discrete dual solutionOne computes the dual solution (uh)∗ on the same mesh for simplicity. Using differencequotients or gradient recovery (see Section 6.2.8), one can replace ||u∗ − (uh)∗||L2(T ) byh2

T ||∇(uh)∗||2;T or by ||u∗h,recover − (uh)∗||L2(T ). Another possibility is to compute dual


solutions on a fine (h) and a coarse (H) mesh and let ||u∗ − (uh)∗||L2(T ) ≈ ||(uH)∗ −(uh)∗||L2(T ).

6.3.3 Optimal meshes–A heuristic argument

First recall Section 6.1.5, where we defined an optimal mesh-size distribution as theminimizer of a constraint optimization problem. As before, we letN (h) :=

∫Ωσ/hn and

Q(e) = E(h) =∫Ωh2E2, but now with E2(x) = |∇2u(x)| |∇2u∗(x)|, motivated by the

previous section. Observe that our error criterion E(h) != TOL leads to the Lagrangefunction

L(h, λ) = N (h)− λ(TOL−E(h)

)=∫

Ω

σ

hn− λ(TOL−

∫Ω

h2E2),

which differs from the former definition (6.9) by the exponent of TOL and E(h), andby the formula for E2. Therefore we will conclude, instead of (6.11), that

h2E2 =TOLN(h)

σ

hn.

Extracting h and inserting this into the definition of N (h) we find the relation

N (h) ∼ TOL−n/(2α)

if the integral∫ΩE2n/(n+2α) exists. To compare this to our previous situation we

consider first a smooth solution on a uniform mesh. Since for linear finite elements ||u−uh||L2(Ω) = O

(h2

0

), that is α = 1, the requirement h2

0 ∼ TOL gives N (h0) ∼ TOL−n/2

as above. If we establish a solution by the adaptive finite element method using theenergy estimator (6.20) with tolerance TOL, we need N (h) ∼ TOL−n (cf. Section6.1.4). However, these results do not come with a reliable a posteriori error estimatefor the point values. A rigorous a posteriori error bound, that would very likely givethe optimal complexity, has been given by [513].

A theory of convergence and optimality is the subject of current research. A recentresult is [495].

6.3.4 The general approach

Now we assume that the problem is given by:

seek u ∈ V such that A(u)[φ] := 〈A(u), φ〉V′×V = 0 for all φ ∈ V.Here, V is some Hilbert space and A : V → V ′ has two derivatives A′ and A′′. Forexample, the nonlinear Poisson problem −Δu = f(u) in Ω with homogeneous Dirichletboundary conditions leads to

A(u)[φ] :=∫

Ω

{∇u · ∇φ− f(u)φ

}, A′(u)[φ, ψ] =

∫Ω

{∇ψ · ∇φ− f ′(u)ψφ

}A′′(u)[φ, ψ, χ] =

∫Ω

{− f ′′(u)ψφχ

}.


Now let Vh ⊂ V be a finite element space and assume that integration is performedexactly. Then the discrete problem reads:

seek uh ∈ Vh such that A(uh)[φh] = 0 for all φh ∈ Vh.

Our quantity of interest may be given by some nonlinear, twice differentiable functionalQ : V → R. We define the error functional by

'Q(uh) := Q(u)−Q(uh).

As before, we seek a representation formula for 'Q. We start with a Taylor series (ortwice applying the Mean Value Theorem 1.43): with e := u− uh we get

A(u)[ . ] = A(uh)[ . ] +∫ 1

0

A′(uh + se)[e, . ](s− 1) ds (6.31)

= A(uh)[ . ] + A′(uh)[e, . ]−∫ 1

0

A′′(uh + se)[e, e, . ](s− 1) ds

and

Q(u) = Q(uh) + Q′(uh)[e]−∫ 1

0

Q′′(uh + se)[e, e](s− 1) ds

which gives

'Q(uh) = Q′(uh)[e]−∫ 1

0

Q′′(uh + se)[e](s− 1) ds. (6.32)

Now let u∗ ∈ V be the solution of the dual problem

A′(uh)[φ, u∗] = Q′(uh)[φ] for all φ ∈ V

and (uh)∗ ∈ Vh be the solution of its discrete version

A′(uh)[φh, (uh)∗] = Q′(uh)[φh] for all φh ∈ Vh.

Taking φ = e we obtain by definition of u and uh

'Q(uh) = A′(uh)[e, u∗]−∫ 1

0

Q′′(uh + se)[e, e](s− 1) ds

= A(u)[u∗]−A(uh)[u∗]

+∫ 1

0

(A′′(uh + se)[e, e, u∗]−Q′′(uh + se)[e, e]

)(s− 1) ds

≡ A(u)[u∗]−A(uh)[u∗] + R(uh, e, u∗)

= −A(uh)[u∗ − φh] + R(uh, e, u∗)


for all φh ∈ Vh. Hence

|'Q(uh)| ≤ infφh∈Vh

∣∣A(uh)[u∗ − φh]∣∣+ O

(||e||2V

).

This generalizes Equation (6.30). In the linear case the higher order term R vanishes.In the example from the beginning of this section, the nonlinear Poisson problem, andwith a linear Q, we find the dual problem∫

Ω

{∇φh · ∇u∗ − f ′(uh)φu∗

}= Q[φ] for all φ ∈ V,

whose strong form is −Δu∗ − f ′(uh)u∗ = Q[ . ] in Ω, u∗∣∣∂Ω

= 0. Since 'Q(uh) = Q[e],

∣∣Q[e]∣∣ ≤ inf

φh∈Vh

∣∣∣ ∫Ω

∇uh · ∇(u∗ − φh)− f(uh)(u∗ − φh)∣∣∣+ ∣∣∣ ∫ 1

0

∫Ω

f ′′(uh + se)e2u∗∣∣∣.

By partial integration in the first term we obtain

∑T∈T h

⎧⎨⎩∫

T

f(uh)(u∗ − φh) +∑

E⊂∂T\∂Ω

[∂nu

h]E

(u∗ − φh)

⎫⎬⎭ .

The development of the a posteriori error estimate and the practical aspects are asbefore in Section 6.3.2.

The special case of A=L’ and Q=LThis holds in many physical applications where L is an “energy” and the weak equationis the extremal condition L′(u)[v] ≡ A(u)[v] ≡ 0 for all v ∈ V. Our quantity of interestmay be the value of the energy Q(u) := L(u). Hence 'Q(uh) := Q(u)−Q(uh) =L(u)− L(uh). Then (6.32) translates to

L′(u)[ . ] = L′(uh)[ . ] +∫ 1

0

L′′(uh + se)[e, e, . ](s− 1) ds

= L′(uh)[ . ] + L′′(uh)[e, . ]−∫ 1

0

L′′′(uh + se)[e, e, . ](s− 1) ds

and instead of (6.32) we have the expansion

'Q(uh) =∫ 1

0

L′(uh + se)[e] ds = sL′(uh + se)[e]∣∣∣10−∫ 1

0

sL′′(uh + se)[e, e] ds

= −12(s2 − 1)L′′(uh + se)[e, e]

∣∣∣10

+∫ 1

0

12(s2 − 1)L′′′(uh + se)[e, e, e] ds

= −12L′′(uh)[e, e] +

12

∫ 1

0

(s2 − 1)L′′′(uh + se)[e, e, e] ds.

We now observe that we do not need to solve a dual problem since it now suffices touse −1/2 e as a test function in the expansion of L′(u)[ . ] to eliminate −1

2L′′(uh)[e, e]


and so we get

'Q(uh) = −12L′′(uh)[e, e] +

12

∫ 1

0

(s2 − 1)L′′′(uh + se)[e, e, e] ds

=12L′(uh)[e]− 1

2

∫ 1

0

s(1− s)L′′′(uh + se)[e, e, e] ds

=12L′(uh)[e− φh] + O

(||e||3V

)=

12L′(uh)[u− φh] + O

(||e||3V

)and thus

'Q(uh) = infφh∈Vh

∣∣∣12L′(uh)[u− φh]

∣∣∣+ O(||e||3V

).

We repeat that this holds without a dual solution and that the second term vanishesif L is quadratic! This is especially useful if we consider eigenvalue problems [398].

7

Discontinuous Galerkin methods(DCGMs), with V. Dolejsı

7.1 Introduction

Discontinuous Galerkin methods (DCGMs) represent a class of FEMs. In contrastto the usual FEMs, piecewise polynomials of varying degrees in different elementsK are used. Continuity between neighboring elements and boundary conditions arenot required. These violations are punished by a penalty function, see (7.41), (7.61),increasingly unpleasant for finer mesh sizes h. In some sense, DCGMs are thereforenonconforming FEMs.

So, many of the remarks in the introduction to Chapter 4 remain valid forDCGMs as well. Again we use the general discretization theory in Chapter 3. Theproofs for convergence via stability and consistency techniques are again based uponlinearization.

The original DCGM was introduced by Reed and Hill [546] for the solution of theneutron transport equation. It was analyzed for the first time by Le Saint and Raviart[472]; later improvements were achieved by Wheeler [665] and Johnson and Pitkaranta[426]. These methods were also applied to purely elliptic problems. Examples are theoriginal method of Bassi and Rebay [75] the stabilized version of the original Bassi–Rebay method studied by Brezzi et al. [146], and its generalization, called the localdiscontinuous Galerkin methods (LDGMs), introduced by Cockburn and Shu [194],and further studied by Castillo et al. [166, 190]. In the 1970s, Galerkin methods forelliptic and parabolic equations using discontinuous finite elements were independentlyproposed and a number of variants introduced and studied, cf. Arnold [39], Baker [56]and Wheeler [665]. These DCGMs were called interior penalty (IP) methods, andtheir development remained independent of the development of the DCGMs forhyperbolic equations. A unified analysis and comparison of various discontinuousGalerkin techniques for elliptic problems was developed by Arnold et al. [40]. Fora survey see, e. g. Cockburn et al. [182,192], summarizing the development of DCGMsfor elliptic, parabolic and hyperbolic problems.

Finally, we mention the so-called hybridized discontinuous Galerkin methods pro-posed and developed by Cockburn and co-workers in [163, 183–189]. This approachbelongs among the mixed techniques where the hybridization reduces the globallycoupled unknowns.

456 7. Discontinuous Galerkin methods (DCGMs)

Here and throughout this book, we restrict the discussion to elliptic problems. Inthis chapter, we present a detailed study of a class of DG methods for second-orderelliptic problems which includes most of the above-mentioned methods.

Many papers (e.g. [166,297,319,517]) consider the linear diffusive operator −Δu or∇ · (A∇u), where A is a symmetric matrix independent of u. In addition, lower orderterms are allowed, cf. (7.13). Several generalizations to nonlinear diffusion operatorswere studied. Riviere and Wheeler [552] consider the nonlinear diffusion operator

−∇ · (a(x, u)∇u), a : Ω× R → R, (7.1)

where the function a(x, u) is Lipschitz-continuous with respect to its second variableand there exist constants γ and γ′ such that

0 < γ ≤ a(x, u) ≤ γ′ ∀(x, u) ∈ Ω× R. (7.2)

Extensions and improved energy estimates with applications to single phase flow inporous media are described by Wheeler et al. [554,555].

Houston, Robson and Suli considered in [407], and extended in Houston, Suli andWihler [412], the quasilinear diffusive operator in the form

−∇ · (μ(x, |∇u|)∇u) ,with|∇u| = |∇u|n, (7.3)

where μ ∈ C(Ω× [0,∞)) is a function satisfying the assumption

mμ(t− s) ≤ μ(x, t)t− μ(x, s)s ≤Mμ(t− s), t ≥ s ≥ 0, x ∈ Ω, (7.4)

with positive constants mμ and Mμ.Here we start with εΔu plus a low order flux term in (7.13) and proceed to more

general cases. For quasilinear systems, e.g. the diffusion term takes the form

−∇ · �R(x, �u,∇�u), �R : Ω× Rq × Rn×q → Rq. (7.5)

Among several discontinuous Galerkin techniques two approaches, SIPG (symmetricinterior penalty Galerkin) and NIPG (nonsymmetric interior penalty Galerkin), arevery popular. The main idea, applied to a linear diffusion problem, here to −Δu, isthe following. By applying Green’s theorem, we obtain, for every inner face e and itsunit normal vector ν, an integral∫

e

({∇u}, ν)n[v] dS , (7.6)

appropriately modified for boundary faces. The symbols {·} and [·] denote an averagevalue and jump of a function beyond the face e, cf. (7.32), (7.33). Then we add, byformally exchanging the regular function u and the test function v, an integral in theform

θ

∫e

({∇v}, ν)n[u] dS , (7.7)


where θ = 1 for SIPG and θ = −1 for NIPG. Obviously, the integrals (7.7) vanish fora regular function u ∈ H1(Ω). Moreover, the so-called interior penalty,∫

e

σ[u][v] dS , σ > 0 (7.8)

again vanishing for a regular function, is included in the interior penalty variantsof DCGMs. In Antonietti et al. [33] a different type of stabilization of DCGMs ispresented. Moreover, Burman and Stamm [153] propose a method which requiresstabilization only on a part of the skeleton.

The antisymmetry of the sum of terms in (7.6) and (7.7) (with θ = −1) implies thecoercivity of an extended bilinear form plus the penalty term, cf. (7.57), (7.161), ofthe NIPG technique for any penalty parameter σ > 0. On the other hand, for ensuringthe coercivity of this bilinear form for the other DCGMs it is necessary to choose theparameter σ sufficiently large. However, this pays for the SIPG type discretizations.It yields a symmetric discrete problem. This is important for numerical methods andallows optimal error estimates in the L2 norm.

The SIPG and NIPG techniques can be directly extended to nonlinear diffusionterms as in (7.1)–(7.3). The face integrals corresponding to (7.6) and (7.7) are∫

e

({a(u)∇u}, ν)n[v] dS ,∫

e

({μ(x, |∇u|)∇u}, ν)n[v] dS (7.9)

and

θ

∫e

({a(u)∇v}, ν)n[u] dS , θ

∫e

({μ(x, |∇u|)∇v}, ν)n[u] dS , (7.10)

respectively. The incomplete replacement of u by v from (7.9) to (7.10) maintainsthe essential linearity with respect to v. The linearity with respect to u is lostanyway.

This technique is no longer possible for general quasilinear diffusion terms of theform (7.5). Hence, SIPG as well as NIPG methods are not suitable for the discretizationof quasilinear terms ∇ · �R(x, �u,∇�u), unless �R(x, �u,∇�u) is linear in ∇�u, or specificconditions as for (7.3) are imposed. Applying Green’s theorem, the face integral oftype (7.6) and/or (7.9) yields, differently from before,

−∫

e

({�R(x, u,∇u)}, ν)n[v] dS . (7.11)

Exchanging the arguments u and v as in (7.7) would cause the resulting integral tobecome nonlinear with respect to v. Therefore we employ the so-called incompleteinterior penalty Galerkin (IIPG) method studied by Dawson et al. [269,615,616]. Ourmethod here for elliptic problems has been afterwards, without reference, modifiedand generalized to the parabolic case in [295]. We obtain a nonsymmetric diffusiveform and have to choose the penalty parameter sufficiently large.

As for FEMs with variational crimes, the interplay between the strong and theweak form of the problem plays an essential role. So the original strong problem is


transformed into its weak form. This excludes, for our approach, general fully nonlinearproblems. For smooth enough functions, the strong forms of our operators can be easilydetermined, cf. (7.1)–(7.5). The corresponding weak forms we systematically define viatesting in the appropriate spaces, cf. e.g. (7.19)–(7.22).

We start discussing these ideas for the important model problem (7.13)–(7.15). Foru ∈ H2(Ω) ∩ L∞(Ω), (7.13) is transformed into its weak form (7.23), valid for u ∈H1(Ω) ∩ L∞(Ω). These convection–diffusion problems are among the most interestingapplications of the DCGMs. Here the diffusion term is multiplied by a small diffusioncoefficient ε > 0.

In most of the chapter, we deal with convection–diffusion problems. For a reactionterm f(u) instead of

∑ns=1 ∂fs(u)/∂xs, (7.13) represents a reaction–diffusion prob-

lem. Sometimes a reaction term is added to the equation, e.g. for obtaining betterproperties. However, it is not present in the equations of fluid dynamics.

The terms in (7.6)–(7.11) are combined with the piecewise or broken Sobolev normsin (7.29)–(7.31) on the triangulation T h for u ∈ Hk(T h) for all k ≥ 0 for convergenceresults. Penalty terms, e.g. (7.8), measure the transition errors along the interior andboundary edges for broken bilinear and nonlinear forms and penalty terms, cf. (7.41).Thus the original problem, e.g. (7.23), with its boundary condition, is translatedinto the discrete (7.70) or (7.71), cf. (7.57)–(7.61). For the strong form, (7.13), withu ∈ H2(Ω), v ∈ L2(Ω), this often requires u ∈ L∞(Ω) for the nonlinearity. For the tran-sition to the weak form, and the DCGMs, we have to impose conditions u ∈ H1(T h) oru, v ∈ H3/2+ε(T h). This up and down between u, v ∈ H2(T h), u, v ∈ H3/2+ε(T h), andu, v ∈ H1(T h), all u ∈ L∞(Ω), sometimes complicates the presentation, cf. Remarks7.1 and 2.1, (2.156), and Theorem 7.5.

As indicated, we extend the DCGMs from the semilinear model problem (7.13) tomore general quasilinear systems with diffusive terms in (7.5). Furthermore, we study,besides the standard numerical flux for convection diffusion equations, other types offluxes as well, cf. (7.45)–(7.50). As nonconforming FEMs, the DCGMs are possible forlinear to quasilinear, but not for fully nonlinear problems. The modification from oursecond order equations to the different types of semilinear or quasilinear systems istotally analogous to that for the previous FEMs. So we do not formulate the DCGMsfor all these types of elliptic problems, but only for some order 2 including generalquasilinear cases.

The results are presented in the Hilbert space setting, although extensions to theBanach space setting would be essentially possible.

Extensions to order 2m are problematic. The main tool for second order problems,the appropriate relation between the strong an weak forms of the problem, wouldbecome highly technical. So we omit problems of order 2m,m > 1.

Recall that the following problems are only discussed for FEMs: General convergencetheory for monotone operators, cf. Section 4.5, variational methods for eigenvalueproblems, cf. Section 4.7, methods for nonlinear boundary conditions, cf. Section 5.3,quadrature approximate methods, cf. Section 5.4. They remain valid for DCGMs aswell. Our techniques for fully nonlinear elliptic problems, cf. Section 5.2, are notpossible for DCGMs. However, the monotone operator approach, cf. Section 4.5, allowsDCGMs for problems of order 2m,m ≥ 1.

7.2. The model problem 459

7.2 The model problem

Let Ω ⊂ Rn (here n = 2 or 3) be a bounded polyhedral domain with closure Ω andboundary ∂Ω = ∂ΩD ∪ ∂ΩN , ∂ΩD ∩ ∂ΩN = ∅. We consider the following boundaryvalue problem, for the exact solution u0 in Ω :

u0 : Ω → R, u0 ∈ H1(Ω) ∩ L∞(Ω),Ω ⊂ Rn,Ω ∈ C0,1, open bounded, (7.12)

G(u0) :=n∑

s=1

∂fs(u0)∂xs

− εΔu0 − g = 0 in Ω, (7.13)

u0 |∂ΩD= uD on ∂ΩD, (7.14)

∂u0

∂ν|∂ΩN

= gN on ∂ΩN , (7.15)

cf. Remark 7.1, with an appropriate flux term. We require the following conditions:

(a) fs ∈ C1(R), fs(0) = 0, s = 1, . . . , n; (7.16)

(b) ε > 0;

(c) g ∈ L2(Ω), or below g ∈ H−1(Ω) with 〈g, v〉 := 〈g, v〉H−1(Ω)×H1(Ω);

(d) uD is the trace of some u∗ ∈ H1(Ω) ∩ L∞(Ω) on ∂ΩD;

(e) gN ∈ L2(∂ΩN ).

This u∗ ∈ H1(Ω) ∩ L∞(Ω) allows trivial boundary conditions by substituting u :=u− u∗ and updating g. We avoid that here to demonstrate the role of the violatedboundary conditions in DCGMs more clearly, cf. (7.41).

Equation (7.13) shows that the assumption fs(0) = 0, s = 1, . . . , n, does not causeany loss of generality. The functions fs, called fluxes, represent convective terms, ε > 0is the diffusion coefficient. The diffusion term can be more complicated, in some caseseven nonlinear, see Section 7.5. For our model problem in (7.13)–(7.15), we considermixed Dirichlet–Neumann boundary conditions. The same technique is possible forthe following general cases as well. There we only present the Dirichlet condition onthe whole boundary.

A sufficiently regular function, u0 ∈ C2(Ω), satisfying (7.13) – (7.15) pointwise iscalled a classical solution. Let

Vb = {v ∈ H1(Ω), v|∂ΩD= 0}. (7.17)

Multiplying (7.13) with v ∈ Vb and partial integration yields, via the first Green’sformula, (2.9), (2.10), and with (7.15), and the Euclidean product (., .)n in Rn,∫

Ω

(−Δu) vdx =∫Ω

(∇u,∇ v)ndx−∫

∂ΩD

∂u

∂νvdS −

∫∂ΩN

gN vdS. (7.18)


Now we have to discuss, e.g. the∫

∂ΩD∂u/∂ν vdS, vanishing for v ∈ Vb. The restriction

v|∂Ω for v ∈ H1(Ω) yields v|∂Ω ∈ H1/2(∂Ω) in the trace sense. This is, for our Ω in(7.12), an immediate consequence of Theorems 1.37, 1.39.

Remark 7.1. For Neumann conditions (7.15), and for the evaluation of the ∂u/∂ν|∂ΩD

in (7.18) and the formulation of the DCGMs, the situation is more complicated.An arbitrary u ∈ H1(Ω) does not allow a meaningful restriction to ∂Ω or to e for∂u/∂ν ∈ L2(∂Ω) or ∂u/∂ν ∈ L2(e).

There are two ways out of this dilemma. Either Theorems 1.37, 1.39 are used forreplacing the above u ∈ H1(Ω) by u ∈ H3/2+ε(Ω) with any ε > 0. Then ∂u/∂ν|∂Ω ∈Hε(∂Ω) ⊂ L2(∂Ω) or ∂u/∂ν|e ∈ L2(e) is valid. Hence, the

∫∂Ω

v∇udS, and∫

ev∇udS,

are well defined, cf. (7.42). Or u ∈ H1(Ω) is a function in the domain, e.g. of theLaplacian, then again

∫ev∇udS are well defined. This problem of boundary traces has

attracted some attention recently, cf. Taylor [618], Chapter 4, Proposition 4.5, andJonsson and Wallin [425], Jerison and Kenig [419], Schwab [575] and Grisvard [373].

Readers not so familiar with the H3/2+ε(Ω) results can in this chapter nearly every-where replace H3/2+ε(Ω) by H2(Ω), with ∂u/∂ν|∂Ω ∈ L2(∂Ω) and ∂u/∂ν|e ∈ L2(e),thus requiring a bit more than necessary.

So we assume u ∈ H3/2+ε(Ω), hence∫

∂ΩD∂u/∂ν vds is well defined, and∫

∂ΩD∂u/∂ν vdS = 0 for our v ∈ Vb. This allows the transition, motivated by (7.18),

to the following well-defined forms in their appropriate spaces:

a(u, v) = ε

∫Ω

(∇u,∇v)n dx, u ∈ U := H1(Ω) ∩ L∞(Ω), v ∈ Vb cf. (7.17): (7.19)

b(u, v) =∫

Ω

n∑s=1

∂fs(u)∂xs

v dx, u ∈ U , v ∈ L2(Ω), cf. (7.16) (a), (7.20)

�(v) = (g, v)L2(Ω) − (gN , v)L2(∂ΩN ), v ∈ L2(Ω), v|∂Ω ∈ L2(∂Ω). (7.21)

These a(u, v), b(u, v), �(v) allow introducing the nonlinear operator G :

G : u ∈ D(G) ⊂ U → U ′ : 〈G(u), v〉 − b(u, v)− a(u, v) + �(v) = 0 ∀ v ∈ Vb. (7.22)

A weak solution, u0, of (7.13) – (7.15) satisfies the weak equation

u0 ∈ U : 〈G(u0), v〉 = 0 ∀ v ∈ Vb and u0 |∂ΩD= uD on ∂ΩD (7.23)

∂u0

∂ν|∂ΩN

= gN on ∂ΩN .

A regular solution u0 ∈ H2(Ω) of (7.23) solves (7.13)–(7.15) as well.In Chapter 2, we discussed existence and uniqueness results for a large class of

problems, generalizing (7.13), (7.14), (7.23). A classical solutions solves (7.23) for∀ v ∈ Vb. Vice versa, if (7.23) has smooth input data and a smooth enough solution,u0 ∈ H2(Ω), then its solves (7.13),(7.14). So we assume a unique solution u0 of (7.23)with boundedly invertible G′(u0). A similar discussion applies to the more generalcases, (7.74), (7.78), (7.79) and (7.90), (7.91).

7.3. Discretization of the problem 461

7.3 Discretization of the problem

7.3.1 Triangulations

We formulate the DCGM on a polyhedral domain

Ω ⊂ Rn, n ≥ 2, bounded polyhedral with boundary ∂Ω ∈ C0,1. (7.24)

Generalizations to curved domains Ω are possible, admitting

Ω ⊂ Rn, n ≥ 2, bounded curved domain with boundary ∂Ω ∈ C0,1, (7.25)

if we require that, except at the edges of ∂Ω, the solution u0 ∈ Hk(Ω) can be extendedinto u0 ∈ Hk(Ωe) with an Ωe ⊃ Ω. Under this condition, we can employ the techniquein Section 5.2, Theorem 5.4, for extending DCGMs on polyhedral to curved domains.We can obtain at most second order accuracy, if we choose a

sequence of approximating polygonal Ωh for Ω, s.t. dist(∂Ω, ∂Ωh) ≤ Ch2. (7.26)

Higher order versions would require Theorems 7.6 and 7.5 also for curved elements.Theorem 7.6 is proved for this case; for Theorem 7.5 we did not succeed.

So we restrict the formulation of the following DCGMs to polyhedral domains. LetT h (h > 0) be a partition of the domain Ω into a finite number of open n-dimensional(convex or nonconvex) mutually disjoint polyhedra K. In contrast to triangles T ∈ T h,for the standard FEMs, we denote the often more general elements here as K. In 2Dproblems we usually choose K ∈ T h as triangles or quadrilaterals, or as in Figure 7.1.In 3D, K ∈ T h can be, e. g. tetrahedra, pyramids or hexahedra. But we can constructeven more general elements K, as dual finite volumes from [314]. In the analysis carriedout in Section 7.7 it is important to assume star-shaped elements K ∈ T h. Then thestandard error and inverse estimates in Theorems 7.6, 7.7 are valid, cf. Theorems 4.17,4.19, and [141], p. 104. The exact assumptions on T h will be given in Section 7.7.1.We modify the notation for a triangulation, T h, of Ω as

∀K ∈ T h : K is open, ∀K1 �= K2 ∈ T h : K1 ∩K2 = ∅, Ω = ∪K∈T hK, (7.27)

T h = {K}K∈T h with hK = diam(K), h = maxK∈T hhK and {e}e∈T h ,

the set of all faces of T h. Then for any e ∈ T h there are two possibilities:

(i) e ∈ T h \ ∂Ω is an inner face, so there exist two elements Kl, Kr ∈ T h, Kl �= Kr

sharing e, i.e. e ⊂ Kl ∩Kr.45 If Kl ∩Kr is not a straight face we split it intoa minimal possible number of straight faces, see Figure 7.1, showing a possible2D situation. It is arbitrary which one of the elements sharing the inner face isdenoted as Kl and as Kr, but this choice should be kept.

(ii) e ⊂ ∂Ω, often denoted as e ∈ ∂Ω, is a boundary face. Then there exists anelement Kl ∈ Th such that e ⊂ ∂Kl.

By νe := ν := (ν1, . . . , νn) we denote the unit normal on the face e, which is theouter normal to the element Kl sharing e (see Figure 7.1). Moreover, the symbols

45 Subscripts l and r indicate “left” and “right” elements sharing e.


K l Kre1

e2

νc1

νc2

Figure 7.1 Neighboring elements Kl, Kr sharing two faces e1, e2 ∈ T h.

|K| and |e| mean the n-, and (n− 1)-dimensional Lebesgue measure of K, and of e,respectively, (i. e. the length and the area of e, if n = 2 and n = 3), and diam(e) itsdiameter. It is obvious that for K ∈ T h and its face e ⊂ K we have

|K| ≤ C1hnK ≤ C1h

n, |e| ≤ C1hn−1K ≤ C1h

n−1, d(e) := diam(e) ≤ hK ≤ C1h. (7.28)

7.3.2 Broken Sobolev spaces

We recapitulate the broken Sobolev space, norms and seminorms over the triangula-tion T h only for the case of Hilbert spaces, see (4.35), (4.36).

Hk(T h) = {v : v|K ∈ Hk(K) ∀K ∈ T h}, (7.29)

‖v‖Hk(T h) =

⎛⎝ ∑K∈T h

‖v‖2Hk(K)

⎞⎠1/2

, (7.30)

|v|Hk(T h) =

⎛⎝ ∑K∈T h

|v|2Hk(K)

⎞⎠1/2

. (7.31)

For v ∈ H1(T h), we recall jumps and means, cf (5.82): for a face e ∈ T h ∩K we extendv to K; for a boundary face e ∈ ∂Ω, or an inner face with e = Kl ∩Kr, let

vl = v|Kl, vr = v|Kr

, [v] = [v]e = vl|e − vr|e, {v} = {v}e = (vl|e + vr|e)/2, (7.32)

[v] = [v]e = {v} = {v}e = v|∂Ωh along e ∈ ∂Ωh, v arbitrary in Rn\Ωh. (7.33)

Obviously, the value [v]e depends upon the setting of which of the sharing element isKl and which is Kr. On the other hand, the interesting term νe[v]e is independent ofthe choice of Kl and Kr. In the following, omitting the index, e, we agree∑

e∈T h

∫e

[v]({∇u}, ν)n means∑

e∈T h

∫e

[v]e({∇u}e, νe)n etc.


7.3.3 Extended variational formulation of the problem

In order to derive the discrete problem, we start from the strong solution u0 ∈H2(Ω) ∩ L∞(Ω) of (7.13). We consider (7.13) on each K, multiply it by an arbitraryv ∈ H1(T h), integrate over each K ∈ T h, apply Green’s theorem and summarize overK ∈ T h. Note that we no longer require ∈ H1

0 (T h). Whenever defined, we obtain theidentity∑

K∈T h

(−∫

K

(�f(u0),∇v)n dx +∫

∂K

(�f(u0), ν)n v dS (7.34)

+ ε

∫K

(∇u0, ∇v)n dx− ε

∫∂K

(∇u0, ν)n v dS −∫

K

g v dx

)= 0 ∀v ∈ H1(T h)

with (�f(u), ν)n =n∑

s=1

fs(u) νs, and (�f(u),∇v)n =n∑

s=1

fs(u) ∂sv.

We combine (7.32), (7.33) with (5.259), obtaining for the sum of the second boundaryintegrals in (7.34), whenever defined,

−ε∑

K∈T h

(∫∂K

(∇u, ν)n v dS

)= (7.35)

− ε∑

e∈T h\∂Ω

∫e

([v]({∇u}, ν)n + {v}([∇u], ν)n

)dS − ε

∑e∈∂Ω

∫e

v(∇u, ν)ndS.

Similarly, we obtain as in (5.259), (5.260),∑K∈T h

∫∂K

(�f(u), ν)n v dS (7.36)

=∑

e∈T h\∂Ω

∫e

([v]({�f(u)}, ν)n + {v}([�f(u)], ν)n

)dS +

∑e∈∂Ω

∫e

v(�f(u), ν)ndS.

We have to discuss the question of when the terms in (7.34), (7.35), (7.36) aredefined. The extended variational form (7.34) is certainly not defined for the originalu0 ∈ U as in (7.19). Instead (7.34) motivates spaces u0, u ∈ Uo

e = H3/2+ε(Ω) for theoriginal problem, or even an extension to T h, allowing ∂u/∂ν|e, ∂v/∂ν|e ∈ L2(e).Similarly we will choose Ve = Ue for (7.42) ff. We use Vb in (7.17) and introduce,always restricting U , and for simplicity the V as well, to ∩L∞(Ω),

U = V = H1(Ω) ∩ L∞(Ω), U ′ ⊂ H−1(Ω), Vb,Ub = H10 (Ω) ∩ L∞(Ω), (7.37)

Uoe := Vo

e := H3/2+ε(Ω) ∩ L∞(Ω),Ue := Ve := H3/2+ε(T h) ∩ L∞(T h), . . . ,

and for the strong forms Us = H2(Ω) ∩ L∞(Ω), Us,e := H2(T h) ∩ L∞(Ω).

We are going to introduce DCGMs. They are defined on Sh ⊂ H3/2+ε(T h) ∩ Ue. Soa combination now of the exact solution u0 ∈ Ue, with v ∈ Ue, the Neumann condition


(7.15), and (7.34), (7.35), (7.36), yields the following equation:

u0 ∈ Uoe ⊂ Ue : −

∑K∈T h

∫K

(�f(u0),∇v)ndx (7.38)

+∑

e∈T h\∂Ω

∫e

([v]({�f(u0)}, ν)n + {v}([�f(u0)], ν)n

)dS +

∑e∈∂Ω

∫e

v(�f(u0), ν)ndS

+ ε∑

K∈Th

∫K

(∇u0, ∇v)n dx− ε∑

e∈T h\∂Ω

∫e

([v]({∇u0}, ν)n + {v}([∇u0], ν)n

)dS

− ε∑

e∈∂ΩD

∫e

v(∇u0, ν)n dS −∫

Ω

g v dx− ε

∫∂ΩN

gN v dS = 0 ∀v ∈ U .

There are different ways for further modifying the diffusion terms: The unchangedε∑

e∈T h

∫e[v]({∇u}, ν)ndS, based upon the vanishing ([∇u0], ν)n, is used by Wheeler

[665], p. 153, extended to the case of nonlinear diffusion in (7.1).Here we use the fact that for the exact solution u0 ∈ H3/2+ε(Ω) ⊂ Ue the terms

[�f(u0)], [∇u0] and (u0 − uD)|∂ΩDin (7.38) vanish. Furthermore, preparing the

DCGMs, we first omit the terms {v}([�f(u0)], ν)n, {v}([∇u0], ν)n.Then we add to (7.38), for e ∈ T h \ ∂Ω, 0 = θε({∇v}, ν)n[u0], and, for e ∈ ∂ΩD,

0 = θε(∇v, ν)n(u0 − uD). This is used in the integration over the edges e in (7.38).The standard choices for θ, are, see below,

θ = −1, or θ = 1, or θ = 0, sometimes even θ ∈ [−1, 1].

For the DCGMs we need the function σ, defined by, cf. (7.28),

σ :⋃

e∈T h

e→ R : σ|e =cw

d(e), with d(e) = diam(e), cw > 0. (7.39)

Another possibility is to define σ, e.g. as

σ|e =cw

max{diam(Ke1),diam(Ke2)}or σ|e =

cw min{|Ke1 |, |Ke2 |}d(e)

, (7.40)

where Ke1 and Ke2 are neighboring elements sharing face e, and Ke1 = Ke2 for e ⊂ ∂Ω.These choices of interior penalties lead to equivalent results. Moreover, we should notethat cw may depend on the degree of the polynomial approximation. Finally, we addthe following penalty terms to the last two lines of (7.38).

ε∑

e∈T h\∂Ω

∫e

σ [u] [v] dS, ε∑

e∈∂ΩD

∫e

σ (u − uD) v dS. (7.41)

This allows the desired coercivity results for DC piecewise polynomial test functionsvh ∈ Sh, cf. (7.63), Theorems 7.16–7.19. They are based upon the penalty norm


‖v‖JH1(T h) = ‖v‖, see (7.153). But the original ‖v‖H1(T h) and the new ‖v‖J

H1(T h) arenonequivalent norms on H1(T h) for h→ 0, see Theorem 7.12. We will formulate thecorresponding consistency results in Theorems 7.21–7.33, tested by vh ∈ Sh, linearand bounded with respect to ‖v‖J

H1(T h).

All the terms in (7.38), (7.41), including those evaluating u,∇u0, v,∇v along edges,e, are well defined ∀u, v ∈ Ue, e ∈ T h, and many of them vanish or equal uD for e ∈ T h,for the exact solution u0 ∈ H3/2+ε(Ω), cf. Theorem 1.37 on trace operators. This wouldbe incorrect for a u0 ∈ U .

We use (7.32), (7.33), (7.35), and collect∑

e∈T h\∂Ω and∑

e∈∂ΩDinto

∑e∈T h\∂ΩN

,

now for u0, v ∈ H3/2+ε(Ω) = Uoe . Then we obtain

u0 ∈ Uoe : −

∑K∈T h

∫K

(�f(u0), ∇v)n dx +∑

e∈T h

∫e

[v]({�f(u0)}, ν)ndS (7.42)

+ ε∑

K∈Th

∫K

(∇u0, ∇v)n dx− ε∑

e∈T h\∂ΩN

∫e

([v]({∇u0}, ν)n + θ({∇v}, ν)n[u0]

)dS,

+ ε

⎛⎝ ∑e∈T h\∂ΩN

∫e

σ[u0] [v] dS −∑

e∈∂ΩD

∫e

σuD v dS

⎞⎠−(∫

Ω

g v dx + ε

∫∂ΩN

gN v dS − εθ∑

e∈∂ΩD

∫e

(∇v, ν)n uD dS

)= 0 ∀v ∈ Ue.

For the exact solution u = u0 of (7.23), satisfying the boundary conditions (7.14),and the test functions v ∈ H3/2+ε(Ω) ⊂ Ue we obtain

[u0] = [u0]e = 0, {∇u0} = {∇u0}e = ∇u0|e, and similarly for v, (7.43)

thus we get the weak form (7.23), and the strong form (7.13) for u = u0 ∈ Us.For the “convective” face integrals∑

e∈T h

∫e

[v]({�f(u0)}, ν)ndS, (7.44)

the more sophisticated concept of a numerical flux in DCGMs is employed. Thisis motivated, e.g. by finite volume techniques for conservation laws. There the useof the numerical flux ensures the stability of numerical schemes using explicit timediscretization, see, e.g. [313, 315]. For the different choices of numerical fluxes thepresent type of problems plays a crucial role. For example, the center numerical flux({�f(u0)}, ν)n is unconditionally unstable for explicit discretization of time dependentproblems. However, for implicit time discretization we have no stability restriction,but we have to solve a system of nonlinear algebraic equations at each time step. Ourresults below show stability and convergence for elliptic problems.


In our context the numerical flux emphasizes the evaluation of the term (�f(u), ν)n

appearing as the integrand in the first line of (7.34). We consider the edge, e, again asthe face of Kl and Kr, e ⊂ Kl ∩Kr given by (7.27) ff., (7.32) with the outer normal,ν. The numerical flux H(ul|e, ur|e, ν) approximates the evaluation along e as the faceof Kl in the first argument of H(·, ·, ·) and indicates the impact of e as the face of Kr

in its second argument.A famous example is the Lax–Friedrich numerical flux in the form

H(ul|e, ur|e, ν) =12

(P (ul|e, ν) + P (ur|e, ν)− λ(ur|e − ul|e)

), λ > 0, (7.45)

(e.g.) =12

((�f(ul|e), ν)n + (�f(ur|e), ν)n − λ(ur|e − ul|e)

)(7.46)

=(({�f(u|e)}, ν)n −

12λ[u|e]

)(7.47)

hence, P (u, ν) := (�f(u), ν)n, for (7.46).Another example is the upwind scheme given by

H(ul|e, ur|e, ν) =

{(�f(ul|e), ν)n =

∑ns=1 fs(ul|e)νs, if A > 0

(�f(ur|e), ν)n =∑n

s=1 fs(ur|e)νs, if A ≤ 0,(7.48)

where

A =n∑

s=1

f ′s({u}e)νs = (�f ′({u}e), ν)n. (7.49)

The face integrals (7.44) correspond to the central numerical flux

H(ul|e, ur|e, νe) = ({�f(u|e)}, νe)n ∀e ∈ T h. (7.50)

For further examples of numerical fluxes we refer to, e.g. [313].We shall assume that the numerical flux has the following properties:

Assumptions (H):

1. H(u, v, ν) is defined in R2 ×B1, where B1 = {ν ∈ Rn; |ν| = 1}, and Lipschitz-continuous with respect to u, v:

|H(u, v, ν)−H(u∗, v∗, ν)| ≤ C2(|u− u∗|+ |v − v∗|), (7.51)

u, v, u∗, v∗ ∈ R, ν ∈ B1.

2. H(u, v, ν) is flux-consistent:

H(u, u, ν) = (�f(u), ν)n, u ∈ R, ν ∈ B1. (7.52)

3. H(u, v, ν) is conservative:

H(u, v, ν) = −H(v, u,−ν), u, v ∈ R, ν ∈ B1. (7.53)


Moreover, (7.51) and (7.52) imply that the function �f is Lipschitz-continuous withconstant Lf = 2C2 and from (7.16) (a) and (7.52) we see that

H(0, 0, ν) = 0 ∀ ν ∈ B1. (7.54)

Obviously, numerical fluxes given by (7.45), (7.48) and (7.50) satisfy Assumptions(H).

So we obtain in the first line of (7.34), (7.42) with (7.33) the terms∑K∈T h

(∫∂K

(�f(u), ν)n v dS

)

=∑

e∈Th\∂Ω

∫e

(H(ul|e, ur|e, νe)vl|e + H(ur|e, ul|e,−νe)vr|e

)dS

+∑

e∈∂Ω

∫e

(H(ul|e, ur|e, νe)vl|e

)dS (7.55)

=∑

e∈Th\∂Ω

∫e

H(ul|e, ur|e, νe)(vl|e − vr|e) dS +∑

e∈∂Ω

∫e

(H(ul|e, ur|e, νe)vl|e

)dS

=∑e∈Th

∫e

H(ul|e, ur|e, νe)[v]e dS, ∀v ∈ H1(T h).

Consequently (7.42) has the form

u0 ∈ Ue : −∑

K∈T h

∫K

(�f(u0), ∇v)n dx +∑e∈Th

∫e

H(u0,l|e, u0,r|e, νe)[v]e dS (7.56)

+ ε∑

K∈Th

∫K

(∇u0, ∇v)n dx− ε∑

e∈T h\∂ΩN

∫e

([v]({∇u0}, ν)n + θ({∇v}, ν)n[u0]

)dS

+ ε∑

e∈T h\∂ΩN

∫e

σ[u0] [v] dS −∫

Ω

g v dx−∫

∂ΩN

gN v dS

−(ε∑

e∈∂ΩD

∫e

σuD v dS − εθ∑

e∈∂ΩD

∫e

(∇v, ν)n uD dS

)= 0 ∀v ∈ Ue.

On the basis of equations (7.42) and (7.56) we introduce:

∀u, v ∈ Ue : ah(u, v) := ε∑

K∈Th

∫K

(∇u, ∇v)n dx (7.57)

− ε∑

e∈T h\∂ΩN

∫e

(({∇u}, ν)n [v] + θ({∇v}, ν)n[u]

)dS


bh(u, v) = −∑

K∈T h

∫K

({�f(u)},∇v)n dx +∑

e∈T h

∫e

({�f(u)}, ν)n[v]dS (7.58)

ch(u, v) := −∑

K∈T h

∫K

({�f(u)},∇v)n dx +∑e∈Th

∫e

H(ul, ur, ν)[v] dS (7.59)

�h(v) := 〈g, v〉+ (gN , v)L2(∂ΩN ) − εθ∑

e∈∂ΩD

∫e

(∇v, ν)n uD dS (7.60)

+ ε∑

e∈∂ΩD

∫e

σuD v dS with

Jσh (u, v) :=

∑e∈T h\∂ΩN

∫e

σ[u] [v] dS, and J1h(u, v) :=

∑e∈T h\∂ΩN

∫e

[u] [v] dS. (7.61)

In the discrete problem the form Jσh represents a penalty function for interior

discontinuities along the edges e. The boundary errors are collected in the termsε∑

e∈∂Ω

∫eσ(uD v − u v)dS obtained from εJσ

h (u, v)− �h(v). These terms replace therequirement of continuity and boundary conditions of the approximate solution inconforming FE methods. Note that for increasingly finer T h the σ in (7.39) increases.Thus violated conditions are punished more strongly. For some of the later discussionswe mainly consider uh, vh ∈ Sh, with bounded Jσ

h (v, v), e.g. those well approximatingu, v ∈ U . This is achieved by introducing the so-called penalty norms, cf. Definition7.11.

So we have modified the above (7.38). These (7.57)–(7.61) allow formulating differentDCGMs. As mentioned above, the standard choices for θ, are

θ = −1, or θ = 1, or θ = 0, sometimes even θ ∈ [−1, 1], (7.62)

with θ = −1 for the so-called nonsymmetric form of the interior penalty Galerkinmethod, NIPG cf. Riviere et al. [552, 553, 555], θ = 1 for the symmetric form, SIPG,cf. Arnold [39], θ = 0 for the incomplete form, IIPG, cf. Dawson et al. [269, 615, 616],and −1 ≤ θ ≤ 1 for the general form for the general penalty Galerkin method, GIPG,cf. Houston et al. [407]. Hence ah(u, v) �= ah(v, u) or, for θ = 1, the ah(u, v) = ah(v, u)is obtained.

Our analysis includes all four cases. Standard numerical methods essentially useθ = −1, 0, 1. For θ = −1, we only have to impose cw > 0. For θ = 0, 1, and θ ∈ (−1, 1]we have to observe lower bounds for cw > 0, cf. Theorem 7.16 ff. There we will comeback to the discussion of appropriate choices of the cw > 0.

These different choices have “pros and cons”: The symmetric form maintains thesymmetry of the previous and later a(u, v) = a(v, u) with desirable consequences fornumerical methods. However, the cw in (7.39) has to be chosen, for the symmetricform, the incomplete form and for θ ∈ (−1, 1] according to (7.164), (7.180), (7.194).This implies coercive principal parts of the corresponding linearized discrete operators.This is valid for the unsymmetric form for any cw > 0.


The symmetric bilinear form ah(·, ·) allows deriving optimal a priori error estimatesin the L2 norm using the well-known Aubin–Nitsche trick, cf. Lemma 5.73 andTheorem 5.74 for FEMs and [40] for DCGMs and the references therein. Neverthe-less, the methods with θ = 1 give in some situations optimal experimental order ofconvergence, namely for odd degrees of polynomial approximations when (almost)equidistant partitions are employed. On the other hand, for ensuring the coercivityfor the symmetric form it is necessary to choose the penalty parameter σ, in particularthe cw, more carefully than in the nonsymmetric case, cf. Theorems 7.16, 7.18, 7.19.For general quasilinear problems we only formulate this last type of IIPGM below.

7.3.4 Discretization

Now we introduce the discrete problem approximating (7.56) via the DCGM. Wedefine the space of discontinuous piecewise polynomial functions

Sh := Sd−1,−1(T h) := {v : T h → R; v|K ∈ Pd−1(K) ∀K ∈ T h}, d ≥ 2, (7.63)

where Pd−1(K) denotes the space of all polynomials on K of degree ≤ d− 1 ∈ N.We derive the discrete problem in such a way that u0 in (7.56) is approximated by

uh0 ∈ Sh and tested by vh ∈ Sh. In the open K ∈ T h, ∇uh

0 ,∇vh and their restrictionsto e ⊂ K are well defined for uh, vh ∈ Sh ⊂ H3/2+ε(T h) ⊂ H1(T h). Hence the formsah(u, v), ch(u, v), bh(u, v), �h(v), Jσ

h (u, v), cf. (7.57)–(7.61) make sense for u, v ∈ Ue

and u = uh, v = vh ∈ Sh.We have seen that extended analytical problems require u, v ∈ Ue. So it is surprising

and advantageous that for the DCGMs a combination of Jσh (v, v) and |v|2H1(T h) is

sufficient for proving the boundedness, and consistency of the previous ah(·, ·), bh(·, ·),ch(·, ·), �h(·), and the coercivity of the principal part of and the stability for ah(·, ·)with respect to such a norm. This so-called penalty norm, cf. (7.61), and Definition7.11, is

‖v‖ := ‖v‖Vh := ‖v‖JH1(T h) :=

(Jσ

h (v, v) + |v|2H1(T h)

)1/2. (7.64)

This norm is chosen for Uh = Vh = Sh, cf (7.153)

uh, vh ∈ Uh = Vh = Sh with ‖vh‖Uh = ‖vh‖Vh := ‖vh‖JH1(T h) = ‖v‖. (7.65)

It is not equivalent to the broken Sobolev norm for h→ 0, cf. Theorem 7.12.Before we formulate the DCGMs, we discuss the relations between the original and

discrete operators, G and Gh. This is essentially the same idea for all the followingproblems. So we demonstrate it for our model problem. With the a(u, v), . . . in(7.19)–(7.21), the ah(u, v), . . . , Jσ

h (u, v) in (7.57)–(7.61), and, e.g. the 〈Gh(u), v〉 =〈Gh(u), v〉U ′

e×Ue, we define, for a given fixed u or uh, the weak operators G,Gh, G

h,

by testing with v or vh. We indicate this in (7.66), e.g. by u,∀v ∈ U . For G we usethe b(u, v) in (7.20), and in Gh we allow replacing ch(u, v) by bh(u, v) in (7.66). ForG we have two possibilities: either we impose the Dirichlet condition directly as in


(7.66) (b), or we enforce them for h→ 0 as in (7.66) (c) by the penalty terms.

(a) G : D(G) ⊂ U → U ′, Gh : D(Gh) ⊂ Ue → U ′e, G

h : D(Gh) ⊂ Uh → U ′h, (7.66)

(b) 〈G(u), v〉 − a(u, v)− b(u, v) + �(v) = 0 ∀ v ∈ Vb, (u− uD)|∂ΩD= 0, u ∈ U ,

(c) 〈G(u), v〉 − ah(u, v)− bh(u, v)− εJσh (u, v) + �h(v) = 0, u,∀ v ∈ Uo

e ,

(d) 〈Gh(u), v〉 − ah(u, v)− bh(u, v)− εJσh (u, v) + �h(v) = 0, u,∀ v ∈ Ue,

(e) 〈Gh(uh), vh〉 − ah(uh, vh)− bh(uh, vh)− εJσh (uh, vh) + �h(vh) = 0, uh,∀vh

∈ Sh, ah(uh, vh), bh(uh, vh) := ah(uh, vh), bh(uh, vh)|Sh×Sh , �h(vh) := �h(vh)|Sh ,

or we may replace in (c)–(e) the bh by ch. In (c) we either have to consider u,∀v ∈ Uoe ,

or u,∀v ∈ U , with [u] = [v] = 0 for interior faces e and such that u|∂Ω, v|∂Ω ∈ L2(∂Ω).For relating G,Gh, and Gh we use the previous Uo

e = H3/2+ε(Ω) ∩ L∞(Ω), cf. (7.37).For all u, v ∈ Uo

e , many terms in Gh cancel:

u, v ∈ Uoe and [u]e = [v]e = 0, for all interior faces e ∈ T h, imply

ah(u, v) := ε∑

K∈Th

∫K

(∇u, ∇v)n dx− ε∑

e∈∂ΩD

∫e

((∇u, ν)nv + θ(∇v, ν)nu

)dS

bh(u, v) = −∑

K∈T h

∫K

(�f(u),∇v)n dx +∑

e∈∂Ω

∫e

(�f(u), ν)nvdS

ch(u, v) := −∑

K∈T h

∫K

(�f(u),∇v)n dx +∑

e∈∂Ω

∫e

H(ul, ur, ν)v dS

�h(v) := 〈g, v〉+ (gN , v)L2(∂ΩN ) − εθ∑

e∈∂ΩD

∫e

(∇v, ν)n uD dS + ε∑

e∈∂ΩD

∫e

σuD v dS

Jσh (u, v) :=

∑e∈T h

∫e

σ[u] [v] dS.

We collect these terms in 〈Gh(u), v〉 and find

∀u, v ∈ Uoe : 〈Gh(u), v〉 = ah(u, v) + bh(u, v) + εJσ

h (u, v)− �h(v)

= 〈G(u), v〉+∑

e∈∂Ω

∫e

(�f(u), ν)nvdS (7.67)

− ε∑

e∈∂ΩD

∫e

((∇u, ν)nv + θ(∇v, ν)nu

)dS + ε

∑e∈∂ΩD

∫e

σu v dS

+ εθ∑

e∈∂ΩD

∫e

(∇v, ν)n uD dS − ε∑

e∈∂ΩD

∫e

σuD v dS

= 〈G(u), v〉∀ u ∈ Uoe , (u− uD)|∂ΩD

= 0 ∀ v ∈ Ub.


Note that we used for the last equality in (7.67) the

b(u, v) = bh(u, v) = −∑

K∈T h

∫K

({�f(u)},∇dxv)n

+∑

e∈∂ΩN

∫e

(�f(u), ν)nvdS ∀ v ∈ Ub. (7.68)

This allows summarizing these relations as

〈Gh(u), v〉 = ah(u, v) + bh(u, v) + εJσh (u, v)− �h(v)∀u, v ∈ Uo

e , (7.69)

= 〈G(u), v〉 = a(u, v) + b(u, v)− �(v) ∀u : (u− uD)|∂ΩD= 0 ∀ v ∈ Ub,

= 〈G(u), v〉 ∀u ∈ U : (u− uD)|∂ΩD= 0, ∀ v ∈ Ub, so, with

Uoe,b := {u ∈ Uo

e , (u− uD)|∂ΩD= 0} , Ub := {u ∈ U , (u− uD)|∂ΩD

= 0} :

Gh = Gh|Sh , and Gh|Uoe,b

= G|Uoe,b

= G ↑Ub.

Remark 7.2. This last equality in (7.69), Gh|Uoe,b

= G|Uoe,b

= G ↑Ub, indicates the

restriction from Gh, defined in Uoe to Uo

e,b, and the extension of G, defined in Uoe,b, to G

defined in the original Ub ⊃ Uoe,b. Note that we have to test Gh = Gh|Sh by v ∈ Sh, Ghu

by v ∈ Ue, Gh|Uoe

= G|Uoe

by v ∈ Uoe , and G ↑Ub

by v ∈ Ub. We observe an analogoussituation for all the following problems as well.

We define the nonstandard DCGM and its discrete approximate solution as afunction uh

0 : T h → R satisfying the conditions, cf. (7.66), (7.69),

uh0 ∈ Sh;

⟨Gh(uh

0

), vh⟩

= 0 ∀ vh ∈ Sh. (7.70)

Similarly, we define the discrete approximate solution of a standard DCGM, byreplacing bh in (7.58), (7.69) by ch in (7.59), for a function uh

0 : Ω → R satisfying

(a) uh0 ∈ Sh, s.t. ∀ vh ∈ Sh, (7.71)

(b)⟨Gh(uh

0

), vh⟩

:= ah(uh

0 , vh)

+ ch(uh

0 , vh)

+ εJσh

(uh

0 , vh)− �h(vh) = 0.

Many terms in (7.71) cancel, in particular Jσh (u, v) = 0, and we have obtained

Gh|Uoe

= G|Uoe

= G ↑U , in (7.66). Proceeding as in (7.67), (7.68), (7.69), the exactsolution u0 ∈ U , obviously satisfies the original (7.23) as

〈G(u0), v〉 =(ah(u0, v) + ch(u0, v) + εJσ

h (u0, v))− �h(v) = 0 ∀ v ∈ Vb. (7.72)

With this regular exact solution u0 we find the identity

〈Gh(u0), vh〉 := ah(u0, vh) + ch(u0, v

h) + εJσh (u0, v

h)− �h(vh) = 0 ∀vh ∈ Sh (7.73)


by (7.57). This implies the so-called Galerkin orthogonality property of the error. Thismeans that, for this and the following cases,⟨Gh(uh

0

), vh⟩

= 0 = 〈Gh(u0), vh〉 ∀vh ∈ Sh ⇒⟨Gh(uh

0

)−Gh(u0), vh

⟩= 0 ∀vh ∈ Sh.

7.4 General linear elliptic problems

Now we are ready to formulate more general problems. For simplicity, we consideronly the Dirichlet boundary condition on the whole boundary of Ω in this and thefollowing sections, i.e. ∂Ω = ∂ΩD. A possible generalization to a case with Neumannboundary conditions on a part of the boundary can be carried out simply as in theprevious section.

We have fully formulated FEMs for elliptic linear or semi- and quasilinear secondorder equations and systems. So we restrict the presentation here to linear equationsand to semilinear equations and quasilinear systems in Section 7.5. The linear boundaryvalue problem and its DCGMs are discussed briefly. We give more details for thefollowing semilinear second order equations. Since there are strong similarities wedo not repeat all details for extending (7.13), (7.14) ff. to general linear equations.Compared to (7.13), we replace the original Δu, ∂u/∂ν by the general Au, (Ba∇u, ν)n

or Au, (Ba∇0u, ν)n in the following (7.74), (7.75) and omit the flux terms, and therestriction to L∞(Ω) in U , . . . , in (7.37). As usual, multiplying the strong operator,As, with v ∈ U , we obtain by partial integration and Green’s formula, the relation tothe weak operator, A, cf. (5.278): Compute46

u0 ∈ U := H10 (Ω) : a(u0, v) = 〈Au0, v〉V′×V = 〈f, v〉V′×V ∀v ∈ U , with (7.74)

a(u, v) = 〈Au, v〉V′×V =∫

Ω

⎛⎝ n∑i,j=0

aij∂iu ∂jv

⎞⎠ dx with maxx∈Ω

{|aij(x)|} ≤ C0,

=∫

Ω

⎛⎝ n∑i,j=0

(−1)j>0∂j (aij∂

iu)

⎞⎠ vdx +∫

∂Ω

(Ba∇0u, ν)nvds

=: (Asu, v)L2(Ω) +∫

∂Ω

(Ba∇0u, ν)nvds

∀v ∈ U , u ∈ Us = H2(Ω), if aij ∈W 1−δj0,∞(Ω),

with (Ba∇0u, ν)n :=j=1,...,n∑i=0,...,n

νjaij∂i u, and

n∑i,j=1

aij(·)ξiξj ≥ λ|ξ|2n , λ > 0, for our elliptic operators.

46 We require maxx∈Ω{|aij(x)|} ≤ C0 instead of the usual aij ∈ L∞(Ω), since we need bounds for

‖aij‖L∞(e) along e ∈ T h as well, not available for aij ∈ L∞(Ω).

7.4. General linear elliptic problems 473

This (Ba∇0u, ν)n depends upon ∇0u = (∂iu)ni=0, e.g. Asu = −Δu yields (Ba∇0u,

ν)n = (Ba∇u, ν)n = ∂u/∂ν = (∇u, ν)n.Again we replace Hk(Ω) by Hk(T h) and obtain with Green’s formula as for (7.34)

ff., with Uh,Vh in (7.63), (7.65), and with (7.32), (7.33),

〈Ahu, v〉H−1(T h)×H1(T h)

:=∑

K∈T h

⎛⎝∫K

n∑i,j=0

aij∂i u∂j v

⎞⎠ dx ∀u, v ∈ Ue ⊃ Uh

=∑

K∈T h

∫K

⎡⎣(As,h u)v :=n∑

i,j=0

(−1)j>0∂j(aij∂

i u) v

⎤⎦ dx (7.75)

+∑

e∈T h\∂Ω

∫e

([v]{(Ba∇0u, ν)n} + {v}

[(Ba∇0u, ν)n

] )ds

+∑

e∈∂Ω

∫e

v(Ba∇0u, ν)n ds

=: as,h(u, v) +∑

e∈T h

∫e

([v]{(Ba∇0u, ν)n}+ {v}

[(Ba∇0u, ν)n

] )ds.

As above, for preparing the DCGMs, we consider several terms, which vanish forthe exact solution u0 = u ∈ Ue, and vh = v ∈ Vh. We omit, for all e ∈ T h, the{vh}[(Ba∇0u, ν)n]. We add, for all e ∈ T h \ ∂Ω, 0 = θ{(Ba∇0v, ν)n}[u], and fore ∈ ∂Ω, 0 = θ(Ba∇0v, ν)n(u− uD). Again the nonsymmetric, symmetric, incompleteand general form are defined by θ = −1, θ = 1, θ = 0 and θ ∈ [−1, 1], respectively. Forall u, v ∈ Ue we use the Jσ

h (u, v) in (7.61), and introduce, cf Remark 7.2:

ah(u, v) :=∑

K∈Th

⎛⎝∫K

n∑i,j=0

aij∂iu ∂jv

⎞⎠ dx−∑

e∈T h

∫e

({(Ba∇0u, ν)n}[v] (7.76)

+ θ{(Ba∇0v, ν)n}[u])dS for u, v ∈ Ue

�h(v) := 〈g, v〉 − θ∑

e∈∂Ω

∫e

((Ba∇0v, ν)n)uD dS +∑

e∈∂Ω

∫e

σuD v dS and

〈Ahu, v〉 − ah(u, v)− Jσh (u, v) + �h(v) = 0 for a fixed u ∈ Ue,∀ v ∈ Ue,

with ah(·, ·) := ah(·, ·)|Uh×Vh , a(·, ·) = ah(·, ·)|U×V ,

ahs (·, ·) := as,h(·, ·)|Uh×L2(Ω), as(·, ·) = as,h(·, ·)|Us×L2(Ω)

�h(·) := �h(·)|Vh , �(·) = �h(·)|U , and again,

〈Ahu, v〉 = 〈Au, v〉 ∀u, v ∈ Ue and = 〈Au, v〉 ∀u, v ∈ U .


This yields for linear elliptic equations the DCGM and its discrete approximatesolution as a function uh

0 ∈ Sh satisfying the conditions

uh0 ∈ Sh :

⟨Ah(uh

0

), vh⟩

= 0 ∀ vh ∈ Sh = Vh. (7.77)

The Galerkin orthogonality results in (7.73) remain correct with the obvious modi-fications.

Another interesting type, the advection–diffusion–reaction problem, is studied byAntonietti and Houston [34]

7.5 Semilinear and quasilinear elliptic problems

7.5.1 Semilinear elliptic problems

Next we discuss semilinear boundary value problems, from now on only with Dirichletboundary conditions. Determine the exact solution u0 : Ω → R such that

G(u0) :=n∑

s=1

∂sfs(u0)− ε

n∑i,j=1

(∂j(aij(x, u0)∂iu0

)− g(x, u0) = 0 in Ω, (7.78)

u0 |∂Ω = uD on ∂Ω. (7.79)

We choose, similarly to (7.13),∑n

i,j=1 instead of an equally possible∑n

i,j=0,and ∂ΩD = ∂Ω. We do not formulate the obvious modifications for

∑ni,j=0. This

generalizes (7.13), (7.14). Its linearized second order principal part of the form∑ni,j=1 aij(x, u0(x))∂j∂iu) is a special case of (7.75). The special case

∇ · a(x, u)∇u =n∑

i=1

∂i(a(x, u)

)∂iu

of (7.78) has been discussed by Riviere and Wheeler [552] for parabolic problems. Theellipticity of (7.78) is defined by its linearized principal part, cf. (7.80) (e). We assumethe following conditions in a form that is simply to handle, e.g. (c):

(a) fs ∈ C1(R), fs(0) = 0, s = 1, . . . , n, (7.80)

(b) ε > 0,

(c) aij : Ω× R → R, i, j = 1, . . . , n, are bounded by |aij(x, u)| ≤ Λa,

(d) aij are Lipschitz-continuous with respect to u, i.e. ∃L > 0 ∀x ∈ Ω, u1, u2 ∈ R :

|aij(x, u1)− aij(x, u2)| ≤ L|u1 − u2|, i, j = 1, . . . , n,

(e) Ellipticity: For u ∈ U , with ‖u− u0‖U ≤ ρ there exist λ,Λ ∈ R+ such that

0 < λ|ϑ|2 ≤n∑

i,j=1

aij(x, u(x))ϑj ϑi ≤ Λ|ϑ|2 ∀x ∈ Ω, ϑ = (ϑ1, . . . , ϑn) ∈ Rn,

7.5. Semilinear and quasilinear elliptic problems 475

(f) aij(·, u(·)) ∈ C1(Ω) for u ∈ C2(Ω) in (7.78), and

∈ L∞(Ω) for u ∈ U in (7.81),

(g) g : Ω× R → R, g(·, u(·)) ∈ L2(Ω) or ∈ U ′ for u ∈ U ,(h) g ∈ U ′ is Lipschitz-continuous with respect to u, i.e.

∃L′ > 0 ∀x ∈ Ω ∀u1, u2 ∈ R : |g(x, u1)− g(x, u2)| ≤ L′|u1 − u2|,(i) uD is the trace of some u∗ ∈ U on ∂Ω.

A classical solution u0 ∈ Us satisfies (7.78)–(7.79) pointwise. For the weak solutions,u, v ∈ U , we use the notation, cf (7.19), (7.20), (7.74):

a(u, v) = ε

∫Ω

n∑i,j=1

aij(x, u)∂iu∂jv dx = ε

∫Ω

(Ba(u)∇u,∇v)ndx,with (7.81)

Ba(u)∇u :=

(n∑

i=1

(aij(x, u)∂iu

)n

j=1

,

b(u, v) =∫

Ω

n∑s=1

∂sfs(u) v dx, with boundary condition (7.79) u, v ∈ U . (7.82)

Thus we update the above∇u, and a(u, v) in (7.19), by Ba(u)∇u, and a(u, v) in (7.81).A weak solution of (7.78) – (7.79) is defined by the updated (7.23).

7.5.2 Variational formulation and discretization of the problem

In analogy to (7.34), we obtain ∀K ∈ T h, by replacing ∇u by Ba(u)∇u,∑K∈T h

(−∫

K

(�f(u), ∇v)n dx +∫

∂K

(�f(u), ν)n v dS + ε

∫K

(Ba(u)∇u,∇v)ndx (7.83)

−ε∫

∂K

(Ba(u)∇u, ν)nv dS −∫

K

g(x, u) v dx)

= 0 ∀v ∈ H1(T h),

or we replace∫

Kg(x, u) v dx by 〈g(x, u), v 〉H−1(K)×H1(K). Further considerations

follow (7.38), (7.41)–(7.42): Here, we again omit the terms {v}([�f(u0)], ν)n,{v}([Ba(u0)∇u0], ν)n in (7.83) and add

ε

∫e

σ [u] [v] dS, θε

∫e

({Ba(u)∇v}, ν)n[u] dS, for e ∈ T h \ ∂Ω and (7.84)

ε

∫e

σ (u − uD )v dS, θε

∫e

(Ba(u)∇v, ν)n(u− uD) dS, for e ∈ ∂Ω,

with θ = −1 θ = 1, θ = 0, θ ∈ [−1, 1] for the NIPG, SIPG, IIPG, GIPG methods,respectively.


So we obtain, instead of (7.42), again with (7.32), (7.33), and u0 replacing u,

u0 ∈ Uoe = H3/2+ε(Ω) : −

∑K∈T h

∫K

(�f(u0), ∇v)n dx +∑

e∈T h

∫e

[v]({�f(u0)}, ν)ndS

+ ε∑

K∈Th

∫K

(Ba(u0)∇u0, ∇v)n dx− ε∑

e∈T h

∫e

(({Ba(u0)∇u0}, ν)n[v] (7.85)

+ θ({Ba(u0)∇v}, ν)n[u0])dS + ε

∑e∈T h

∫e

σ[u0] [v] dS

−∫

Ω

g v dx− ε∑

e∈∂Ω

(∫e

σuD v − θ(Ba(u0)∇v, ν)nuD

)dS = 0 ∀v ∈ Uo

e ,

cf. (7.37). Now we return to the exact solution u = u0 ∈ Ue, of (7.78), satisfying theboundary conditions, (7.79), and the test functions v ∈ Uo

e ∩ Ub. Along the edges, e,we have

[u0]e = 0, {Ba(u0)∇u0}e = Ba(u0)∇u0|e ∀e ∈ T h \ ∂Ω and

{Ba(u0)∇u0}e = Ba(u0)∇u0|e, [u0]e = u0|e = uD ∀e ∈ ∂Ω;

similarly for v, we end up again with the weak form, the modified (7.23).On the basis of the above considerations and notation, we introduce, similarly to

(7.42), (7.56), the strong and a provisional and the final weak discrete semilinear forms,as,h(·, ·), ah(·, ·), and ah(·, ·), by including the transition terms, as

as,h(u, v) := −ε∑

K∈T h

∫K

n∑i,j=1

(∂j(aij(x, u)∂iu

)vdx, (7.86)

ah(u, v) := ε∑

K∈T h

∫K

[(Ba(u)∇u, ∇v)n =

n∑i,j=1

(aij(x, u)∂iu ∂jv

)]dx

ah(u, v) := ε∑

K∈T h

∫K

(Ba(u)∇u, ∇v)n dx (7.87)

− ε∑

e∈T h

∫e

(({Ba(u)∇u}, ν)n [v] + θ({Ba(u)∇v}, ν)n [u]

)dS,

�h(u, v) := (g(x, u), v)L2(Ω) + ε∑

e∈∂Ω

∫e

(σuD v − θ(Ba(u)∇v, ν)n u

)dS (7.88)

∀ u, v ∈ Ue ⊃ Sh, and ch(u, v), Jσh (u, v) as in (7.59), (7.61).

The as,h(u, v) and ah(u, v) even require u ∈ H2(T h), v ∈ L2(T h), and u, v ∈H1(T h). Or (g(x, u), v) may be replaced by 〈g(x, u), v 〉 = 〈g(x, u), v 〉H−1(T h)×H1(T h).


This yields

as,h(u, v) = as(u, v)∀u ∈ Us, v ∈ L2(Ω).

Totally analogously to (7.66) we define the operators G,Gh, Gh, by replacing those

ah(u, v), �h(v) in (7.58), (7.59), (7.61), by the new terms in (7.87), (7.88). We do notexplicitly update (7.66), cf (7.89) for Gh.

We define the standard DCGM in NIPG, SIPG, IIPG, and GIPG forms and itsdiscrete approximate solution as a function uh

0 : Ω → R satisfying the conditions

(a) uh0 ∈ Sh, s.t. ∀ vh ∈ Sh (7.89)

(b)⟨Gh(uh

0

), vh⟩

:= ah(uh

0 , vh)

+ ch(uh

0 , vh)

+ εJσh

(uh

0 , vh)− �h

(uh

0 , vh)

= 0 with

ah(uh, vh), ch(uh, vh), �h(uh, vh) := ah(uh, vh), ch(uh, vh), �h(uh, vh)|Sh×Sh .

The nonstandard DCGM in NIPG, SIPG, IIPG, and GIPG form is again definedby replacing in the last equation ch

(uh

0 , vh)

by bh(uh

0 , vh).

A particularly interesting problem is the stationary incompressible Navier-Stokesequation. This has to be discretized by mixed formulations of DCGMs, namely by theso-called LDG, the local discontinuous Galerkin method. It could be treated similarlyto Section 4.6 on the basis of the Ladyzenskaja–Brezzi–Babuska condition, cf. (4.214),for the Stokes operator and for DCGMs, e.g. [191,434].

7.5.3 Quasilinear elliptic systems

Previously we discussed equations. The following quasilinear problems are formulatedas systems of q equations of second order. Again we start with the standard divergentquasilinear systems in strong form in the Hilbert space setting. We distinguish thedirect generalization of (7.78) and (7.79) in (7.90), (7.91) and its further generalizationin (7.93). We determine the solution �u0 ∈ D(G) ⊂ H2(Ω,Rq) ∩ L∞(Ω,Rq) of theproblem in its strong form, such that,

G(�u0) :=n∑

s=1

∂sfs(�u0)− ε

n∑k=1

∂kAk(·, �u0,∇�u0)− g(·, �u0) = 0 ∈ Rq in Ω, (7.90)

�u0 |∂Ω = �uD on ∂Ω, with smooth enough Ak : Ω× Rq × Rn×q → Rq. (7.91)

In this case we maintain the conditions for �f, ε, g in (7.80). The precise condition forthe Ak will be discussed below. For our quasilinear problem we introduce, with anappropriate D(G), the spaces, cf. (7.37),

U = H1(Ω,Rq) ∩ D(G), Vb = H10 (Ω,Rq),Ue := Ve := H3/2+ε(T h,Rq) ∩ D(G),

Ub = H10 (Ω,Rq) ∩ D(G), U ′ ⊂ H−1(Ω,Rq), . . . , , and for the (7.92)

strong forms Us = H2(Ω,Rq) ∩ D(G), Us,e := H2(T h,Rq) ∩ D(G).


The general quasilinear weak form is

�u0 ∈ U : 〈G(�u0), �v〉 :=∑

K∈T h

∫K

n∑k=0

(Ak(x, �u0,∇�u0), ∂k�v

)qdx = 0 ∀�v ∈ Vb,

�u0 |∂Ω = �uD on ∂Ω. (7.93)

For equations we admitted u0 ∈ D(G) ⊂W 2,α(Ω) in Chapter 2, Section 4.4 and thefollowing results would have to be modified accordingly. The necessary conditionsfor the Nemickii operators Ai(x, uh,∇uh), e.g. for bounded forms and operators,ah(uh, vh) and Gh, are formulated in Sections 2.5, 2.6, 4.4 and (4.174). We have studiedgrowth conditions for systems for Ak(x, �u,∇�u) =

(ai

k(x, �u,∇�u))qi=1

in Subsection 2.6.4,cf. e.g. (2.361) ff., (2.368) ff. and (2.372). They allow for the existence, uniquenessand regularity results for the solutions of our quasilinear systems, e.g., Theorems 2.89,2.96–2.101. Their bounded linearizations with the Legendre condition and the coerciveprincipal part are discussed in Section 2.7.4, cf. Theorem 2.125.

For the Ak, we modify (7.80) into (7.94) on the basis of these results. As an examplewe consider the so-called natural growth conditions, cf. (2.380). The controllable growthconditions in (2.379) can be treated in a similar way.

We summarize one type of these natural growth conditions. The ellipticity of (7.93)is defined by the ellipticity of its linearized principal part. This can be guaranteed by(7.94) (e). With �u = (u1, . . . ,un)T ∈ Rn×q, we need the partials of Ak(x, �u, �u) withrespect to x ∈ Rn, �u,ul ∈ Rq, evaluated in (x, �u,∇�u) and, e.g. applied to �ϑk, �ϑl ∈ Rq,so the (∂Ak/∂ul)(x, �u,∇�u)�ϑl ∈ Rq or

((∂Ak/∂ul)(x, �u,∇�u)�ϑk, �ϑl

)q∈ R, or ∂Ak/∂u =

(∂Ak/∂u1, . . . , ∂Ak/∂un). We do not distinguish Ak : Ω× Rq × Rn×q → Rq, and Ak :H1(Ω,Rq) → H−1(Ω,Rq), and omit the details in the norms, e.g. write ‖∂Ak/∂ul‖,instead of ‖∂Ak/∂ul‖Rq←↩Rq2 .

(a) Ak : w := (x, �u, �u) ∈ Ω× Rq × Rn×q → Rq, k = 0, 1, . . . , n, (7.94)

(b) Ak : �u ∈ D(G) ⊂ D(Ak) ⊂ H1(Ω,Rq) → Ak(�u) := Ak(·, �u,∇�u) ∈ H−1(Ω,Rq),

(c) ‖Ak‖, ‖∂Ak/∂x‖, ‖∂Ak/∂�u‖ ≤ L(1 + ‖�u‖), ‖∂Ak/∂�u‖ ≤ L ∀k > 0,

(d) ‖A0‖, ‖∂A0/∂x‖, ‖∂A0/∂�u‖ ≤ L(1 + ‖�u‖2), ‖∂A0/∂�u‖ ≤ L(1 + ‖�u‖),∀‖�u‖ ≤M, all terms in c), d) are evaluated in w = (x, �u, �u),

(e) Ellipticity: For �u ∈ H1(Ω,Rq), with ‖�u− �u0‖H1(Ω,Rq) ≤ ρ ∃λ,Λ ∈ R ∀x ∈ Ω,

0 < λ‖�ϑ‖2≤n∑

k,l=1

(∂Ak

∂ul(x, �u,∇�u)�ϑk, �ϑl

)q

< Λ‖�ϑ‖2∀0 �= �ϑ = (�ϑ1, . . . , �ϑn) ∈ Rn×q,

(f) �uD is the trace of some �u∗ ∈ U on ∂Ω.

7.5.4 Discretization of the quasilinear systems

Now we start formulating the DCGMs for these divergent quasilinear equations ofsecond order in (7.93), cf. the results for conforming FEMs in Theorem 5.81. We


choose d ≥ 2, generalizing (7.63),

Uh = Vh = Sh := Sd−1,−1q (T h) := {v : T h → Rq; v|K ∈ Pd−1(K) ∀K ∈ T h}. (7.95)

By the standard multiplication of (7.93) with �vh, partial integration over K, andsummation, we relate the strong and weak discrete quasilinear forms as,h(·, ·), andah(·, ·) and the final weak discrete form, ah(·, ·), in (7.99). We define these strongand weak forms for

∀�uh, �vh ∈ Ue with �uh ∈ Us,e, �vh ∈ L2(Ω,Rq) for as,h(�uh, �vh), (7.96)

as,h(�uh, �vh) := (Gs,h(�uh), �vh) :=∑

K∈T h

∫K

n∑k=0

((−1)k>0∂

kAk(x, �uh,∇�uh), �vh)qdx,

ah(�uh, �vh) := 〈Gh(�uh), �vh〉 :=∑

K∈T h

∫K

n∑k=0

(Ak(x, �uh,∇�uh), ∂k�vh)qdx

where we use the abbreviations (Gs,h(�uh), �vh) = (Gs,h(�uh), �vh)L2(Ω,Rq), 〈Gh(�uh), �vh〉 =〈Gh(�uh), �vh〉V′

e×Ve. Since the summation in (7.96) starts with k = 0 in

∑K∈T h

∫K, we

choose the above notation for ah(�uh, �vh) in contrast to the semilinear case. This yieldsthe relation for

∀�uh ∈ Us,e,∀�vh ∈ Ue with BA(�uh,∇�uh) :=(Ak(x, �uh,∇�uh)

)nk=1

∈ Rn×q, (7.97)

as,h(�uh, �vh) = ah(�uh, �vh)−∑

e∈T h\∂Ω

∫e

((({BA(�uh,∇�uh)}, ν)n, [�vh]

)q

+(([BA(�uh,∇�uh)], ν)n, {�vh}

)q

)dS −

∑e∈∂Ω

∫e

((BA(�uh,∇�uh), ν)n, �v

h)qdS.

cf. (5.286). All the previous face integrals∫

edisappear for �uh = �u ∈ H3/2+ε(Ω,Rq),

�vh = �v ∈ H10 (Ω,Rq) implying for the exact solution Gs,h(�u0) = 0 and Gh(�u0) = 0.

We proceed as from (7.83)–(7.88): we replace Ba(u)∇u by BA(�u,∇�u),(Ba(uh)∇uh, ν)nv

h by((BA(�uh,∇�uh), ν)n, �v

h)q

etc. We omit the flux terms,and

∫e

({�v}, ([BA(�u0,∇�u0)], ν)n

)q, vanishing in (7.97) for �uh = �u0, and add the

penalty terms in (7.84). We only consider the IIPG with θ = 0, otherwiseθ(({BA(�vh,∇�vh)}, ν)n, [�uh]

)q

would cause a nonlinearity with respect to �vh. Further-more, we collect −∑e∈T h\∂Ω and −∑e∈∂Ω into −∑e∈T h

∫e. So (7.87), (7.88), is

generalized, with �uh, �vh as in (7.96), as∑K∈Th

∫K

n∑k=0

(Ak(x, �uh,∇�uh), ∂k�vh)q dx−∑

e∈T h

∫e

(({BA(�uh,∇�uh)}, ν)n, [�vh]

)qdS

+

⎛⎝∑e∈T h

∫e

σ([�u], [�v] )qdS −∑

e∈∂Ω

∫e

σ(�uD, �v )qdS

⎞⎠ = 0. (7.98)


Again the exact solution �u = �u0 ∈ H3/2+ε(Ω,Rq), of (7.93), (7.91), with the testfunctions �v ∈ H1

0 (Ω,Rq) satisfy (7.98).With this notation, we introduce, similarly to (7.57)–(7.61), the quasilinear, linear

and bilinear forms, ah(·, ·), �h(·), and Jσh (�u,�v), cf. (7.97), and rewrite (7.98), cf. (7.96),

as

ah(�uh, �vh) := ah(�uh, �vh)−∑

e∈T h

∫e

(({BA(�uh,∇�uh)}, ν)n, [�vh]

)qdS, (7.99)

�h(�u) :=∑

e∈∂Ω

∫e

σ(�uD, �v)q dS and Jσh (�u,�v) :=

∑e∈T h

∫e

σ([�u], [�v])q dS. (7.100)

Again analogously to (7.66) we define G,Gh, Gh, by replacing those ah(u, v),

ch(u, v), bh(u, v), �h(v), Jσh (u, v) in (7.58), (7.59), (7.61), by the new terms

ah(�u,�v), �h(�v), Jσh (�u,�v), ch(�u,�v) = bh(�u,�v) = 0,

in (7.99), (7.100). Instead of explicitly updating (7.66), we refer to (7.89) for Gh.We only consider the IIPG form, so θ = 0, of the DCGM for the quasilinear problem

(7.93), (7.91). Its discrete approximate solution �uh0 : Ω → Rq is defined as

(a) �uh0 ∈ Sh, cf. (7.95), (7.101)

(b)⟨Gh(�uh

0

), �vh⟩

:= ah(�uh

0 , �vh)

+ Jσh

(�uh

0 , �vh)− �h(�uh, �vh) = 0 ∀�vh ∈ Sh

with ah(·, ·) := ah(·, ·)|Sh×Sh , and �h(·) := �h(·)|Sh .

A direct generalization of the NIPG, SIPG, GIPG form of the DCGM would yieldproblematic forms, nonlinear in v. Possible linearizations seem to be pretty artificial.

The quasilinear problem and their DCGMs, studied by Houston et al. [407] somehowrepresent a combination of our approaches for semilinear and quasilinear problems.As for semilinear problems they consider all types of DCGM with θ = −1,= 0,= 1,∈[−1, 1], but for a quasilinear equation of the special form

G(u) := −∇ · (μ(x, |∇u|)∇u) := −n∑

i=1

∂i(μ(x, |∇u|)∂iu

)= g(x, u). (7.102)

Here μ ∈ C(Ω× [0,∞)) is a real function satisfying the assumption, with mμ < Mμ ∈R+,

mμ(t− s) ≤ μ(x, t)t− μ(x, s)s ≤Mμ(t− s), t ≥ s ≥ 0, x ∈ Ω (7.103)

and for t > s = 0 =⇒ 0 < mμ ≤ μ(x, t) ≤Mμ.

In addition, we obtain μ(·, |∇u(·)|) ∈ L∞(Ω) for u ∈W 1,∞(Ω). Garret and Liu [338]proved that (7.103) implies, for symmetric matrices, X,Y = (Yij)n

i,j=1 ∈ Rn2, or for

x, y ∈ R,

|μ(x, |Y |)Y − μ(x, |X|)X| ≤ C1|Y −X|, and (7.104)

C2|Y −X|2 ≤(μ(x, |Y |)Y − μ(x, |X|)X, (Y −X)

)n2 ,


with |Y | the Euclidean (Frobenius) scalar product and norm, hence |Y | =(∑n

i,j=1 =

Y 2ij

)1/2. This includes, via diagonal matrices, Y = (Y1, . . . , Yn) ∈ Rn.

This can be used in two ways: linearization of the weak form of G(u) yields

(G′(u0)w, v) =n∑

i=1

∫Ω

μ(x, |∇u0|)∂iw∂ivdx (7.105)

+n∑

i,j=1

∫Ω

μt(x, |∇u0|)∂ju0|∇u0|−1/2∂jw∂iv, with μt(x, t) := (∂μ/∂t)(x, t),

so for |μt(x, |∇u0|)||∇u0|1/2 small compared to mμ, a coercive principal part of thelinearized operator and hence stability of the DCGM as in Theorem 7.18.

Or Houston et al. [407] use (7.103) for the discrete form of the weak Gh, cf. (7.109).They prove that Gh is Lipschitz-continuous and strongly monotone with respectto uh.

Compared to (7.78), here we have chosen ε = 1. Then the lower order flux terms∑ns=1 ∂

sfs(u0) are no longer dominant. Hence we omit them here. For g(x, u) weimpose the conditions in (7.80) (g), (h).

In this special quasilinear equation in (7.102), cf. [407], in (7.78), the second orderterm is replaced by ∇ · (μ(x, |∇u|)∇u). Hence we obtain, cf. (7.3), (7.23),

a(u, v) =∫

Ω

n∑i=1

μ(x, |∇u|)∂iu∂iv dx =∫

Ω

(Ba(u)∇u,∇v)ndx =∫

Ω

g(x, u)vdx

(7.106)

Ba(u)∇u := μ(x, |∇u|)∇u, (Ba(u)∇u,∇v)n = μ(x, |∇u|)n∑

i,=1

(∂iu ∂iv

)∀u, v ∈ U ,

and determine the weak solution u0 ∈ U : a(u0, v) =∫Ωg(x, u0)vdx ∀v ∈ U .

The DCGMs for (7.102) is similar to (7.89). We choose ε = 1 and replace theah(uh, vh) there by (7.106). We avoided, in (7.99), the θ terms

θ∑

e∈T h

∫e

({Ba(u)∇v}, ν)n [u] dS = θ∑

e∈T h

∫e

({μ(x, |∇u|)∇v}, ν)n [u] dS

by choosing θ = 0. To obtain a monotone operator, Houstan et al. [407] employ insteadthe form

θ∑

e∈T h\∂Ω

∫e

({μ(x, h−1|[u]|)∇v}, ν)n [u]dS

+ θ∑

e∈∂Ω

∫e

({μ(x, h−1|u− uD|)∇v}, ν)n (u− uD)dS.


Correspondingly, we reformulate ah(u, v), cf. (7.87), for −1 ≤ θ ≤ 1 as

ah(u, v) :=∑

K∈T h

∫K

(Ba(u)∇u, ∇v)n dx−∑

e∈T h

∫e

(({Ba(u)∇u}, ν)n [v]

)dS

+ θ∑

e∈T h\∂Ω

∫e

(({μ(·, h−1|[u]|)∇v}, ν)n [u]

)dS. (7.107)

Similarly, we modify �h (7.60) and use the Jσh (7.61):

�h(u, v) := (g(·, u), v)L2(Ω) − θ∑

e∈∂Ω

∫e

({μ(x, h−1|u− uD|)∇v}, ν)n (u− uD)dS

+∑

e∈∂ΩD

∫e

σuD v dS (7.108)

Finally, we determine the discrete solution uh0 ∈ Sh, such that⟨

Gh(uh

0

), vh⟩

:= ah(uh

0 , vh)

+ Jσh

(uh

0 , vh)− �h

(uh

0 , vh)

= 0 ∀ vh ∈ Sh. (7.109)

As for equations the generalization to systems is considered by Houston et al. [407].Again either the monotony or the stability of the linearization can be used. Thenthe uh

0 , vh, μ(x, |∇u|) in (7.109) or additionally in (7.105) with �v = (v1, . . . , vn) are

replaced for systems by

�uh0 , �v

h, μ(x, |e(�u)|), e(�u) :=12(∇�u + (∇�u)T

)or e(�u) := (∂1v1, . . . , ∂nvn) (7.110)

with Ba(�v)e(�v) := μ(x, |e(�u)|)e(�v) and finally,⟨Gh(�uh

0

), �vh⟩

:= ah(�uh

0 , �vh)

+ Jσh

(�uh

0 , �vh)− �h

(�uh

0 , �vh)

= 0 ∀�vh ∈ Sh. (7.111)

Again (7.103), (7.104) implies, cf. Houston et al. [407], Lemmas 2.2 and 2.3, thatGh is Lipschitz-continuous and strongly monotone with respect to �uh.

7.6 DCGMs are general discretization methods

For these methods, stability and consistency implying convergence play a crucial role.For realizing this interpretation, we relate the original and discrete function spaceswith the corresponding operators. We only distinguish the spaces H3/2+ε(Ω) andH3/2+ε(Ω,Rq), . . ., where it is necessary, e.g. in the consistency results for ah(�uh, �vh)below, in Theorem 7.33. Most of the other results are valid componentwise, withobvious modifications, for both cases, e.g. Theorems 7.5–7.19.

Directly comparing G and Gh, similarly A and Ah, required a detour via the Gh.This has been shown for the model problem in (7.13)–(7.15), (7.66), and indicatedfor the other problems as well. We recall the situation here for nonlinear operators.Previously we studied U = V,Uh = Vh but will allow, for possible generalizations, in

7.6. DCGMs are general discretization methods 483

this section, U �= V, or Uh �= Vh, as well. So we start with the original

G : D(G) ⊂ U , e.g., U = H1(Ω) → V ′, e.g. V ′ = H−1(Ω), extended as (7.112)

Gh : D(Gh) ⊂ Uoe , e.g., = H3/2+ε(Ω) → (Vo

e )′ , e.g. = H−3/2−ε(Ω), extended as

Gh : D(Gh) ⊂ Ue, e.g., = H3/2+ε(T h) → V ′e, e.g. = H−3/2−ε(T h), restricted as

Gh : D(Gh) ⊂ Uh, e.g., Uh = Sh → V ′h, e.g. V ′h = S′h, in (7.63), (7.95).

For a nonlinear G the restriction to D(G) ⊂ U is important. This D(G) has tobe chosen such that G(u) is well defined and has the desired properties, e.g. listedin (7.16), (7.51) ff. for (7.13), (7.14), or (7.80) for (7.78), (7.79), or (7.94) for (7.90),(7.91). Other possibilities are the conditions imposed for nonlinear operators and theirlinearization discussed in Chapter 2, here Sections 2.5.3 ff. and 2.7.2.

We combine U ,V with the discrete spaces Uh,Vh, e.g. Sh via the operators Ph :=Ih ∈ L(U ,Uh), and Q

′h ∈ L(V ′,V ′h). We may choose the Ph relatively arbitrarily, herethe approximation operator Ph introduced in Theorem 7.7. The Ph, Q

′h have to becombined such that we obtain optimal consistency order. In particular, the Q

′h hasto be chosen to fit the above variational method. Similarly to (7.112), we need localpairs of Ph

e , Q′he . The difference is the piecewise definition, with Ph

e the local form onthe K ∈ T h in Theorem 7.7. For Q

′he , we have to include the integrals along edges and

boundaries. Correspondingly, see (5.294),(5.295), we define Q′h, Q

′he . We start with

the local version Q′he and return to Q

′h in (7.115).

Q′he ∈ L

(V ′

e,V′h)

: Q′he fh − fh ⊥ Vh ⇔

⟨Q

′he fh − fh, v

h⟩

= 0 ∀vh ∈ Vh (7.113)

with 〈·, ·〉 = 〈·, ·〉V′e×Ve

. The general form of these fh is, cf. (2.107):

for f−1 ∈ L2(Ω,Rn), f0 ∈ L2(Ω,R), fe−1, f

e0 ∈ L2(∂T h,R), define 〈fh, v〉V′

e×Ve:=

〈fh, v〉 :=∑

K∈T h

∫K

((f−1,∇ v)n + f0v) dx +∑

e∈T h

∫e

(fe−1 (∇v, ν)n + fe

0 v)dS, (7.114)

fh ∈ V ′e, ∀v ∈ Ve ⊃ V + Vo

e + Vh, with ∂T h := {e ∈ T h}, f = fh|V , fh := fh|Vh .

Examples are the previous cases, fh := Ahuh, uh ∈ Ue = H3/2(T h), and fh :=

Ahu, u ∈ Uoe , induced by the ah(uh, vh) in (7.57), (7.76), (7.87), (7.99). In contrast

to these ah(·, ·), in the original a(·, ·) = ah(·, ·)|U×V , in (7.19), (7.74), (7.81), the∫

eare

missing. The DCGMs for the previous problems have shown that it is impossible: nei-ther evaluating Auh for a uh ∈ Uh, nor testing it nor the corresponding f := Au, u ∈ U ,with vh ∈ Vh ⊂ Ve. So we define, for A and G,

(Q′hA)

∣∣∣Uh := Q′he Ah

∣∣∣Uh

, ΦhA := (Q′hA)

∣∣∣Uh := Q′he Ah

∣∣∣Uh

, and (7.115)

(Q′hG)

∣∣∣Uh := Q′he Gh

∣∣∣Uh

, ΦhG := (Q′hG)

∣∣∣Uh := Q′he Gh

∣∣∣Uh

,


G :A,GU → → → → V ′ tested by

←→ V

Ueo Ah,Gh

Ah,Gh

Ah,Gh

(Veo tested by

←→ Vgo

Φh Ph Q′h

Ue etested by←→ Ve

↓ Peh Q′h ↓

Φh G=Gh Uh → → → → V ′h tested by←→ Vh

)′

V ′

e

Figure 7.2 Original and discrete spaces and operators in DCGMs.

cf. Figure 7.2. The combination of Phe ↙ and Q

′he ↘ in this figure indicate the ranges

Phe Ue = Uh, Q

′he V ′

e = V ′h. Note that by (7.115), the Q′h ∈ L(V ′,V ′h) are only defined

via Q′he ∈ L(V ′

e,V′h) in (7.113),(7.115), including the

∫e.

The uniformly bounded Ph, Q′h and Ph

e , Q′he have the properties required for the

general discretization theory, cf. Definition 3.12, (3.24), Theorems 7.7, Corollary 7.8:

for d ≥ 2 : limh→0

Phu = u ∀ u ∈ U , limh→0

‖Q′hf‖hV′h = ‖f‖V′∀f ∈ V ′,

(7.116)and : lim

h→0Ph

e uh = uh ∀ uh ∈ Ue, limh→0

∥∥Q′he fh

∥∥h

V′h = ‖fh‖V′e∀fh ∈ V ′

e,

with the local degree d− 1, cf. (7.63), (7.95). Thus, the Uh are approximating spacesfor U ,Ue, and their subspaces, similarly for Vh,V,Ve.

Summarizing, we have assigned to T h with h, 0 < h ∈ H ⊂ R, infh∈H h = 0, thesequence of spaces {Uh,Vh}h, 0<h∈H of the same finite dimension. Often we use theshort notation Uh,Vh. Thus we have defined by (7.70), (7.71) and the correspondinggeneralizations (7.77), (7.89), (7.101), an operator Φh transforming the original G intothe discrete Gh = Φh(G), similarly the linear Ah = Φh(A), cf. (7.115),

ΦhG := Gh := (Q′hG)|Uh =

(Q

′he Gh

)|Uh , and ΦhA := Ah := Q

′he Ah|Uh . (7.117)

In contrast to (5.2), we enforce the continuity and boundary conditions in Ub viapenalty terms and penalty norms for Uh, cf. Theorems 7.12, 7.13.

This procedure defines a mapping, Φh, applied to linear or nonlinear operators, Φh :A ∈ L(U ,V ′) → L(Uh,V ′h) or Φh : G ∈ NL(D(G),V ′) → NL(D(Gh),V ′h), accordingto the chosen DCGM. It allows reinterpreting (7.70), (7.71), (7.89), (7.101):

We determine the approximate solution uh0 ∈ Vh as

Ghuh0 = Q

′he Ghu

h0 = 0 ⇔

⟨Ghu

h0 , v

h⟩

= 0 ∀vh ∈ Vh ⇔ Gh(uh

0

)⊥ Vh. (7.118)

Next, we study the relations between operators and their discretizations. Here thepenalty norms for Ue, mentioned before, will play an essential role. In particular, wewill prove stability and consistency.

7.6. DCGMs are general discretization methods 485

Stability is a kind of perturbation insensitivity, cf. Definition 3.19. For a nonlinearoperator Gh : D(Gh) ⊆ Uh → V ′h, and uh ∈ D(Gh) we assume

Br(uh) = {vh ∈ Uh : ‖vh − uh‖Uh < r} ⊂ D(Gh) ∀h < h0, h ∈ H.

Furthermore, let h0, r, S ∈ R+ be fixed constants, such that, uniformly in h ∈ H,h < h0:

uhi ∈ Br(uh), i = 1, 2,=⇒

∥∥uh1 − uh

2

∥∥Uh ≤ S‖Gh

(uh

1

)−Gh

(uh

2

)‖V′h . (7.119)

Then Gh are called stable in uh. For a linear Ah or for Ah − fh this reduces to(Ah)−1 ∈ L(V ′h,Uh) exists and ||(Ah)−1||Uh←V′h ≤ S.

Stability is proved using Theorems 3.23 and 3.29 in Chapter 3. It suffices to studystability for the linearized operator. It is here always a compact perturbation of acoercive operator, the latter defining a stable discretization. We assumed the linearizedoperator to be boundedly invertible. Then the stability of the linearized operatorfollows as in Section 5.5 by proving the coercivity for a perturbed principal part.This follows with the penalty norm from our Theorems 7.16–7.19. Combined with anearly verbatim copy of the arguments in Lemma 5.77 and Theorem 3.29, it yieldsstability for (G′(u0))h with respect to the penalty norm ‖v‖, cf. Theorem 7.12, (7.64),(7.153), for general boundedly invertible linearizations, G′(u0). This is left as anexercise for the interested reader. We will summarize the exact conditions and results inTheorem 7.40.

The classical consistency error has to converge to 0. It has, for a linear or nonlinearoperator, A or G, the following general form, cf. (5.313), (3.34), Chapter 3, (3.33).These GhPhu− (Q

′hG)u, cf. (7.117), have to be tested by vh ∈ Sh:

AhPhu− (Q′hA)u and GhPhu− (Q

′hG)u tested by 0 �= vh ∈ Sh yield (7.120)

|〈GhPhu− (Q′hG)u, vh〉|

‖vh‖ → 0 or ≤ C(u)hk with C(u) ∈ R+, k = k(u) ∈ N;

these C(u), k = k(u) depend on the regularity of u and the degree d− 1 of the chosenpolynomials in Vh, cf Theorem 7.40.

The classical consistency error can be estimated by combining (7.71) or the corre-sponding generalizations (7.77), (7.89), (7.101), with the Lipschitz-continuity of G andGh and the penalty norm, cf. Theorem 7.21 ff. The combination of (7.71) with Q

′h in(7.113) yields for our first example (7.23) for any u ∈ U and vh ∈ Sh, cf. (7.118),

〈GhPhu−Ghu, vh〉 =

(ah(Phu, vh) + bh(Phu, vh) + εJσ

h (Phu, vh) (7.121)

−�h(Phu, vh))−(ah(u, vh) + bh(u, vh) + εJσ

h (u, vh)

−�h(u, vh))∀ vh ∈ Sh.

For our next examples (7.74), (7.78), and (7.93) we simply replace ah(Phu, vh),�h(Phu, vh) in (7.57), (7.60) by the updated forms in (7.76), (7.87), (7.88), and in(7.99), (7.100). The bh(Phu, vh) is skipped in (7.74), (7.93), (7.99). All these differences


of corresponding pairs can be estimated by combining the interpolation errors inTheorem 7.7 and Corollary 7.8 on the K and ∂K with specific properties of theah(Phu, vh), . . . , �h(Phu, vh) in Theorems 7.21–7.33.

Continuity, consistency plus stability imply the unique existence of the discretesolution and its convergence. For the reader’s convenience we reformulate the essentialTheorem 3.21 in the following theorem. It applies to the original problems, and theirdiscretizations, (7.13), (7.14), (7.70), (7.71), and (7.74), (7.77); (7.78), (7.79), and(7.89); (7.93), and (7.101).

Theorem 7.3. Unique existence and convergence of discrete solutions: Let the origi-nal problem have the exact solution u0. Let Gh : D(Gh) ⊂ Uh → V ′h be its discretiza-tion, cf. (7.118), and Definitions 3.5, 3.6, 3.12 (here nonlinear Φh are allowed) andsatisfy the following conditions

1. Gh = Φh(G) : D(Gh) ⊂ Uh → V ′h is defined and continuous in Br(Phu0) ⊂D(Gh) with r > 0 independent of h;

2. Gh is (classically) consistent with G in Phu0, in the sense of (7.120);3. Gh is stable for Phu0 with respect to ‖ · ‖.

Then the discrete problem Gh(uh) = 0 possesses a unique solution uh0 ∈ D(Gh) ⊂ Uh

near u0 for all sufficiently small h ∈ H and uh0 converges to u0. If Gh is consistent

and consistent of order p, then uh0 converges and converges of order p, hence

‖uh0 − u0‖ ≤ S‖Gh(Phu0)‖V′h and ≤ O(hp), respectively (7.122)

For all the cases considered here, Gh and Gh are differentiable simultaneously withG. They even satisfy consistent differentiability, cf. (5.165), so ∀u, v ∈ Br(u0) ∩ Us:

‖(QhG′(u))v −QhG′(Phu)Phv‖Vh′ ≤ C2hp′

(1 + ‖u‖Us)‖v‖Us

, (7.123)

with a “smooth” norm ‖v‖Us. This is important for the mesh independence princi-

ple. It guarantees the quadratic convergence of the Newton method for solving thecorresponding discrete nonlinear systems, essentially independent of the mesh size,cf. Section 7.13.

So the program for the next sections is the proof for stability, consistency, andconsistent differentiability for our preceding problems.

7.7 Geometry of the mesh, error and inverse estimates

In this section we shall be concerned with the necessary approximation results. Theadvantage of the approach in this section compared to the usual FE approach isthe avoided condition of quasiuniform subdivisions. For FEMs this is possible bymodifying the results for boundary integral methods of Graham et al. [362], cf.Chapter 4. For DCGMs some unusual conditions upon the geometry of the subdivisionhave to be imposed. Then a multiplicative trace inequality is combined with inverseestimates.

7.7. Geometry of the mesh, error and inverse estimates 487

7.7.1 Geometry of the mesh

Let us consider a system {T h}h∈(0,h0), h0 > 0, of partitions of the domain Ω, i.e.T h = {K ∈ T h}. We assume throughout the chapter that the system {T h}h∈(0,h0) hasthe following properties:

(A0) T h is a covering partition of the domain Ω = ∪K∈T hK.(A1) Each element K ∈ T h, h ∈ (0, h0), is a star-shaped domain with respect to at

least one interior point xK = (xK1 , . . . , xKn) of K. We assume that

(i) there exists a constant κ > 0 independent of K and h such that

maxx∈∂K |x− xK |minx∈∂K |x− xK |

≤ κ ∀K ∈ T h, h ∈ (0, h0); (7.124)

(ii) each K can be divided into a finite number of closed simplices:

K =⋃

S∈S(K)

S ; (7.125)

there exist positive constants C3, κ independent of K, S and h such that

hS

ρS≤ C3 ∀S ∈ S(K) (shape regularity) (7.126)

where hS is the diameter of S, ρS is the radius of the largest n-dimensionalball inscribed into S and, moreover,

1 ≤ hK

hS≤ κ <∞ ∀S ∈ S(K). (7.127)

(A2) There exists a constant C4 > 0 such that

hK ≤ C4 d(e) ∀K ∈ T h, e ⊂ ∂K, h ∈ (0, h0). (7.128)

Remark 7.4. The conditions (A1) and (A2) imply a bound on the maximum numberof edges, i.e. there exists a constant ce such that #(e ∈ K) ≤ ce for all K ∈ T h.Moreover, it follows that the number of hanging nodes for each element is bounded.

Let us note that these properties can be verified, e.g. in the case of dual finite volumesconstructed over a regular simplicial mesh. Furthermore, nondegenerate subdivisionsin FEMs satisfy these conditions as well, see [314]. Figure 7.1 satisfies (A0)–(A2), it isstar-shaped and the K are nonconvex. Figure 7.3 shows a type of mesh used in [314].For practical computation these grids are not very suitable.

We cite some technical results formulated as Theorems 7.5–7.7.

7.7.2 Inverse and interpolation error estimates

Under the above assumptions, (A1), (A2) the following results can be established.


–0.4

–0.2

0

0.2

0.4

–0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4

‘dmesh’

Figure 7.3 Dual mesh.

Theorem 7.5. Multiplicative trace inequality: Assume (A1), (A2). Then there existsa constant C5 > 0 independent of v, h and K such that

‖v‖2L2(∂K) ≤ C5

(‖v‖L2(K) |v|H1(K) + h−1

K ‖v‖2L2(K)

), (7.129)

K ∈ T h, v ∈ H1(K), h ∈ (0, h0).

Proof. Using the approach from [296] and [297], we prove the theorem in two steps.(i) Let K ∈ Th and S ∈ S(K) be arbitrary but fixed. We denote by xS the center of

the largest d-dimensional ball inscribed into S. Without loss of generality we supposethat xS is the origin of the coordinate system. We start from the following relation, aconsequence, e.g. of (2.9),∫

∂S

v2x · ν dS =∫

S

∇ · (v2x) dx, v ∈ H1(S). (7.130)

Let νe be the unit outer normal to S on the face e ⊂ ∂S. Then

x · νe = |x||νe| cosα = |x| cosα = ρS x ∈ e, (7.131)

since ρS is the distance of xS from the face e, see Figure 7.4.From (7.131) we have∫

∂S

v2x · ν dS =∑

e⊂∂S

∫e

v2x · νij dS (7.132)

=∑

e⊂∂S

te

∫e

v2 dS ≥ ρS

∑e⊂∂S

∫e

v2 dS = ρS‖v‖2L2(∂S).

7.7. Geometry of the mesh, error and inverse estimates 489

xS

S

ax

ne

eρS

Figure 7.4 Simplex S with its face e and the distances te.

Moreover,

∫S

∇ · (v2x) dx =∫

S

(v2∇ · x + x · ∇v2

)dx (7.133)

= d

∫S

v2 dx + 2∫

S

vx · ∇v dx ≤ d‖v‖2L2(S) + 2∫

S

|vx · ∇v| dx.

With the aid of the Cauchy inequality the second term of (7.133) is estimated as

2∫

S

|vx · ∇v| dx ≤ 2 supx∈S

|x|∫

S

|v||∇v| dx ≤ 2hS‖v‖L2(S)|v|H1(S). (7.134)

Then (7.130), (7.132), (7.133) and (7.134) give

‖v‖2L2(∂S) ≤1ρS

[2hS‖v‖L2(S)|v|H1(S) + d‖v‖2L2(S)

](7.135)

≤ C3

[2‖v‖L2(S)|v|H1(S) +

d

hS‖v‖2L2(S)

],

where C3 is the constant given by (7.126) and is independent of S, K, v and h.(ii) Let K ∈ Th and S ∈ S(K). Then (7.135) yields

‖v‖2L2(∂S) ≤ C(‖v‖L2(S) |v|H1(S) + h−1

S ‖v‖2L2(S)

), (7.136)

S ∈ S(K), v ∈ H1(K),


where C = C3 max{2, d}. Using the inclusion ∂K ⊂ ⋃S∈S(K) ∂S, relations (7.136) and(7.127) and the Cauchy inequality, we find that

‖v‖2L2(∂K) ≤∑

S∈S(K)

‖v‖2L2(∂S)

≤ C

⎧⎨⎩ ∑S∈S(K)

(‖v‖L2(S) |v|H1(S) + h−1

S ‖v‖2L2(S)

)⎫⎬⎭ (7.137)

≤ C

{( ∑S∈S(K)

‖v‖2L2(S)

)1/2( ∑S∈S(K)

|v|2H1(S)

)1/2

+ κh−1K ‖v‖2L2(K)

}

= CM

(‖v‖L2(K) |v|H1(K) + h−1

K ‖v‖2L2(K)

),

where CM = κC = κC3 max{2, d}. �

Theorem 7.6. Inverse inequality: Under the conditions (A1), (A2), there exists aconstant C6 = C6(d− 1) > 0 independent of v, h and K such that

|v|H1(K) ≤ C6h−1K ‖v‖L2(K), v ∈ Pd−1(K), K ∈ T h, h ∈ (0, h0). (7.138)

The proof of Theorem 4.35 remains valid for Theorem 7.6 as well. According toSchwab [575], the constant C6 is O((d− 1)2).

Important are the approximation properties of the space Sh constructed over themesh T h. The following theorem is for the FE situation, a consequence of Definition4.8 and Theorem 4.17; for the DCG situation it is proved by Verfurth [656].

Theorem 7.7. Error estimates: Assume (A1), (A2). Then there exists a constantC7 > 0 independent of v and h and a mapping Ph : H1(K) → Pd−1(K) such that∀ k = 0, 1, 2, v ∈ Hs(T h), K ∈ T h, h ∈ (0, h0),

|Phv − v|Hk(K) ≤ C7hmin{d,s}−kK |v|Hmin{d,s}(K), with | · |H0(K) = | · |L2(K)

|Phv − v|Hk(T h) ≤ C7

⎛⎝ ∑K∈T h

(h

min{d,s}−kK |v|Hmin{d,s}(K)

)2

⎞⎠1/2

(7.139)

By Veeser and Verfurth [652], and Thrun [628], (7.139) remain correct, if theHk,Hmin{d,s} are replaced by W k,∞,Wmin{d,s},∞.

If in K, polynomials of degree dK are employed we replace in Theorem 7.21 ff.

hmin{d,s}|u|Hmin{d,s}(Ω) by h{min d,min s}

⎛⎝ ∑K∈T h

h2(dK−d)K |u|2

Hmin{dK ,sK}(K)

⎞⎠1/2

.

(7.140)

7.8. Penalty norms and consistency of the Jσh 491

The precise conditions for these modified results, used in adaptive and hp- DCGMs,are described in Section 7.14.

Proof. According to [656], we denote by

πKϕ ≡ 1|K|

∫K

ϕdx (7.141)

the mean value of any integrable function ϕ on K. For any v ∈ Hs(K) we recursivelydefine polynomials pd−1,K(v), pd−2,K(v), . . . , p0,K(v) in Pd−1(K) by

pd−1,K(v) ≡∑

α∈Nd−1|α|=d−1

1(d− 1)!

xαπK(Dαv) (7.142)

and, for k = d− 1, d− 2, . . . , 1

pk−1,K(v) ≡ pk,K(v) +∑

α∈Nd−1|α|=k−1

1(d− 1)!

xαπK (Dα(v − pk,K(u))) . (7.143)

Finally, we set

Phv ≡ p0,K(v). (7.144)

For the proof of the approximation properties (7.139) we refer to [656]. �

As a consequence of the Trace Theorem 1.38 we obtain

Corollary 7.8. There exist a constant C8 > 0 independent of v and h0 such that

|Phv − v|Hk(∂K) ≤ ‖Phv − v‖Hk(∂K) ≤ C7hmin{d,s}−k−1/2K |v|Hmin{d,s}(K), (7.145)

for all v ∈ Hs(K), K ∈ T h, 0 ≤ k ≤ s, and h ∈ (0, h0), h0 < 1.

7.8 Penalty norms and consistency of the Jσh

The need for the penalty norms has been mentioned several times before. Their formhere differs from the energy norms in Chapters 2 and 4. So we maintain the notationof penalty norms for the DCGMs here.

The coercivity, boundedness and even more so the consistency properties in thenext two sections are not possible with respect to broken Sobolev norms, but onlywith respect to these penalty norms. For the following considerations we assume, cf.(7.39),

σmin := mine∈T h

{σ|e = 1/d(e)} ≥ 1. (7.146)

The consistency equation (7.121) and the estimates in the following theorems showthat a successful convergence approach requires carefully monitoring the Jσ

h (uh, vh).Beginning with Theorem 7.16, a naive approach to the following coercivity and


consistency estimates would yield unbounded results with respect to broken Sobolevnorms, ‖v‖H1(Th). This is caused by the Jσ

h (vh, vh), strongly growing for arbitraryvh and increasingly finer meshes, T h. We compare, e.g. (7.121) with the con-sistency results for FEMs with boundary crimes in (5.348). This motivates, e.g.for the ah(uh, vh) in (7.57), nonlinear in uh, but linear and bounded in vh, acondition

|ah(Phu, vh)− ah(u, vh)| ≤ C(u)dist(u, Sh)‖vh‖?, (7.147)

with appropriate norms in dist(u, Sh) and ‖vh‖? and C(u) usually depending upon u.dist(u, Sh) is well estimated by the interpolation errors in Theorem 7.7, Corollary 7.8.However it neglects the violated continuity and boundary conditions. For Vh = Sh

the ‖vh‖? will be chosen as a combination of the standard broken Sobolev norm andJσ

h (vh, vh), allowing the necessary (7.147).In our general discretization approach we are mainly interested in those uh, vh ∈ Ue,

approximating the elements u, v in a bounded subset of H3/2+ε(Ω). By Proposition 7.9and Theorem 7.12, for smooth v and the interesting vh ≈ Phv ∈ Sh or Ue the above‖vh‖? is very close to a standard broken Sobolev norm.

Proposition 7.9. Essential properties of Jσh : Choose a bounded B ⊂ Hs(Ω), with

s ≥ 1, and the local polynomial degree, d− 1 > 0. Then the Jσh (·, ·) : B × B → R is

continuous. With eT h = maxK∈T h{#(e ∈ K)}, d = minK∈T h{dK}, a constant C14 :=22eT hcwC4C

27 > 0 exists, such that ∀u, v ∈ B∣∣Jσ

h (Phu− u, Phv − v)∣∣ ≤ C14h

2 min{d,s}−2|u|Hmin{d,s}(Ω)|v|Hmin{d,s}(Ω). (7.148)

Proof. Corollary 7.8, the Cauchy–Schwarz inequality, and (7.128) imply for e ⊂ ∂K∫e

σ([u− Phu] [v − Phv]dS ≤ σ|e22‖u− Phu‖L2(∂K)‖v − Phv‖L2(∂K) (7.149)

≤ σ|e(2C7h

min{d,s}−1/2K

)2

|u|Hmin{d,s}(K)|v|Hmin{d,s}(K)

≤ cwC422C27h

2 min{d,s}−2K |u|Hmin{d,s}(K)|v|Hmin{d,s}(K).

The final∑

e∈T h , the discrete Cauchy–Schwarz inequality, and a local degree dK = dshow, with C14 = cwC422C2

7 maxK∈T h{#(e ∈ K)},∣∣Jσh (Phu− u, Phv − v)

∣∣ ≤ C14

∑K∈T h

h2 min{dK ,s}−2K |u|Hmin{dK ,s}(K)|v|Hmin{dK ,s}(K)

≤ h2 min{d,s}−2C14

∑K∈T h

|u|Hmin{d,s}(K)|v|Hmin{d,s}(K)

with d = minK∈T h

{dK} (7.150)

hence (7.148) with (7.150). �

7.8. Penalty norms and consistency of the Jσh 493

Theorem 7.10. Consistency estimates for the Jσh : We obtain for u ∈ Hs(Ω), vh ∈

Hs(T h), the local degree, d− 1 > 0, and with C15 := 2√C5cwC7.∣∣Jσ

h (Phu, vh)− Jσh (u, vh)

∣∣ ≤ hmin{d,s}−1C15|u|Hmin{d,s}(Ω)‖vh‖∀h ∈ (0, h0). (7.151)

where Phu is the previous Sh-interpolant of u.

Proof. From (7.61), the Cauchy inequality, (7.149)–(7.150), the multiplicative traceinequality (7.129) and (7.139) (but (7.317) for hp methods) we have∣∣Jσ

h (u, vh)− Jσh (Phu, vh)

∣∣ (7.152)

≤∑

e∈T h

∫e

σ|[u− Phu]| |[vh]| dS

≤ Jσh (vh, vh)1/2

⎛⎝∑e∈T h

∫e

σ[u− Phu]2 dS

⎞⎠1/2

≤ ‖vh‖∑

K∈T h

(cwh

−1K ‖u− Phu‖2L2(∂K)

)1/2

≤√C5cw‖vh‖

⎛⎝ ∑K∈T h

h−1K ‖u− Phu‖L2(K)|u− Phu|H1(K) + h−2

K ‖u− Phu‖2L2(K)

⎞⎠1/2

≤√

2C5cw‖vh‖C2AC

27

×( ∑

K∈Th

(1hK

hmin{d,s}K h

min{d,s}−1K +

1h2

K

h2 min{d,s}K

)|u|2Hmin{d,s}(K)

)1/2

≤ 2√C5cwC7h

2 min{d,s}−1K

( ∑K∈Th

|u|2Hmin{d,s}(K)

)1/2

‖vh‖,

thus (7.151) with C15 := 2√C5cwC7. �

Before we discuss DCGMs, we introduce the announced penalty norm. It penalizeselements v ∈ Hs(T h), violating continuity and boundary conditions.

Definition 7.11. For σ in (7.39), (7.146), we introduce the penalty norms for 1 ≤s ∈ R,

‖v‖JHs(T h) :=

(Jσ

h (v, v) + |v|2Hs(T h)

)1/2,∀v ∈ Hs(T h), and for (7.153)

s = 1 : ‖v‖ := ‖v‖JH1(T h) used as ‖vh‖Vh := ‖vh‖ in Sh, and finally

‖v‖J‖Hs(T h)‖ :=

(Jσ

h (v, v) + ‖v‖2Hs(T h)

)1/2, used e.g. in subsection 7.11.4.


Theorem 7.12. Banach spaces with respect to ‖v‖: For s ≥ 1 and σ in (7.146), weobtain

∃ce > 1 : (ce)−1‖v‖J‖H1(T h)‖ ≤ ‖v‖J

H1(T h) ≤ ce‖v‖J‖H1(T h)‖ ∀v ∈ H1(T h), (7.154)

hence, equivalent norms in Hs(T h), a Banach-space with respect to both norms. Inparticular, we obtain for s = 1, ‖ · ‖ = ‖v‖J

H1(T h). The original ‖ · ‖Hs(T h) and thenew ‖ · ‖J

Hs(T h) are nonequivalent.

Proof. Obviously ‖ · ‖JHs(T h) are norms for Hs(T h), nonequivalent with the original

norm. A Cauchy sequence {vν}∞ν=1 with respect to ‖ · ‖JHs(T h) is Cauchy sequence with

respect to ‖ · ‖Hs(T h) as well, hence has a limit element v0 ∈ Hs(T h) with respect to‖ · ‖Hs(T h). Since Jσ

h is continuous in Hs(T h), this v0 is limit element with respect to‖ · ‖J

Hs(T h) as well, hence Hs(T h) is a Banach space with respect to both norms. Theequivalence of both norms ‖v‖J

Hs(T h) and ‖v‖J‖Hs(T h)‖ is an immediate consequence

of Proposition 1.25. �

Theorem 7.13. Interpolation errors with respect to ‖ · ‖JHk(T h) and the equivalence

of these norms:

1. There exists a constant C16 > 0 independent of v and h, such that

‖Phv − v‖JHk(T h) ≤ C16h

min{d,s}−(1/2+k)|v|Hmin{d,s}(T h), k = 0, 1, 2 ≤ s (7.155)

for all v ∈ Hs(T h) or v ∈ Hs(Ω) and h ∈ (0, h0).2. For a given u ∈ Hs(Ω), d ≥ s ≥ 1, define Bh

u ∈ Sh as

Bhu := {vh ∈ Hs(T h) ∩ Sh : ‖u− vh‖J

Hk(T h) ≤ 1/(4|u|Hs(T h)), k = 0, 1, 2 ≤ s}

with small enough h. Then all the norms ‖ · ‖JHs(T h), and ‖ · ‖Hs(T h) are equiva-

lent in Bhu, and limh→0 ‖Phu‖J

Hs(T h) = ‖u‖Hs(Ω).

Proof. Inequality (7.155) is a consequence of Theorem 7.7, and Corollary 7.8. Claim2 follows from Proposition 7.9 and (7.155). �

7.9 Coercive linearized principal parts

7.9.1 Coercivity of the original linearized principal parts

Existence and uniqueness results for the original previous problems are available bythe Fredholm alternative, if the linearized operators are compact perturbations ofbounded coercive operators, cf. Chapter 2. This is essential for the proof of stabilityas well. So we summarize these results.

The bilinear form for the Laplacian is, cf. (7.19),

c(u, v) := a(u, v) = ε

∫Ω

(∇u, ∇v)n dx for u, v ∈ U , (7.156)

7.9. Coercive linearized principal parts 495

or a(u, v) in (7.74) for the general bilinear form. For the more general forms ofa(u, v), nonlinear in u, in (7.81), and (7.96) ff., the linearized principal parts, the ∂ap/∂u, evaluated in u in the neighborhood of u0 and applied to w, v, are

c(w, v) :=(∂ap

∂u(u))

(w, v) =(G′

p(u)w, v)

= ε

∫Ω

n∑i,j=1

aij(x, u)∂iw ∂jv dx and

(7.157)

c(�w,�v) :=(∂ap

∂�u(�u))

(�w,�v) =(G′

p(�u)�w,�v)q

=∫

Ω

n∑k,l=1

(∂Ak

∂ul(x, �u,∇�u)∂l �w, ∂k�v

)q

(7.158)

for u,w, v ∈ U and �u, �w,�v ∈ H1(Ω,Rq). The corresponding conditions are listed in(7.80) and (7.94), in particular the ellipticity conditions (e).

Theorem 7.14. Coercivity of the original (linearized) principal parts: For a polygonalΩ the bilinear principal forms c(·, ·) in (7.156)–(7.158), and its generalization to(7.74), are bounded and coercive with respect to the energy (semi-) norm. For thenonlinear problems in (7.81) and (7.96) ff., we have to impose the additional conditions(7.80) and (7.94):

c(v, v) ≥ Cε|v|2U , |c(u, v)| ≤ C ′ε|u|U |v|U ∀u, v ∈ U = H1(Ω) (7.159)

c(�v,�v) ≥ C|�v|2U , |c(�u,�v)| ≤ C ′|�u|U |�v|U ∀�u,�v ∈ U = H1(Ω,Rq).

For u, v ∈ H10 (Ω) and �u,�v ∈ H1

0 (Ω,Rq) this bounded coercivity (7.159) holds for thenorm ‖ · ‖H1

0 (Ω,Rq), q ≥ 1, the equivalent standard Sobolev norm.

7.9.2 Coercivity and boundedness in Vh for the Laplacian

Stability for the previous discrete problems can be guaranteed by two properties.We need the bounded invertibility of the corresponding linearized operator andthe bounded coercivity in Vh of its linearized principal part Ah(u, v) + Jσ

h (u, v) orAh(u;w, v) :=

(∂ah/∂u(u)

)(w, v) + Jσ

h (w, v) in the following subsections.

Remark 7.15. Boundedness and coercivity in Vh always means equiboundedness andequicoercivity with respect to h.

For the above different principal parts c(u, v), we define the corresponding ch(u, v),and

Ah(u, v) := ch(u, v) + εJσh (u, v) with Jσ

h (·, ·) in (7.61). (7.160)

For the nonlinear cases, let ch(w, v), ch(�w,�v) indicate the corresponding discretelinearizations analogous to c(w, v), c(�w,�v) in (7.157), (7.158). Then, cf. (7.99),

Ah(u;w, v) := ch(w, v) + εJσh (w, v) and Ah(�u; �w,�v) := ch(�w,�v) + Jσ

h (�w,�v). (7.161)


As above, we discuss the three types of DCGMs, the nonsymmetric interior, NIPG,with θ = −1, the symmetric interior, SIPG, with θ = 1, and the incomplete interiorpenalty Galerkin, IIPG, methods with θ = 0. For completeness, we extend it tothe general interior penalty Galerkin, GIPG, method with −1 ≤ θ ≤ 1, cf. Houstonet al. [407]. We prove the coercivity and boundedness of these Ah(·, ·) with respectto ‖ · ‖

For our model problem, these bilinear forms can be written in the form

ah(u, v) = ε∑

K∈Th

∫K

(∇u, ∇v)n dx− ε∑

e∈T h\∂ΩN

∫e

({∇u}, ν)n[v] (7.162)

+ θ({∇v}, ν)n[u])dS ∀u, v ∈ Ue for − 1 ≤ θ ≤ 1.

The multiplicative trace and the inverse inequality (7.129) and (7.138) yield

‖v‖2L2(∂K) ≤ C5(1 + C6)h−1K ‖v‖2L2(K) ∀v ∈ Vh. (7.163)

Theorem 7.16. Bounded and coercive ah(u, v) for the Laplacian: Choose an arbitraryconstant cw > 0 in (7.39) for the NIPG method. For the SIPG, IIPG, and GIPGmethods (with θ = 1, 0,∈ [−1, 1]) choose

cw ≥ C5(1 + C6)(1 + θ)2 (7.164)

where C5 and C6 are given by (7.138), respectively. Then the corresponding bilinearforms Ah(·, ·) given by (7.160), (7.162), are bounded and coercive with respect to thenorm, ‖v‖, such that

Ah(v, v) ≥ ccε‖v‖2, (7.165)

|Ah(u, v)| ≤ cbε‖u‖ ‖v‖ ∀u, v ∈ Vh. (7.166)

Remark 7.17. Theoretically it would be possible to determine the constants, definingcw in (7.164): Before starting computations, check all elements K ∈ Th and find themaximal values of κ, C3, κ, C4 appearing in (7.124)–(7.128). This is combined withthe C5, C6 in (7.129), (7.138). However, we doubt that anybody uses this approach.However, Epshteyn and Riviere [309] present computable lower bounds of the penaltyparameters in the SIPG method. In particular, they derive the explicit dependence ofthe coercivity constants with respect to the polynomial degree and the angles of themesh elements.

Usually, a suitable value of cw is sought empirically. Furthermore, a too high valueof cw leads to bad computational properties of the corresponding discrete problem.Some numerical experiences for the choice of cw for the system of the compressibleNavier-Stokes equations, cf. (7.343), are given in [294]. Houston et al. use a similarapproach, see their [407], text between relations (2.4) and (2.5). They have an explicitrelation containing constants from a multiplicative trace inequality (Cd) and theinverse inequality (C3).


Proof.

1. Coercivity of the NIPG method: Inequality (7.165) for Ah immediately followsfrom the definition (7.160), (7.162) with θ = −1, cc = 1.

2. Coercivity of the SIPG, IIPG, and GIPG methods: By (7.160) and (7.162) wehave

Ah(v, v) = ε|v|2H1(T h) − ε(1 + θ)∑

e∈T h\∂ΩN

∫e

({∇v}, ν)n[v] dS + εJσ

h (v, v).

(7.167)

This, the Cauchy inequality and Young’s inequality in the form,

ab ≤ ca2 + b2/(4c) for a, b, c ∈ R, c > 0, (7.168)

imply that for c = 1 and any δ > 0,

Ah(v, v) ≥ ε|v|2H1(T h) + εJσh (v, v)− ε(1 + θ)

⎛⎝1δ

∑e∈T h\∂ΩN

∫e

d(e){∇v}2dS

⎞⎠1/2

×

⎛⎝δ∑

e∈T h\∂ΩN

∫e

1d(e)

[v]2dS

⎞⎠1/2

(7.169)

≥ ε‖v‖2 − εω − εδ

cwJσ

h (v, v),

where

ω =(1 + θ)2

4δ

∑e∈T h\∂ΩN

∫e

d(e)|{∇v}|2dS. (7.170)

Further, from (7.129), (7.163), we get for v ∈ Vh∑K∈Th

hK‖∇v‖2L2(∂K) ≤ C5

∑K∈Th

hK

(|v|H1(K)|∇v|H1(K) + h−1

K |v|2H1(K)

)(7.171)

≤ C5(1 + C6)|v|2H1(T h)

and thus

ω ≤ (1 + θ)2

4δ

∑K∈Th

hK‖∇v‖2L2(∂K) ≤C5(1 + C6)(1 + θ)2

4δ|v|2H1(T h). (7.172)

Now let us choose

δ = C5(1 + C6)(1 + θ)2/2. (7.173)


Then it follows from (7.164), (7.167) and (7.169)–(7.173) that

Ah(v, v) ≥ ε‖v‖2 − ε

2|v|2H1(T h) −

εC5(1 + C6)(1 + θ)2

2cwJσ

h (v, v) (7.174)

≥ ε‖v‖2 − 12ε(|v|2H1(T h) + Jσ

h (v, v))

=ε

2‖v‖2,

which proves (7.165) with cc = 1/2.3. Boundedness: In view of the definition of the form Ah, we can write

Ah(u, v) = ϑ1 + ϑ2 + ϑ3, (7.175)

ϑ1 = ε∑

K∈Th

∫K

(∇u, ∇v)n dx

ϑ2 = −ε∑

e∈T h\∂ΩN

∫e

({∇u}, ν)n[v] + θ({∇v}, ν)n[u]

)dS,

ϑ3 = εJσh (u, v),

where θ = −1 for NIPG, θ = 1 for SIPG, θ = 0 for IIPG, and θ ∈ [−1, 1] forGIPG methods.

4. Using the Cauchy inequality, we get the bound for ϑ1 and ϑ3

|ϑ1| ≤ ε|u|H1(T h)|v|H1(T h) ≤ ε ‖u‖ ‖v‖, and (7.176)

|ϑ3| ≤ ε (Jσh (u, u))1/2 (Jσ

h (v, v))1/2 ≤ ε ‖u‖ ‖v‖.For estimating ϑ2, we apply the Cauchy inequality, with |θ| ≤ 1 yielding

|ϑ2| ≤ ε

⎛⎝ ∑e∈T h\∂ΩN

d(e)cw‖∇u‖2L2(e)

⎞⎠1/2⎛⎝ ∑e∈T h\∂ΩN

cw

d(e)‖[v]‖2L2(e)

⎞⎠1/2

(7.177)

+ ε

⎛⎝ ∑e∈T h\∂ΩN

d(e)cw‖∇v‖2L2(e)

⎞⎠1/2⎛⎝ ∑e∈T h\∂ΩN

cw

d(e)‖[u]‖2L2(e)

⎞⎠1/2

≤ ε√cw

( ∑K∈Th

hK‖∇u‖2L2(∂K)

)1/2

Jσh (v, v)1/2

+ε√cw

( ∑K∈Th

hK‖∇v‖2L2(∂K)

)1/2

Jσh (u, u)1/2.

We combine (7.171), (7.177), and the discrete Cauchy–Schwarz inequality

s1r2 + s2r1 ≤(s21 + s2

2

)1/2 (r21 + r2

2

)1/2, s1, s2, r1, r2 ∈ R


to obtain

|ϑ2| ≤ ε

√C5(1 + C6)

cw×(|u|H1(T h)J

σh (v, v)1/2 + |v|H1(T h)J

σh (u, u)1/2

)

≤ ε

√C5(1 + C6)

cw×(|u|2H1(T h) + Jσ

h (u, u))1/2 (

|v|2H1(T h) + Jσh (v, v)

)1/2

≤ ε

√C5(1 + C6)

cw‖u‖ ‖v‖. (7.178)

Now from (7.175), (7.176), (7.178) we immediately get (7.166) with cb = 2 +ε√C5(1 + C6)/cw. �

7.9.3 Coercivity and boundedness in Vh for the general linear and thesemilinear case

As a consequence of (7.89), we omit the lower order compact perturbations termsbh, ch, �h and confine the discussion here to the principal part in (7.179). Let, foru,w, v ∈ Ue, and Dirirchlet boundary conditions on Ω,

Ah(u;w, v) = εJσh (w, v) + ε

∑K∈Th

∫K

n∑i,j=1

aij(x, u) ∂iw ∂jv dx (7.179)

− ε∑

e∈T h

∫e

n∑i,j=1

({aij(x, u) ∂iw νj} [v] + θ{aij(x, u) ∂iv νj} [w]

)dS

with θ = −1, 1, 0,∈ [−1, 1] for the NIPG, SIPG, IIPG, and GIPG methods and Jσh in

(7.61). For general linear problems, we replace the aij(x, u) by aij(x) in (7.179).

Theorem 7.18. Bounded and coercive ah(u, v) for general linear and semilinearproblems: For the Λa, λ in (7.74), (7.80), and λ = min(1, λ), we choose cw arbitrarilypositive for the NIPG and, for the SIPG, IIPG or GIPG methods, as

cw ≥C5(1 + C6)(1 + θ)2 Λ2

an2

λ2 (7.180)

with C5 and C6 in (7.129) and (7.138), respectively. Then the corresponding bilinearforms Ah(·, ·), given by (7.179) or by replacing in (7.179) the aij(x, u) by aij(x), arebounded and coercive with respect to the norm, ‖v‖, such that

Ah(u, v, v) ≥ ccε‖v‖2, (7.181)

|Ah(u,w, v)| ≤ cbε‖w‖ ‖v‖ ∀u,w, v ∈ Vh. (7.182)


Proof. We formulate the proof only for semilinear problems.

1. The coercivity of the NIPG method again is a consequence of (7.179) with θ = −1and (7.80) (e):

Ah(u, v, v) = εJσh (v, v) + ε

∑K∈Th

∫K

n∑i,j=1

aij(x, u) ∂iv ∂jv dx (7.183)

≥ ε(Jσ

h (v, v) + λ|v|2H1(T h)

),

which proves (7.181) with cc := min(1, λ).

2. Coercivity of SIPG, IIPG, and GIPG methods: Let ρ = (ρ1, . . . , ρn), ϑ =(ϑ1, . . . , ϑn) ∈ Rn and aij(·, ·) satisfy assumption (7.80) (c). Then using anappropriately reordered discrete Chebyshev inequality (1.43), and the Cauchy–Schwarz inequality (1.44), for p = q = 2, we obtain∣∣∣∣∣∣

n∑i,j=1

figj

∣∣∣∣∣∣ =∣∣∣∣∣∣

n∑i=1

fi

n∑j=1

gj

∣∣∣∣∣∣ ≤ n

⎛⎝ n∑j=1

f2j

⎞⎠1/2⎛⎝ n∑j=1

g2j

⎞⎠1/2

= n|f |n|g|n = n|f ||g|.

(7.184)

This yields

n∑i,j=1

aij(·, ·)ρiϑj ≤

∣∣∣∣∣∣n∑

i,j=1

aij(·, ·)ρiϑj

∣∣∣∣∣∣ ≤ Λa

n∑i,j=1

|ρi| |ϑj | (7.185)

≤ nΛa

(n∑

i=1

ρ2i

)1/2 ( n∑i=1

θ2i

)1/2

= nΛa |ρ| |ϑ|.

From (7.179), (7.80) (e) and (7.185) we get, with θ ∈ (−1, 1],

Ah(u, v, v) = εJσh (v, v) + ε

∑K∈Th

∫K

n∑i,j=1

aij(x, u) ∂iv ∂jv dx

− (1 + θ)ε∑

e∈T h

∫e

n∑i,j=1

{aij(x, u) ∂iv νj} [v] dS

≥ εJσh (v, v) + ε λ|v|2H1(T h) − n(1 + θ)εΛa

∑e∈T h

∫e

|∇v||[v]| dS.

Applying a similar technique as in the proof of Theorem 7.16, relations (7.167)– (7.174) yield

Ah(v, v) ≥ λε‖v‖2 − εω − εδ

cwJσ

h (v, v),


where for v ∈ Vh

ω =(1 + θ)2Λ2

an2

4δ

∑e∈T h

∫e

d(e)|{∇v}|2dS (7.186)

≤ C5(1 + C6)(1 + θ)2Λ2an

2

4δ|v|2H1(T h).

Now, putting

δ =C5(1 + C6)(1 + θ)2Λ2

an2

2λ(7.187)

then (7.167) and (7.169)–(7.173) imply that

Ah(v, v) ≥ ε λ ‖v‖2 − ελ

2|v|2H1(T h) −

εC5(1 + C6)(1 + θ)2Λ2an

2

2λ cw

Jσh (v, v)

≥ ελ‖v‖2 − λε

2

(|v|2H1(T h) + Jσ

h (v, v))

=ελ

2‖v‖2, (7.188)

which proves (7.182) with cc = ελ/2.3. Boundedness: In view of the definition of the form Ah (7.179), we can write

Ah(u,w, v) = ϑ1 + ϑ2 + ϑ3, (7.189)

ϑ1 = ε∑

K∈Th

∫K

n∑i,j=1

aij(x, u) ∂iw ∂jv dx

ϑ2 = −ε∑

e∈T h

∫e

n∑i,j=1

×({aij(x, u) ∂iw νj} [v] + θ{aij(x, u) ∂iv νj} [w]

)dS,

ϑ3 = εJσh (w, v).

Using relation (7.185), we get

|ϑ1| ≤ εΛa n|w|H1(T h)|v|H1(T h). (7.190)

In order to estimate ϑ2, we again use (7.185) and obtain

|ϑ2| ≤ εΛa n∑

e∈T h

∫e

(|{∇w}| [v] + θ|{∇v}| [w]

)dS. (7.191)

With the same technique as in the proof of Theorem 7.16, (7.177)–(7.178), wefind

|ϑ2| ≤ εΛa n√C5(1 + C6)/cw ‖w‖ ‖v‖. (7.192)

Now from (7.189), (7.190), (7.192) and (7.176) we immediately get (7.182) withcb = Λa n(1 +

√C5(1 + C6)/cw) + 1. �


7.9.4 Vh-coercivity and boundedness for quasilinear problems

We have to prove the coercivity and boundedness of the linearized principal part forthe IIPG method, so the summation starts with k, l = 1, instead of k, l = 0 above

Ah(�u; �w,�v) := Jσh (�w,�v) +

∑K∈Th

∫K

n∑k,l=1

(∂Ak

∂ul(x, �u,∇�u)∂l �w, ∂k�v

)q

(7.193)

−∑

e∈T h

∫e

n∑k,l=1

({∂Ak

∂ul(x, �u,∇�u)∂l �w · νk

}, [�v]

)q

dS, for �u, �w,�v ∈ Ue.

Theorem 7.19. Bounded and coercive ah(u, v) for quasilinear systems: With theconstant λ from the assumption (7.94) (e), let λ = min(1, λ). We choose the IIPGmethod (θ = 0), and a constant cw in (7.39) as

cw ≥C5(1 + C6) (nqL) 2

λ2 , (7.194)

where C5, C6 and L are given by (7.129), (7.138) and (7.94) (c), respectively. Thenthe corresponding bilinear form Ah(·, ·) given by (7.193) is bounded and coercive withrespect to �w,�v and the norm, ‖�v‖, such that

Ah(�u,�v,�v) ≥ cc‖�v‖2, (7.195)

|Ah(�u, �w,�v)| ≤ cb‖�w‖ ‖�v‖ ∀�u,�v, �w ∈ H3/2+ε(T h,Rq). (7.196)

Proof. Coercivity: Similarly as in (7.185) we have with L in (7.94) (c)

n∑k,l=1

(∂Ak

∂ul(·, ·, ·)∂l�v, νk[�v]

)q

≤

∣∣∣∣∣∣n∑

k,l=1

(∂Ak

∂ul(·, ·, ·)∂l�v, νk[�v]

)q

∣∣∣∣∣∣ (7.197)

≤ nqL

n∑k,l=1

(|∂l�v, |νk[�v]|)q ≤ nqL

n∑k,l=1

(|∂l�v|, |[�v]|)q ≤ nqL|∇�v|q×n |[�v]|q,

for the n components of the ∂l�v, νk[�v] ∈ Rq, their absolute values, |∂l�v|, |νk[�v]| ∈ Rq

and their norms, |∂l�v|q, |∇�v|q×n, |νk[�v]|q ∈ R. Relations (7.94), (7.193) and (7.197),imply

Ah(�u,�v,�v) = Jσh (�v,�v) +

∑K∈Th

∫K

n∑k,l=1

(∂Ak

∂ul(x, �u,∇�u)∂l�v, ∂k�v

)q

−∑

e∈T h

∫e

n∑k,l=1

({∂Ak

∂ul(x, �u,∇�u)∂l�v · νk

}, [�v]

)q

dS,

≥ Jσh (v, v) + λ|�v|2H1(T h) − nqL

∑e∈T h

∫e

|{∇�v}|q×n |[�v]|q dS.

7.10. Consistency results for the ch, bh, �h 503

With very minor changes the remaining proof of coercivity as well as boundness forthis theorem follows that of Theorem 7.18. �

7.10 Consistency results for the ch, bh, �h

For the general discretization theory we have to prove the consistency of the differentterms, the ah, ch, bh, �h. We formulate the estimates with respect to global norms andstep sizes. Local versions are possible as well, cf. Section 7.14.

Depending upon the different choices of ch or bh we obtain the standard ornonstandard version, for θ = −1 or θ = 1 or θ = 0 or θ ∈ [−1, 1] in the NIPG or SIPGor IIPG or GIPG version of the DCGMs. So the following results again cover all thesecases.

Since the consistency for the ch, bh, �h is much easier, at least for the case ∂ΩD = ∂Ω,we delay the consistency for the ah, ah to the next section. The complementary case∂ΩN �= 0, requires for ch, bh a much more complicated proof, cf. [394]. The exact resultis presented in Theorem 7.22. The forms ch, bh are independent of θ. �h depends on θ.So the results in Section 7.10.2 are valid for θ ∈ [−1, 1].

7.10.1 Consistency of the ch and bh

We start with the standard numerical flux ch for ∂ΩD = ∂Ω. The results for the moregeneral ch in (7.59) are valid for its special case bh in (7.58) as well. Let (7.16) (a),(A1), (A2) in Subsection 7.7.1 and, for ch, Assumptions (H) in Subsection 7.3.3 besatisfied. We shall be concerned with the Lipschitz-continuity of the form ch. Thereason for requiring ∂ΩD = ∂Ω is the last inequality in (7.202). Jσ

h is defined in (7.61)as a sum over interior and Dirichlet boundary edges but not the Neumann edges, i.e.∑

K∈T h

∑e⊂∂K

∫e

[v]2

d(e)dS ≤ 2

∑e∈T h

∫e

[v]2

d(e)dS �≤ 2Jσ

h (v, v) = 2∑

e∈T h\∂ΩN

∫e

[v]2

d(e)dS.

Proposition 7.20. Under Assumption (H) and for ∂ΩD = ∂Ω, there exist con-stants C8 > 0, C9 > 0 such that for ch, and bh,

|ch(u, v)− ch(u, v)| ≤ C8‖v‖

⎛⎝‖u− u‖L2(Ω) +

( ∑K∈Th

hK‖u− u‖2L2(∂K)

)1/2⎞⎠,

∀ u, u ∈ Ue, v ∈ H1(T h), h ∈ (0, h0). (7.198)

For ∀uh, uh, vh ∈ Sh, the last term in (7.198) is replaced by∑K∈T h

(hK‖u− u‖L2(K)‖u− u‖H1(K)

)1/2. (7.199)

These estimates remain valid for |bh(u, v)− bh(u, v)|, |bh(uh, vh)− bh(uh, vh)| as well,if f is Lipschitz-continuous.


Proof. By (7.59), for u, u, v ∈ H1(T h),

ch(u, v)− ch(u, v) = −∑

K∈T h

∫K

(�f(u)− �f(u),∇v

)dx︸︷︷︸

:=σ1

(7.200)

+∑

e∈T h

∫e

(H(ul, ur, ν)−H(ul, ur, ν)

)[v] dS︸︷︷︸

:=σ2

.

From the Lipschitz-continuity of the function �f , we have

|σ1| ≤ Lf

∑K∈T h

∫K

|u− u| |∇v| dx ≤ Lf‖u− u‖L2(Ω)|v|H1(T h). (7.201)

The Lipschitz-continuity (7.51) of H, the Cauchy inequality, d(e) in (7.28) imply

|σ2| ≤ C2

∑e∈T h

⟨∫e

|[v]|(|ul − ul|+ |ur − ur|

)dS

⟩(7.202)

≤ C2

∑K∈T h

⟨ ∑e⊂∂K

∫e

|[v]| |u− u| dS⟩

≤ 2C2

∑K∈T h

⟨( ∑e⊂∂K

∫e

[v]2

d(e)dS

)1/2( ∑e⊂∂K

d(e)∫

e

(u− u)2 dS

)1/2⟩

≤ 4C2 Jσh (v, v)1/2

⎛⎝ ∑K∈T h


⎞⎠1/2

.

Now from (7.201) and (7.202) we get (7.198) with C8 = max(Lf , 4C2).We want to prove (7.199). In (7.198) we set v := vh ∈ Sh. Using the multiplicative

trace and the inverse inequalities (7.129) and (7.138),

|σ2| ≤ CJσh (vh, vh)1/2

∑K∈T h

(hK‖u− u‖2L2(K) + ‖u− u‖2H1(K)

)1/2. (7.203)

From (7.200), (7.201) and (7.203) we get the final result (7.198).In this proof we only have to replace C2 by Cf for obtaining the estimates for

|bh(u, vh)− bh(u, vh)|. We do need the Lipschitz-continuity for f. �

Theorem 7.21. Consistency of the ch, and bh: Under Assumption (H) and for∂ΩD = ∂Ω, there exists a constant C10 > 0 such that for ch, and bh,

|ch(u, vh)− ch(Phu, vh)| ≤ C10hmin{d,s}‖vh‖|u|Hmin{d,s}(Ω),

u ∈ Hs(Ω), vh ∈ Sh, h ∈ (0, h0), (7.204)

7.10. Consistency results for the ch, bh, �h 505

or (7.140) with dK ≥ d and where Phu is the Sh-interpolant of u from Theorem 7.7.This estimate remains valid for |bh(u, vh)− bh(Phu, vh)|, as well.

Proof. We start from (7.198) with u ∈ Hp+1(Ω), u := Phu and v := vh ∈ Sh. Usingthe multiplicative trace inequality (7.129), the discrete Cauchy inequality and theapproximating properties (7.139), we obtain∑

K∈T h

hK‖u− Phu‖2L2(∂K) (7.205)

≤∑

K∈T h

C5hK‖u− Phu‖L2(K)‖u− Phu‖H1(Ki)

≤∑

K∈T h

C5C27hKh

min{dK ,s}K h

min{dK ,s}−1K |u|2

Hmin{dK ,s}(K)

≤ C5C27h

2 min{d,s}

⎛⎝ ∑K∈T h

h2(dK−d)K |u|2

Hmin{dK ,s}(K)

⎞⎠1/2

≤ 2C5C27h

2 min{d,s}|u|2Hmin{d,s}(Ω).

This estimate, (7.139) and (7.198) yield (7.204) with C10 = C7C8(√

2C5 + 1). �

In the following theorem, we recall results presented in [394], where the consistencyof the forms ch and bh, both are independent of θ, is proved in the case when ∂ΩN �= 0.

Theorem 7.22. Consistency of the ch, and bh: Under Assumption (H) and for∂ΩN �= 0, there exists a constant C ′

10 > 0 such that for ch, and bh,

|ch(u, vh)− ch(Phu, vh)| ≤ C ′10h

min{d,s}−1/2‖vh‖L2(Ω)(1 + ‖vh‖)|u|Hmin{d,s}(Ω),

u ∈ Hs(Ω), vh ∈ Sh, h ∈ (0, h0), (7.206)

or (7.140) with dK ≥ d and where Phu is the Sh-interpolant of u from Theorem 7.7.This estimate remains valid for |bh(u, vh)− bh(Phu, vh)|, as well.

Proof. See [394], Lemma 3. �

7.10.2 Consistency of the �h

For the model problem (7.13), (7.14), �h(v) in (7.60) is independent of u and hasthe form, with θ = 1,= −1, 0, and θ ∈ [−1, 1], for the SIPG, NIPG, IIPG, and GIPGmethods, respectively

�h(v) = 〈g, v 〉+ (gN , v)L2(∂ΩN ) − ε∑

e∈∂ΩD

∫e

uD

(θ(∇v, ν)n + σ v

)dS,

�h(vh) = �h(vh) . (7.207)


For all the other problems (7.77), (7.78), (7.79), (7.100), we have ∂ΩD = ∂Ω, and omitthe (gN , v)L2(∂ΩN ) term and, for (7.78), we replace g(·) by g(·, u):

�h(u, v) = 〈g(·, u), v(·) 〉 − ε∑

e∈∂ΩD

∫e

θ(Ba(u)∇v, ν)nu + σuD v dS. (7.208)

The �h(�u,�v) in (7.100) is

�h(�vh) =∑

e∈∂Ω

∫e

σ(�uD, �vh )q dS, �h(�vh) = �h(�vh) . (7.209)

These �h(vh) = �h(vh), �h(�vh) = �h(�vh) show by (7.121), the following:

Theorem 7.23. Consistency of the �h: For the problems (7.13), (7.14), (7.77), with�h(v) = �h(v), in (7.207), and the quasilinear problem in (7.90), (7.100) with �h(�v) =�h(�v) in (7.209), the consistency errors vanish for the SIPG, NIPG, IIPG, and GIPGmethods.

However for the more general semilinear problem (7.78), (7.79) we have to con-sider the �h(u, v) in (7.208), with (Ba(u)∇v, ν)n as in (7.81) and 〈g(x, u), v 〉 =〈g(x, u), v 〉H−1×H1(T h):

�h(u, v) = ε∑

e∈∂Ω

∫e

σuD v − θ(Ba(u)∇v, ν)nudS + 〈g(x, u), v 〉. (7.210)

Theorem 7.24. Consistency for the semilinear �h: For the �h(u, vh) in (7.88),(7.208), with properties (7.80), a constant, C13 = (n + 1)max{L,Λa}C7(1 +√

2C5(C6 + 1)) > 0, exists such that ∀h ∈ (0, h0), h0 < 1,

|�h(Phu, vh)− �h(u, vh)| ≤ C13

⎛⎝ ∑K∈T h

(h

min{d,s}−1k |u|Hmin{d,s}(K)

)2

⎞⎠1/2

(7.211)

(1 + ‖u‖L∞(∂Ω))‖vh‖H1(T h) ∀u ∈ Hs(T h), vh, Phu ∈ Sh.

Proof. We combine the Lipschitz-continuity of the gj(x, u), j = 0, . . . , n, and aij(x, u),i, j = 1, . . . , n, with

∓aij(x, Phu)u = −aij(x, Phu)u + aij(x, Phu)u = 0 = |�h(Phu, vh)− �h(u, vh)|

≤

∣∣∣∣∣∣∑

K∈T h

∫K

n∑j=0

(gj(x, Phu)− gj(x, u)

)∂jvh dx

∣∣∣∣∣∣︸︷︷︸:=σ9

+ ε

∣∣∣∣∣∣∑

e∈∂Ω

∫e

n∑i,j=1

(aij(x, Phu)Phu∓ aij(x, Phu)u− aij(x, u)u

)∂ivh νjdS

∣∣∣∣∣∣︸︷︷︸:=σ10

.

7.11. Consistency properties of the ah 507

Obviously we obtain with the Lipschitz constant, L, for the gj

|σ9| ≤ (n + 1)L‖Phu− u‖L2(T h)‖vh‖H1(T h)

≤ (n + 1)LC7

⎛⎝ ∑K∈T h

(h

min{d,s}k |u|Hmin{d,s}(K)

)2

⎞⎠1/2

‖vh‖H1(T h).

We estimate by combining (7.129) and (7.138)

hK‖∇vh‖2L2(∂K) ≤ C5hK‖∇vh‖L2(K)|∇vh|H1(K) + C5‖∇vh‖2L2(K) (7.212)

≤ 2C5(C6 + 1)‖∇vh‖2L2(K)

hence,

1C6 + 1

∑K∈T h

hK‖∇vh‖2L2(∂K) ≤ 2C5

∑K∈T h

‖∇vh‖2L2(K) = 2C5|vh|2H1(T h). (7.213)

Then the Lipschitz constant, L, and the bound Λa for the aij , yield

|σ10|/(n(Λa + L‖u‖L∞(∂Ω))

)(7.214)

≤∑

e∈∂Ω

∫e

∣∣∇vh∣∣ | [u− Phu

]| dS

≤

⎛⎝∑e∈T h

∫e

hK |∇vh|2 dS

⎞⎠1/2⎛⎝∑e∈T h

∫e

h−1K ([u− Phu])2 dS

⎞⎠1/2

≤

⎛⎜⎝⎛⎝ ∑

K∈T h

hK‖∇vh‖2L2(∂K)

⎞⎠1/2⎛⎝ ∑K∈T h

h−1K ‖(u− Phu)‖2L2(∂K)

⎞⎠1/2⎞⎟⎠

≤√

2C5(C6 + 1)C7|vh|H1(T h)

⎛⎝ ∑K∈T h

(h


)2

⎞⎠1/2

.

Adding the estimates for σ9 and σ10 proves the claimed (7.211). �

7.11 Consistency properties of the ah

7.11.1 Consistency of the ah for the Laplacian

We prove, for ∂ΩD ∩ ∂ΩN = ∂Ω, the consistency for ah in (7.215). The necessarychanges for semilinear and quasilinear ah in (7.87) and (7.99) are discussed in


Proposition 7.29 and Proposition 7.32. The ah for the model problem have the form

ah(u, v) := ε∑

K∈T h

∫K

(∇u, ∇v)n dx (7.215)

−ε∑

e∈T h\∂ΩN

∫e

[v]({∇u}, ν)n + θ[u]({∇v}, ν)n dS,

where θ = −1 for NIPG, θ = 1 for SIPG, θ = 0 for IIPG, and θ ∈ [−1, 1] for the GIPGmethods.

Proposition 7.25. These ah(u, v) are bounded with respect to u, v, such that u, v,u ∈ Ue

|ah(u, v)− ah(u, v)| ≤ ε |u− u|H1(T h)|v|H1(T h) (7.216)

+ ε 2n

⎛⎜⎝( 1cw

)1/2

(Jσh (v, v))1/2

⎛⎝ ∑K∈T h

hK |u− u|2H1(∂K)

⎞⎠1/2

+

⎛⎝ ∑K∈T h

hK‖∇v‖2L2(∂K)

⎞⎠1/2⎛⎝ ∑K∈T h

h−1K ‖(u− u)‖2L2(∂K)

⎞⎠1/2⎞⎟⎠ .

The last line vanishes for the IIPG method.

Proof. We consider u, u, v ∈ Ue as arguments of ah(·, ·) and estimate the differentterms σj in (7.217) ff.:

|ah(u, v)− ah(u, v)| ≤ ε

∣∣∣∣∣∣∑

K∈T h

∫K

(∇(u− u),∇v)n dx

∣∣∣∣∣∣︸︷︷︸:=σ1

(7.217)

+ε

∣∣∣∣∣∣∑

e∈T h\∂ΩN

∫e

(({∇(u− u)

}, ν)n[v]

)dS

∣∣∣∣∣∣︸︷︷︸:=σ2

+ε

∣∣∣∣∣∣θ∑

e∈T h\∂ΩN

∫e

(({∇v}, ν)n[(u− u)]

)dS

∣∣∣∣∣∣︸︷︷︸:=σ3

.

σ1 is estimated with both Cauchy–Schwarz inequalities (1.44), for p = q = 2, (1.45),

|σ1| ≤∑

K∈T h

|u− u|H1(K)|v|H1(K) ≤ |u− u|H1(T h)|v|H1(T h). (7.218)


For σ2 the same inequalities (1.44), (1.45), and d(e) in (7.28) yield

|σ2| ≤ n∑

e∈T h\∂ΩN

∫e

|[v]| |∇u−∇u| dS (7.219)

≤ n

⎛⎝ ∑e∈T h\∂ΩN

∫e

[v]2

d(e)/cwdS

⎞⎠1/2⎛⎝ ∑e∈T h\∂ΩN

d(e)cw

∫e

|∇u−∇u|2 dS

⎞⎠1/2

≤ 2n(

1cw

)1/2

(Jσh (v, v))1/2

⎛⎝ ∑K∈T h

hK |u− u|2H1(∂K)

⎞⎠1/2

.

σ3 vanishes for the IIPG method. Otherwise, again (1.44), (1.45) yield

|σ3| ≤ n∑

e∈T h\∂ΩN

∫e

|∇v| | [u− u] | dS (7.220)

≤ n

⎛⎝ ∑e∈T h\∂ΩN

∫e

hK |∇v|2 dS

⎞⎠1/2⎛⎝ ∑e∈T h\∂ΩN

∫e

h−1K ([u− u])2 dS

⎞⎠1/2

≤ 2n

⎛⎝ ∑K∈T h

hK‖∇v‖2L2(∂K)

⎞⎠1/2⎛⎝ ∑K∈T h

h−1K ‖(u− u)‖2L2(∂K)

⎞⎠1/2

.

Adding the estimates for σ1 to σ3 in (7.218)–(7.220) proves (7.216). �

Proposition 7.25 implies with u = Phu, and for ∀vh ∈ Sh = Vh as in Theorem 7.21

Theorem 7.26. Consistency of the Laplacian ah: The ah(Phu, vh) in (7.87) areconsistent with a(u, v) in (7.81) ∀ u ∈ Ue, ∀vh ∈ Sh = Vh, so a constant, C17 :=max{C7, 2n

√2C5C7((1/cw)1/2 + 2C5

√C7(C6 + 1)} > 0, exists, such that for the Sh-

interpolant Phu of u in Theorem 7.7,

|ah(u, vh)− ah(Phu, vh)| ≤ ε

⎛⎝ ∑K∈T h

(h


)2

⎞⎠1/2

(7.221)

(C7|vh|H1(T h) + 2n

(c−1/2w

√2C5C7/cw

(Jσ

h (vh, vh))1/2

+ 2C5

√C7(C6 + 1)|vh|H1(T h)

))≤ ce C17h

min{d,s}−1|u|Hmin{d,s}(T h)‖vh‖.

The 2C5

√C7(C6 + 1)|vh|H1(T h) can be deleted for the IIPG method.


Proof. For (7.221) and by (7.216) we get by (7.129)–(7.139). The following first andthird lines are straightforward

|Phu− u|H1(T h) ≤ C7

⎛⎝ ∑K∈T h

(h


)2

⎞⎠1/2

, by (7.139),

(7.222)⎛⎝ ∑K∈T h

hK |u− Phu|2H1(∂K)

⎞⎠1/2

≤√

2C5C7

⎛⎝ ∑K∈T h

(h


)2

⎞⎠1/2

,

⎛⎝ ∑K∈T h

hK‖∇vh‖2L2(∂K)

⎞⎠1/2

≤√

2C5(C6 + 1) |vh|H1(T h), by (7.129), (7.138),

⎛⎝ ∑K∈T h

h−1K ‖u− Phu‖2L2(∂K)

⎞⎠1/2

≤√

2C5C7

⎛⎝ ∑K∈T h

(h


)2

⎞⎠1/2

.

For the term in the second line of (7.222) we get, by (7.129), (7.139),

hK |u− Phu|2H1(∂K)

≤ C5hK |u− Phu|H1(K)|u− Phu|H2(K) + C5|u− Phu|2H1(K)

≤ C5C7(|u|Hmin{d,s}(K))2(hKh

min{d,s}−1K h

min{d,s}−2K + h

2(min{d,s}−1)K

)≤ 2C5C7

(|u|Hmin{d,s}(K)h

(min{d,s}−1)K

)2

implying the second line in (7.222). The third line coincides with (7.213).For the last line in (7.222) we need, with (7.129), (7.138), and (7.139),

h−1K ‖u− Phu‖2L2(∂K) (7.223)

≤ C5h−1K ‖u− Phu‖L2(K)|u− Phu|H1(K) + C5h

−2K ‖u− Phu‖2L2(K)

≤ C5C7(|u|Hmin{d,s}(K))2(h−1

K hmin{d,s}K h

min{d,s}−1K + h−2

K h2(min{d,s})K

)≤ 2C5C7

(|u|Hmin{d,s}(K)h

(min{d,s}−1)K

)2.

Relation (7.222) is applied in this order to the different error terms u− Phu in (7.216):

|ah(u, vh)− ah(Phu, vh)| ≤ ε

⎛⎝ ∑K∈T h

(h


)2

⎞⎠1/2

(C7|vh|H1(T h) + 2n

(c−1/(2w

√2C5C7

(Jσ

h (vh, vh))1/2

+ 2C5

√C7(C6 + 1)|vh|H1(T h)

)).


This shows, with (7.154) and Proposition 7.9, the estimate (7.221). Finally, Theorems7.12 and 7.13 imply the last estimate in (7.221). �

7.11.2 Consistency of the ah for general linear problems

In Section 7.4 we have introduced for general linear elliptic problems the bilinear form,ah(u, v), cf (7.74),(7.76), for θ = −1, θ = 1, θ = 0, θ ∈ [−1, 1],

ah(u, v) :=∑

K∈Th

⎛⎝∫K

n∑i,j=0

aij∂i u∂j v

⎞⎠ dx−∑

e∈T h

∫e

(({Ba∇0u}, ν)0n[v] (7.224)

+ θ ({Ba∇0v}, ν)n[u])dS for u, v ∈ Ue

with

(Ba∇0u,∇0v)0n :=n∑

i,j=0

aij∂i u∂j v, (Ba∇0u, ν)n =

j=1,...,n∑i=0,...,n

νjaij∂i u.

We prove its consistency.

Proposition 7.27. These bilinear ah(u, v) are bounded with respect to u, v, such thatu, v, u ∈ Ue

|ah(u, v)− ah(u, v)| ≤ C0 ‖u− u‖H1(T h)‖v‖H1(T h) (7.225)

+ 2nC0

⎛⎜⎝( 1cw

)1/2

(Jσh (v, v))1/2

⎛⎝ ∑K∈T h

hK |u− u|2H1(∂K) + hK‖u− u‖2L2(∂K)

⎞⎠1/2

+

⎛⎝ ∑K∈T h

hK

(‖∇v‖2L2(∂K) + ‖v‖2L2(∂K)

)⎞⎠1/2⎛⎝ ∑K∈T h

h−1K ‖(u− u)‖2L2(∂K)

⎞⎠1/2⎞⎟⎠.

The last line vanishes for the IIPG method.

Proof. The boundedness follows from (7.225) for u = 0. We consider u, u, v ∈ Ue

as arguments of ah(·, ·) and use, cf. (7.74), the bound |ai,j(x)| ≤ C0. We estimate thedifferent terms σj in (7.226) ff.:

|ah(u, v)− ah(u, v)| ≤ ε

∣∣∣∣∣∣∑

K∈T h

∫K

(Ba∇0(u− u),∇0v)0n dx

∣∣∣∣∣∣︸︷︷︸:=σ1

(7.226)

+ε

∣∣∣∣∣∣∑

e∈T h

∫e

(({Ba∇0(u− u)

}, ν)n[v]

)dS

∣∣∣∣∣∣︸︷︷︸:=σ2


+ ε

∣∣∣∣∣∣θ∑

e∈T h

∫e

(({Ba∇0v

}, ν)n[(u− u)]

)dS

∣∣∣∣∣∣︸︷︷︸:=σ3

.

σ1 is estimated with both Cauchy–Schwarz inequalities (1.44), for p = q = 2,(1.45),

|σ1| ≤ C0

∑K∈T h

‖u− u‖H1(K)‖v‖H1(K) ≤ C0‖u− u‖H1(T h)‖v‖H1(T h). (7.227)


|σ2| ≤ n∑

e∈T h

∫e

|[v]| | Ba∇0u−Ba∇0u| dS (7.228)

≤ n

⎛⎝∑e∈T h

∫e

[v]2

d(e)/cwdS

⎞⎠1/2⎛⎝∑e∈T h

d(e)cw

∫e

| Ba∇0u−Ba∇0u|2 dS

⎞⎠1/2

≤ 2nC0

(1cw

)1/2

× (Jσh (v, v))1/2

⎛⎝ ∑K∈T h

(hK |u− u|2H1(∂K) + hK‖u− u‖2L2(∂K)

)1/2

⎞⎠ .

σ3 vanishes for the IIPG method. Otherwise, again (1.44), (1.45) yield

|σ3| ≤ n∑

e∈T h

∫e

∣∣Ba∇0v∣∣ | [u− u] | dS (7.229)

≤ nC0

⎛⎝∑e∈T h

∫e

hK(|∇v|2 + |v|2) dS

⎞⎠1/2⎛⎝∑e∈T h

∫e

h−1K ([u− u])2 dS

⎞⎠1/2

≤ 2nC0

⎛⎝ ∑K∈T h

hK

(‖∇v‖2L2(∂K) + ‖v‖2L2(∂K)

)⎞⎠1/2

×

⎛⎝ ∑K∈T h

h−1K ‖(u− u)‖2L2(∂K)

⎞⎠1/2

Adding the estimates for σ1 to σ3 in (7.227)–(7.229) proves (7.225). �

Proposition 7.27 implies with u = Phu, as in Theorem 7.21:


Theorem 7.28. Consistency for the general linear ah: The ah(Phu, vh) in (7.76)are consistent with a(u, v) in (7.74) ∀ u ∈ Ue, ∀vh ∈ Sh = Vh. So a constant, C18 :=2max

{C7, 2n

√2C5C7(2(1/cw)1/2 + C5

√C7(C6 + 1)

}> 0, exists, such that for the

Sh-interpolant Phu of u in Theorem 7.7,

|ah(u, vh)− ah(Phu, vh)| ≤ C0

⎛⎝ ∑K∈T h

(h


)2

⎞⎠1/2

(7.230)

2(C7‖vh‖H1(T h) + 2n

(c−1/2w

√2C5C7/cw

(Jσ

h (vh, vh))1/2

+ C5

√C7(C6 + 1)‖vh‖H1(T h)

)≤ ε C18h

min{d,s}−1|u|Hmin{d,s}(T h)‖vh‖.

The C5

√C7(C6 + 1)‖vh‖H1(T h) term can be deleted for the IIPG method.

Proof. The proof is nearly identical with that of Theorem 7.26. For (7.230) and by(7.225) we again use (7.231), however replacing the first two lines by

‖Phu− u‖H1(T h) ≤ 2C7

⎛⎝ ∑K∈T h

(h


)2

⎞⎠1/2

, by (7.139),

×

⎛⎝ ∑K∈T h

hK‖u− Phu‖2H1(∂K)

⎞⎠1/2

(7.231)

≤ 2√

2C5C7

⎛⎝ ∑K∈T h

(h


)2

⎞⎠1/2

,

(7.222), (7.231) are applied to the different error terms u− Phu in (7.225):

|ah(u, vh) − ah(Phu, vh)| ≤ C0

⎛⎝ ∑K∈T h

(h


)2

⎞⎠1/2

(2C7‖vh‖H1(T h) + 4n

(c−1/(2w

√2C5C7

(Jσ

h (vh, vh))1/2

+ 2C5

√C7(C6 + 1)‖vh‖H1(T h)

).



7.11.3 Consistency of the semilinear ah

We turn to the consistency for the semilinear ah in (7.87) where ∂ΩN = ∅. Here it isworthwhile introducing new ah(·; ·, ·), based upon Ba(u)∇w instead of the Ba(u)∇uin (7.87).

For u, v, w ∈ Ue ⊃ Sh, define (7.232)

Ba(u)∇w :=

(n∑

i=1

(aij(x, u)∂iw

)n

j=1

, and (Ba(u)∇w,∇v)n, (Ba(u)∇w, ν)n,

ah(u;w, v) := ε∑

K∈T h

∫K

(Ba(u)∇w, ∇v)n dx

−ε∑

e∈T h

∫e

(({Ba(u)∇w}, ν)n [v] + θ({Ba(u)∇v}, ν)n [w]

)dS.

Proposition 7.29. The ah(u, v) = ah(u;u, v) in (7.232) with properties (7.80) aremodified Lipschitz-continuous with respect to u and linear and bounded with respect tov, such that ah(u;u, v) satisfies ∀ u, v ∈ Ue ∀u ∈W 2,∞(T h)

|ah(u, v)− ah(u, v)|= |ah(u;u, v)− ah(u;u, v)| ≤ ε nΛa|u− u|H1(T h)|v|H1(T h)

+ ε 2nΛa/(cw)1/2 (Jσh (v, v))1/2

⎛⎝ ∑K∈T h

hK |u− u|2H1(∂K)

⎞⎠1/2

(7.233)

+ ε 2nΛa

⎛⎝ ∑K∈T h

hK‖∇v‖2L2(∂K)

⎞⎠1/2⎛⎝ ∑K∈T h

h−1K ‖(u− u)‖2L2(∂K)

⎞⎠1/2

+ ε nL‖u− u‖L2(T h)‖∇u‖L∞(Ω)‖vh‖H1(T h)

+ ε 2nL/(cw)1/2‖u‖W 1,∞(T h) (Jσh (v, v))1/2

⎛⎝ ∑K∈T h


⎞⎠1/2

+ ε 2nL‖u‖L∞(Ω)

⎛⎝ ∑K∈T h

hK‖∇v‖2L2(∂K)

⎞⎠1/2⎛⎝ ∑K∈T h

h−1K ‖(u− u)‖2L2(∂K)

⎞⎠1/2

.

For the special case a(u, v) in (7.19) the last three lines drop out and u ∈W 2,∞(T h)is replaced by u ∈ Ue.

Remark 7.30. The modified Lipschitz-continuity with respect to u indicates thedependency upon ‖u‖W 1,∞(T h). The condition u ∈W 2,∞(T h) is imposed for allowing


‖∇u‖L∞(e) ≤ ‖∇u‖L∞(K), cf. (7.202), (7.242). We will need that for u = Phu. By theW k,∞ form of (7.139), u ∈W 2,∞(T h) implies Phu ∈W 2,∞(T h). Moreover,

‖Phu‖W 2,∞(T h) ≤ ‖Phu− u‖W 2,∞(T h) + ‖u‖W 2,∞(T h) ≤ C‖u‖W 2,∞(T h) (7.234)

hence ‖∇Phu‖L∞(e) ≤ ‖∇Phu‖L∞(K) ≤ C‖∇u‖L∞(K).For good convergence we will impose later on u0 ∈ Hs(T h), s ≥ 2, for the exact

solution anyway. For s− 2 ≥ n/2 this implies u0 ∈W 2,∞(T h), cf. Theorem 1.26,otherwise we have to assume it additionally. For the u ∈ Sh with Sh ⊂W 2,∞(T h)it is automatically satisfied.

Proof. We consider u, u, v ∈ Ue as arguments of ah(·; ·, ·):

|ah(u;u, v)− ah(u;u, v)| (7.235)

≤ |ah(u;u, v)− ah(u;u, v)|+ |ah(u;u, v)− ah(u;u, v)|.

and estimate the different terms σj in (7.236) ff. For the first term we obtain with(7.232)

|ah(u;u, v)− ah(u;u, v)| ≤ ε

∣∣∣∣∣∣∑

K∈T h

∫K

(Ba(u)∇(u− u),∇v)n dx

∣∣∣∣∣∣︸︷︷︸:=σ4

(7.236)

+ ε

∣∣∣∣∣∣∑

e∈T h

∫e

(({Ba(u)∇(u− u)}, ν)n [v] + θ({Ba(u)∇v}, ν)n [(u− u)]

)dS

∣∣∣∣∣∣︸︷︷︸:=σ5+θσ5

.

(7.237)

σ4 is estimated with the bounded aij(x, u), cf.(7.80) (c), and an appropriatelyreordered discrete Chebyshev inequality (1.43), and the Cauchy–Schwarz inequality,(1.44), for p = q = 2, cf. (7.185),

|σ4| ≤∑

K∈T h

∫K

|(Ba(u)∇(u− u),∇v)n| dx (7.238)

=∑

K∈T h

∫K

n∑i,j=1

|aij(x, u)∂i(u− u) ∂jv| dx

≤ Λa

∑K∈T h

∫K

n∑i,j=1

|∂i(u− u) ∂jv| dx

≤ nΛa

∑K∈T h

|u− u|H1(K)|v|H1(K)

≤ nΛa|u− u|H1(T h)|v|H1(T h).


Similarly as in (7.202), we obtain for σ5 with the Chebyshev and the Cauchy–Schwarzinequality (1.43), (1.44), for p = q = 2, and d(e) in (7.28),

|σ5| ≤ nΛa

∑K∈T h

∑e⊂∂K

∫e

|[v]| |∇(u− u)| dS (7.239)

≤ nΛa

∑K∈T h

( ∑e⊂∂K

∫e

[v]2

d(e)/cwdS

)1/2( ∑e⊂∂K

d(e)cw

∫e

|∇(u− u)|2 dS)1/2

≤ 2nΛa/(cw)1/2 (Jσh (v, v))1/2

⎛⎝ ∑K∈T h

hK |u− u|2H1(∂K)

⎞⎠1/2

.

The reordered discrete Chebyshev inequality (1.43), and the Holder inequality, (1.44)for p = q = 2, yield for σ5, cf. (7.220),

|σ5| ≤ nΛa

∑e∈T h

∫e

|∇v| | [u− u] | dS (7.240)

≤ nΛa

⎛⎝∑e∈T h

∫e

hK |∇v|2 dS

⎞⎠1/2⎛⎝∑e∈T h

∫e

h−1K ([u− u])2 dS

⎞⎠1/2

≤ 2nΛa

⎛⎝ ∑K∈T h

hK‖∇v‖2L2(∂K)

⎞⎠1/2⎛⎝ ∑K∈T h

h−1K ‖(u− u)‖2L2(∂K)

⎞⎠1/2

.

For the second term in (7.235) we combine the Lipschitz-continuity of the functionsaij , i, j = 1, . . . , n or Ba with (7.232), getting

|ah(u;u, v)− ah(u;u, v)|

≤ ε

∣∣∣∣∣∣∑

K∈T h

∫K

((Ba(u)−Ba(u))∇u,∇v

)ndx

∣∣∣∣∣∣︸︷︷︸:=σ6

(7.241)

+ ε

∣∣∣∣∣∣∑

e∈T h

∫e

(({(Ba(u)−Ba(u))∇u}, ν)n [v] + θ({(Ba(u)−Ba(u))∇v}, ν)n [u]

)dS

∣∣∣∣∣∣︸︷︷︸:=σ7+θσ7

.

Obviously we obtain with the Lipschitz constant, L, for the aij or the Ba with (1.43),

|σ6| ≤ nL‖u− u‖L2(T h)‖∇u‖L∞(Ω)‖vh‖H1(T h). (7.242)


A combination with the ideas for, e.g. (7.202)–(7.203) allows the estimation of σ7 :here we need ‖∇u‖L∞(e) ≤ ‖∇u‖L∞(K) requiring u ∈W 2,∞(T h) by Theorem 1.38.

|σ7| ≤ 2nL

⎛⎝∑e∈T h

‖∇u‖2L∞(e)

∫e

[v]2

d(e)/cwdS

⎞⎠1/2⎛⎝∑e∈T h

d(e)cw

∫e

n∑i=1

(u− u)2 dS

⎞⎠1/2

≤ 2nL/(cw)1/2‖u‖W 1,∞(T h) (Jσh (v, v))1/2

⎛⎝ ∑K∈T h


⎞⎠1/2

. (7.243)

For σ7 we obtain, as in (7.220),

|σ7| ≤ 2nL‖u‖L∞(Ω)

∑e∈T h

∫e

|∇v| |(u− u)| dS (7.244)

≤ 2nL‖u‖L∞(Ω)

⎛⎝ ∑K∈T h

hK‖∇v‖2L2(∂K)

⎞⎠1/2⎛⎝ ∑K∈T h

h−1K ‖(u− u)‖2L2(∂K)

⎞⎠1/2

.

Adding the estimates for σ4 to σ7 in (7.238)–(7.244) proves the claimed (7.233). �

As consequence of Proposition 7.29 we obtain with u = Phu, the Sh-interpolantfrom Theorem 7.7, but now only for ∀vh ∈ Sh = Vh as in Theorem 7.21, compareRemark 7.30.

Theorem 7.31. Consistency for the semilinear ah: The ah(u, vh) in (7.87) withproperties (7.80) are Lipschitz-continuous with respect to u and linear and boundedwith respect to vh. Furthermore the ah(Phu, vh) are consistent with the ah(u, vh). Soconstants C∗

1 , C∗2 exist, cf. (7.247) such that, ∀ h < 1, u ∈W 2,∞(T h), ∀vh ∈ Sh = Vh,

and the Sh-interpolant, Phu ∈W 2,∞(T h), with dK ≥ d,

|ah(u, vh)− ah(Phu, vh)| (7.245)

≤ ε

⎛⎝ ∑K∈T h

(h

min{dK ,s}−1k |u|Hmin{dK ,s}(K)

)2

⎞⎠1/2(C∗

1 + C∗2‖Phu‖W 1,∞(T h)

)‖vh‖

≤ ε hmin{d,s}−1|u|Hmin{d,s}(T h)

(C∗

1 + 2C∗2‖u‖W 1,∞(T h)

)‖vh‖.

The term +2C∗2‖u‖W 1,∞(T h) drops out for a(u, v) in (7.19) and u ∈W 2,∞(T h) is

replaced by u ∈ Ue.

Proof. It proceeds totally analogously to that of Theorem 7.26. We essentially applythe estimates in (7.222) to the terms in (7.246) and obtain, with the equivalence of


the penalty norms in (7.154), (1.43),

|ah(u;u, vh)− ah(Phu;Phu, vh)|

≤ ε

⎛⎝ ∑K∈T h

(h


)2

⎞⎠1/2

×(nΛaC7|vh|H1(T h) + nLC7h‖vh‖H1(T h)‖∇Phu‖L∞(T h)

+ 2n√

2C5C7/cw(Λa + hL‖Phu‖W 1,∞(T h))(Jσ

h (vh, vh))1/2

+ 4nC5

√C7(C6 + 1)(Λa + L‖Phu‖L∞(T h) )|vh|H1(T h)

)

≤ ε

⎛⎝ ∑K∈T h

(h


)2

⎞⎠1/2

(7.246)

‖vh‖(nΛa

√[(C7 + 4nC5

√C7(C6 + 1))2 + 8nC5C7/cw

]+ ‖Phu‖W 1,∞(T h) (1 + ce)nL

√C7

√h2C7 + 8h2C5/cw + 16C2

5 (C6 + 1))

and with the following C∗1 , C

∗2 ,

≤ ε

⎛⎝ ∑K∈T h

(h


)2

⎞⎠1/2 (C∗

1 + C∗2‖Phu‖W 1,∞(T h)

)‖vh‖.

with

C∗1 := nΛa

√[(C7 + 4nC5

√C7(C6 + 1))2 + 8nC5C7/cw

]and (7.247)

C∗2 := (1 + ce)nL

√C7

√h2C7 + 8h2C5/cw + 16C2

5 (C6 + 1)).

Thus we have proved the estimate (7.245). �

7.11.4 Consistency of the quasilinear ah for systems

Again we introduce a new ah(·; ·, ·), based upon BA(�u,∇�w) instead of the BA(�u,∇�u)in (7.97), (7.99). We update the notation in (7.92), with (T h,Rq) replacing (T h), anddefine, e.g.

BA(�uh,∇�wh) :=(Ak(x, �uh,∇�wh)

)nk=1

∈ Rn×q, (BA(�uh,∇�wh), ν)n ∈ Rq (7.248)


yielding ∀�uh, �wh ∈ Ue,∀�vh ∈ U1,e := H1(T h,Rq):

ah(�uh;∇�wh, �vh) =∑

K∈T h

∫K

n∑k=0

(Ak(x, �uh,∇�wh), ∂k�vh)qdx

−∑

e∈T h

∫e

(({BA(�uh,∇�wh)}, ν)n, [�vh]

)qdS.

Proposition 7.32. The ah(�u; �u,�v) in (7.248) with properties (7.94) are modifiedLipschitz-continuous with respect to �u and linear in �v, and bounded with respect to ‖�v‖.So ∀ �u,�v ∈ Ve, �u ∈W 2,∞(T h,Rq) and ‖�u‖U1,e

, ‖�u‖U1,e≤M this ah(�u; �u,�v) satisfies

|ah(�u,�v)− ah(�u,�v)| = |ah(�u; �u,�v)− ah(�u; �u,�v)| ≤ L|�u− �u|U1,e|�v|U1,e

(7.249)

+L(1 + |�u|W 1,∞(T h,Rq))|�u− �u|U1,e|�v|L2(T h,Rq)

+nL/(cw)1/2 (Jσh (�v,�v))1/2

⎛⎝ ∑K∈T h

hK |�u− �u|2H1(∂K)

⎞⎠1/2

+L‖�u− �u‖L2(T h,Rq)

(1 + ‖�u‖W 1,∞(T h,Rq)

+ ‖�u‖2W 1,∞(T h,Rq)

)‖�vh‖U1,e

+nL/(cw)1/2

⎛⎝ ∑K∈T h

hK‖�u− �u‖2L2(∂K)

⎞⎠1/2

×Jσh (�v,�v))1/2(1 + ‖�u‖W 1,∞(T h,Rq)).

Proof. We consider �u, �u, �w,�v ∈ Ue as arguments of ah(·; ·, ·) and estimate

|ah(�u; �u,�v)− ah(�u; �u,�v)| ≤ |ah(�u; �u,�v)− ah(�u; �u,�v)| (7.250)

+ |ah(�u; �u,�v)| − ah(�u; �u,�v)|.For the second term we obtain with (7.248)

|ah(�u; �u,�v) − ah(�u; �u,�v)| (7.251)

≤

∣∣∣∣∣∣∑

K∈T h

∫K

n∑k=0

((Ak(x, �u,∇�u)−Ak(x, �u,∇�u)

), ∂k�v

)qdx

∣∣∣∣∣∣︸︷︷︸:=σ9=σ9k>0+σ9k=0

+

∣∣∣∣∣∣∑

e∈T h

∫e

([�v], ({BA(�u,∇�u )}, ν)n − ({BA(�u,∇�u )}, ν)n )q dS

∣∣∣∣∣∣︸︷︷︸:=σ10

.


Since the conditions for the partials with respect to �u of the Ak in (7.94) for k = 0 andk > 0 are different, σ9 and σ11 below are split. σ9k>0 , obtained by

∑nk=1, is estimated

with the Lipschitz-continuous Ak(x, �w,∇�u), cf. (7.94) (c), and the Cauchy–Schwarzinequalities (1.44) for p = q = 2, (1.45),

|σ9k>0 | ≤∑

K∈T h

∫K

n∑k=1

((Ak(x, �u,∇�u)−Ak(x, �u,∇�u)

), ∂k�v

)qdx (7.252)

≤ L∑

K∈T h

∫K

‖∇�u−∇�u‖n×q‖∇�v‖n×q dx

≤ L∑

K∈T h

|�u− �u|H1(K)|�v|H1(K)

≤ L|�u− �u|U1,e|�v|U1,e

.

For σ9k=0 , we combine (7.94) (d), with (1.44), (1.45),

|σ9k=0 | ≤∑

K∈T h

∫K

((A0(x, �u,∇�u)−A0(x, �u,∇�u)

), �v)

qdx

≤ L∑

K∈T h

∫K

(1 + ‖∇�u‖n×q)‖∇�u−∇�u‖n×q‖�v‖q dx

≤ L∑

K∈T h

(1 + |�u|W 1,∞(K))|�u− �u|H1(K)|�v|L2(K)

≤ L(1 + |�u|W 1,∞(T h,Rq))|�u− �u|U1,e|�v|L2(T h,Rq). (7.253)

Since BA(�u,∇�u) is independent of A0 we do not need a σ10k=0 , but only σ10. Similarlyas in (7.238), (7.252), we obtain with d(e) in (7.28),

|σ10| ≤ nL∑

e∈T h

∫e

(‖ [�v] ‖q‖∇�u−∇�u‖n×q

)dS

≤ nL

⎛⎝∑e⊂T h

∫e

‖ [�v] ‖2qd(e)/cw

dS

⎞⎠1/2⎛⎝∑e⊂T h

d(e)cw

∫e

(‖∇�u−∇�u‖n×q)2 dS

⎞⎠1/2

≤ nL/(cw)1/2 (Jσh (�v,�v))1/2

⎛⎝ ∑K∈T h

hK |�u− �u|2H1(∂K)

⎞⎠1/2

.


For the first term in (7.250) we combine the bounds for the partials of the functionsAk, k = 1, . . . , n in (7.94) with (7.248), hence σ11 is split:

|ah(�u; �u,�v)| − ah(�u; �u,�v)| (7.254)

≤

∣∣∣∣∣∣∑

K∈T h

∫K

n∑k=0

((Ak(x, �u,∇�u)−Ak(x, �u,∇�u)

), ∂k�v

)qdx

∣∣∣∣∣∣︸︷︷︸:=σ11=:σ11k>0+σ11k=0

+

∣∣∣∣∣∣∑

e∈T h

∫e

([�v], ({BA(�u,∇�u )} − {BA(�u,∇�u )}, ν)n )q) dS

∣∣∣∣∣∣︸︷︷︸:=σ12

.

We obtain with the different bounds for the partials of the functions Ak for k = 0 andk = 1, . . . , n, and for �u ∈W 1,∞(T h,Rq) for the σ11k>0 and σ11k=0

|σ11k>0 | ≤ L‖�u− �u‖L2(T h,Rq)(1 + ‖�u‖W 1,∞(T h,Rq))‖�vh‖U1,e(7.255)

|σ11k=0 | ≤ L‖�u− �u‖L2(T h,Rq)(1 + ‖�u‖2W 1,∞(T h,Rq))‖�vh‖L2(T h,Rq).

A combination of (7.94) (c) with the ideas in (7.202)–(7.203) allows the estimationof σ12. In (7.256), we need ‖∇�u‖L∞(e) ≤ ‖∇�u‖L∞(K) valid for �u ∈W 2,∞(T h,Rq), cf.Theorem 1.38.

|σ12| ≤ nL∑

K∈T h

( ∑e⊂∂K

∫e

‖[�v] ‖2qd(e)/cw

dS

)1/2

(7.256)

×( ∑

e⊂∂K

d(e)cw

∫e

‖(�u− �u) ‖2q dS)1/2

(1 + ‖∇�u‖L∞(e))

≤ nL/(cw)1/2 (Jσh (�v,�v))1/2

⎛⎝ ∑K∈T h

hK‖�u− �u‖2L2(∂K)

⎞⎠1/2

(1 + |�u|W 1,∞(T h,Rq)).

Adding the estimates for σ11 to σ12 in (7.252)–(7.256), proves the claim. �

As consequence of Proposition 7.32 we obtain with �u = Phu as in Theorem 7.31

Theorem 7.33. Consistency for the quasilinear ah: The ah(�u,�vh) in (7.248) withproperties (7.94) are modified Lipschitz-continuous with respect to �u, and linearand bounded with respect to �vh. Furthermore the ah(Ph�u,�vh) are consistent withthe ah(�u,�vh). So for sufficiently small h < 1, and ‖�u‖U1,e

, ‖�u‖U1,e≤M, a constant

C16 := Lmax{2 + 2n2C5C7(1 + h2)/cw, c

2e

}1/2 exists for ah(�u,�vh), s.t. ∀ �vh ∈ Shq and


∀ �u ∈W 2,∞(T h,Rq), the Sh-interpolant, Ph�u from Theorem 7.7, cf. (7.234):

|ah(�u,�vh)− ah(Ph�u,�vh)| ≤ C16hmin{d,s}−1|�u|Hmin{d,s}(T h,Rq) (7.257)

×(1 + ‖�u‖W 1,∞(T h,Rq) + ‖�u‖2W 1,∞(T h,Rq)

)‖�vh‖.

Proof. Again we apply the slightly modified error estimates in (7.222):

‖Ph�u− �u‖U1,e≤ C7

⎛⎝ ∑K∈T h

(h

min{d,s}−1k |�u|Hmin{d,s}(K,Rq)

)2

⎞⎠1/2

×

⎛⎝ ∑K∈T h

hK |�u− Ph�u|2H1(∂K)

⎞⎠1/2

≤√

2C5C7

⎛⎝ ∑K∈T h

(h

min{d,s}−1k |�u|Hmin{d,s}(K,Rq)

)2

⎞⎠1/2

‖Ph�u− �u‖L2(T h,Rq) ≤ C7

⎛⎝ ∑K∈T h

(h

min{d,s}k |�u|Hmin{d,s}(K,Rq)

)2

⎞⎠1/2

×

⎛⎝ ∑K∈T h

hK‖�u− Ph�u‖2L2(∂K)

⎞⎠1/2

≤√

2C5C7

⎛⎝ ∑K∈T h

(h

min{d,s}k |�u|Hmin{d,s}(K,Rq)

)2

⎞⎠1/2

.

These estimates are applied to the error terms in (7.249) to get

|ah(�u,�vh)− ah(Ph�u,�vh)|

≤ max{C7,√

2C5C7}hmin{d,s}−1|�u|Hmin{d,s}(T h,Rq)

×(L|�vh|U1,e

+ L(1 + |�u|W 1,∞(T h,Rq))|�vh|L2(T h,Rq)

+nL/(cw)1/2√

2C5C7

(Jσ

h (�vh, �vh))1/2

+L(1 + ‖�u‖W 1,∞(T h,Rq) + ‖�u‖2W 1,∞(T h,Rq)

)‖�vh‖U1,e

+nLh/(cw)1/2√

2C5C7

(Jσ

h (�vh, �vh))1/2(1 + ‖�u‖W 1,∞(T h,Rq))

).


This implies with Ph�u ∈W 2,∞(Ω,Rq) and the equivalence of ‖�v‖ and ‖�v‖J‖H1(T h,Rq)‖

with ce in (7.154)

|ah(�u,�vh)− ah(Ph�u,�vh)| ≤ max{C7,√

2C5C7

}hmin{d,s}−1|�u|Hmin{d,s}(T h,Rq)

‖vh‖(L(2 + |�u|W 1,∞(T h,Rq)

)+ nL/(cw)1/2

√2C5C7 + Lce

(1 + ‖�u‖W 1,∞(T h,Rq)

+ ‖�u‖2W 1,∞(T h,Rq)

)+ nLh/(cw)1/2

√2C5C7

(1 + ‖�u‖J

W 1,∞(T h,Rq)

))≤ Lmax

{2 + 2n2/cwC5C7(1 + h2), c2e

}1/2hmin{d,s}−1|�u|Hmin{d,s}(T h,Rq)‖vh‖(

1 + ‖�u‖W 1,∞(T h,Rq) + ‖�u‖2W 1,∞(T h,Rq)

).

This shows the claim (7.257). �

7.11.5 Consistency of the quasilinear ah for the equations of Houston,Robson, Suli, and for systems

We turn to consistency for the DCGMs for the quasilinear problem G(u) = 0 inHouston et al. [407], again only for the principal part. All other terms are equalto those for the semilinear problem. As an essential tool for consistency, we refer toBarett and Liu [68], Lemma 2.1: From (7.103)–(7.106) Houston et al. [407] obtainfor −1 ≤ θ ≤ 1

|(Ba(u)∇u−Ba(v)∇v)(x)| ≤ C|(∇u−∇v)(x)| ∀x ∈ Ω,∀u, v ∈ Ue. (7.258)

We have

Proposition 7.34. The ah(u, v) in (7.107) with properties (7.103), and C in (7.258),and C∗ := 4MμC5(1 + C6), are modified Lipschitz-continuous with respect to u andlinear and bounded with respect to v, such that ah(u, v) satisfies ∀ u, u, v ∈ Ue,

|ah(u, v)− ah(u, v)| ≤ nC|u− u|H1(T h)|v|H1(T h) (7.259)

+ 2nC/(cw)1/2 (Jσh (v, v))1/2

⎛⎝ ∑K∈T h

hK |u− u|2H1(∂K)

⎞⎠1/2

+C∗|v|H1(T h)

⎛⎝ ∑K∈T h

h−1K |u− u|2L2(∂K)

⎞⎠1/2

.

Remark 7.35. In (7.107) we have replaced our usual

θ∑

e∈T h

∫e

({Ba(u)∇v}, ν)n [u] dS = θ∑

e∈T h

∫e

({μ(x, |∇u|)∇v}, ν)n [u] dS (7.260)


by the form [407]

θ∑

e∈T h\∂Ω

∫e

({μ(x, h−1|[u]|)∇v}, ν)n [u] dS + θ∑

e∈∂Ω

∫e

({μ(x, h−1|u− uD|)∇v} . . . .

Our form (7.260) yields the term(∑

K∈T h h−1K |u− u|2H1(∂K)

)1/2 in the last line of(7.259). This would reduce the consistency order by 1, unless we restrict the methodto θ = 0. The above form [407] yields

(∑K∈T h h

−1K |u− u|2L2(∂K)

)1/2, and thus the

desired order of consistency.

Proof.

1. We estimate the different terms σj in (7.261) ff. and obtain with (7.107)

ah(u, v)− ah(u, v)|

≤

∣∣∣∣∣∣∑

K∈T h

∫K

(Ba(u)∇u−Ba(u)∇u,∇v)n dx

∣∣∣∣∣∣︸︷︷︸:=σ4

(7.261)

+

∣∣∣∣∣∣∑

e∈T h

∫e

(({Ba(u)∇u−Ba(u)∇u}, ν)n [v]

)dS

∣∣∣∣∣∣︸︷︷︸:=σ5

+ θ

∣∣∣∣∣∣∑

e∈T h\∂Ω

∫e

(μ(x, h−1|[u]|) [u]− μ(x, h−1|[u]|) [u]

)({∇v}, ν)n dS +

∑e∈∂Ω

. . .

∣∣∣∣∣∣︸︷︷︸:=θσ5

.

2. σ4 and σ5 are estimated, similarly to the previous cases. With (7.258), anappropriately reordered discrete Chebyshev inequality (1.43), and the Cauchy–Schwarz inequality, (1.44), for p = q = 2, cf. (7.185), we obtain

|σ4| ≤∑

K∈T h

∫K

|(Ba(u)∇u−Ba(u)∇u,∇v)n| dx (7.262)

≤ C∑

K∈T h

∫K

|∇u−∇u ||∇v| dx

≤ nC∑

K∈T h

|u− u|H1(K)|v|H1(K)

≤ nC|u− u|H1(T h)|v|H1(T h).


Similarly as in (7.202), we obtain for σ5 with (7.258), the Chebyshev and theCauchy–Schwarz inequalities (1.43), (1.44), for p = q = 2, and d(e) in (7.28),

|σ5| ≤ nC∑

K∈T h

∑e⊂∂K

∫e

|[v]| |∇(u− u)| dS (7.263)

≤ nC∑

K∈T h

( ∑e⊂∂K

∫e

[v]2

d(e)/cwdS

)1/2( ∑e⊂∂K

d(e)cw

∫e

|∇(u− u)|2 dS)1/2

≤ 2nC/(cw)1/2 (Jσh (v, v))1/2

⎛⎝ ∑K∈T h

hK |u− u|2H1(∂K)

⎞⎠1/2

.

3. We start estimating the integrals∫

ein σ5. Then (7.103) implies∣∣∣∣∫

e

(μ(x, h−1|[u]|) [u]− μ(x, h−1|[u]|) [u]

)({∇v}, ν)n dS

∣∣∣∣≤∫

e

∣∣∣∣μ(x, h−1|[u]|) [u]− μ(x, h−1|[u]|) [u]∣∣∣∣|{∇v}| dS

≤∫

e

h

∣∣∣∣μ(x, h−1|[u]|)h−1[u]− μ(x, h−1|[u]|)h−1[u]∣∣∣∣|{∇v}| dS

≤∫

e

hMμh−1|[u]− [u]||{∇v}| dS =

∫e

Mμ|[u− u]||{∇v}| dS.

Summation yields

|σ5| ≤∑

e∈T h\∂Ω

∫e

Mμ|[u− u]||{∇v}| dS

≤ 4Mμ

∑e∈T h\∂Ω

∫e

|u− u||∇v| dS

≤ 4Mμ

⎛⎝ ∑e∈T h\∂Ω

∫e

hK |∇v|2 dS

⎞⎠1/2⎛⎝ ∑e∈T h\∂Ω

∫e

h−1K |u− u|2 dS

⎞⎠1/2

≤ 4MμC5(1 + C6)|v|H1(T h)

⎛⎝ ∑K∈T h

h−1K |u− u|2L2(∂K)

⎞⎠1/2

. (7.264)

For the last estimate(∑

e∈T h\∂Ω

∫ehK |∇v|2 dS

)1/2≤ C5(1 + C6)|v|H1(T h) wehave combined (7.129), (7.138). Similarly for θ

∑e∈∂Ω

∫e({μ(x, h−1|u−

uD|)∇v} . . . .Adding the estimates for σ4, σ5 and σ5 proves the claimed (7.259). �


As a consequence of Proposition 7.34 we obtain the consistency for the quasilin-ear equations. These results can be directly extended to the system of the form,cf (7.102),

−∇ · (μ(x, |∇�u|)∇�u) = −n∑

k=1

∂k(μ(x, |∇�u|)∂k�u

)= �g, (7.265)

where �u = (u1, . . . , uq) : Ω→ Rq, �g ∈ L2(Ω)q and the previous μ ∈ C(Ω× [0,∞)) is areal function satisfying the assumption (7.4). Here we put |∇�u|2 =

∑qi=1 |∇ui|2. Using

Barett and Liu [68], Lemma 2.1, and (7.102)–(7.106) Houston et al. [407] obtain for−1 ≤ θ ≤ 1 similarly as in (7.258)

|(Ba(�u)∇�u−Ba(�v)∇�v)(x)| ≤ C|(∇�u−∇�v)(x)| ∀x ∈ Ω,∀�u,�v ∈ Ue, (7.266)

where Ue is given by (7.92) and

Ba(�v)∇�v = μ(x, |∇�v|)∇�v. (7.267)

Therefore, replacing (7.258) by (7.266) we immediately obtain the consistency resultsrepresented by Proposition 7.34 and Theorem 7.36 also for system (7.265).

Theorem 7.36. Consistency for the quasilinear ah in [407] and for systems:

1. The ah(uh, vh) in (7.106) with properties (7.103) are Lipschitz-continuous withrespect to u and linear and bounded with respect to vh. Furthermore theah(Phu, vh) = ah(Phu, vh) are consistent with the ah(u, vh). So a constant C∗

1

exists, cf. (7.269) such that, ∀ h < 1, ∀ u ∈ Ue, vh ∈ Sh = Vh,

|ah(u, vh)− ah(Phu, vh)| ≤ C∗1h

min{d,s}−1|u|Hmin{d,s}(T h)‖vh‖. (7.268)

2. These results remain correct, if (7.102) is replaced by the system in (7.265) ff.,and the uh, vh, Ba(v)v in ah(u, vh) by �uh, �vh, Ba(�v)�v to obtain ah(�u,�vh).

Proof. The proof proceeds totally analogously to that of Theorem 7.31. With u = Phuwe obtain:

|ah(u, vh)− ah(Phu, vh)| ≤

⎛⎝ ∑K∈T h

(hmin{dK ,s}−1k |u|Hmin{dK ,s}(K))

2

⎞⎠1/2

(7.269)

(nCC7|vh|H1(T h) + 2nC

√2C5C7/cw

(Jσ

h (vh, vh))1/2

+ 4nMμC5(1 + C6)√

2C5C7|vh|H1(T h)

).

7.12. Convergence for DCGMs 527

≤ C∗1

⎛⎝ ∑K∈T h

(h


)2

⎞⎠1/2

‖vh‖

with C∗1 := n

√[(C2C2

7 + 8C2C5C7/cw + 32M2μC

35C7(1 + C6)2

].

Thus we have proved the estimate (7.268). �

7.12 Convergence for DCGMs

We achieve convergence in the following way. The Vh-coercivity and boundedness ofthe discrete principal parts (Gh)′p(u

h) of the linearized operators G′(u) plus the penaltyterm with respect to the penalty norm ‖vh‖, implies its stability, cf. Theorems 7.16–7.19. Its combination with consistency allows the proof of the stability of the discretederivatives (Gh)′(uh), for boundedly invertible G′(u0), and hence of the discretenonlinear Gh. With the consistency and some technical conditions this yields thedesired convergence, cf. Theorem 3.21, 3.23, 3.29.

For consistency, we tested all the ah(uh, vh), bh(uh, vh), ch(uh, vh), �h(uh, vh),Jσ

h (uh, vh), by vh. They are linear and bounded in vh with respect to the ‖ ·‖, but some are nonlinear in uh. The orders of consistency are different forthese forms and the constants in the estimates depend nonlinearly upon u, cf.Theorems 7.21–7.33.

The proof that a boundedly invertible compact perturbation of the previous prin-cipal parts generates a stable discretization is pretty technical. However, we applyexactly the same technique and ideas for DCGMs as for FEMs with variationalcrimes, only indicated here. We use the notation U for H1(Ω,Rq), q ≥ 1, Ue forH3/2+ε(T h,Rq), and Uo

e for H3/2+ε(Ω,Rq) here and below.In a first step towards general stability we define, similarly to nonconforming FEs,

an anticrime transformation from uh ∈ Uh to uh = Ehuh ∈ U . In Lemma 5.77 wehad combined the ah(·, ·)-coercivity for specific a(·, ·), implying the stability, withits consistency, hence the convergence, however here in the opposite direction of theexact solution uh = Ehuh to the chosen discrete uh ∈ Uh. For nonconforming FEswe reduced continuity and boundary conditions along the full edges only to a fewappropriate points along these edges. For the DCG elements we instead consider thepenalty terms.

Lemma 7.37. Anticrime transformation for DCGMs: We choose the boundedlyinvertible Laplacian, with the inhomogeneous A = Δ− g : U → U ′, and Dirichlet con-ditions on ∂Ω. Then its IIPG form with θ = 0, the Ah : Uh → U ′h is stable andconsistent with A for every u ∈ U .

For a given uh ∈ Uh define fh := Ahuh ∈ Ue with ‖fh‖U ′e≤ C‖uh‖, cf. (7.114).

Define Eh : Uh → U by 〈AEhuh, v〉 = 〈fh, v〉∀v ∈ Vb and the boundary condition (uh −uD)|∂Ω = 0. Then we obtain convergence as limh→0 ‖Ehuh − uh‖ = 0.


Proof. In contrast to the previous nonlinear problems, we could omit here the∩L∞(T h) in the definition of Ue = H3/2+ε(T h) ∩ L∞(Ω), etc. That the IIPG formAh : Uh → U ′h is stable and consistent with A is proved above.

With the ah(u, v), . . . , Jσh (u, v) in (7.57)–(7.61), ∂ΩN = ∅, and our notation, e.g.

〈Ahu, v〉 = 〈Ahu, v〉U ′e×Ue

, we define the weak operators A,Ah, Ah, as

(a) A : U → U ′ with (u− uD)|∂Ω = 0, Ah : Ue → U ′e, Ah : Uh = Sh → U ′h, (7.270)

(b) 〈Au, v〉 − a(u, v) + �(v) = 0, (u− uD)|∂Ω = 0 for fixed u ∈ U , ∀ v ∈ Vb,

(c) 〈Ahu, v〉 − ah(u, v)− εJσh (u, v) + �h(v) = 0, fixed u, ∀ v ∈ Ue,

(d) 〈Ahuh, vh〉 − ah(uh, vh)− εJσh (uh, vh) + �h(vh) = 0, fixed uh,∀vh ∈ Sh,

ah(uh, vh) := ah(uh, vh)|Sh×Sh , �h(vh) := �h(vh)|Sh .

Similarly as for nonconforming FEMs, we define, for uh ∈ Sh the fh ∈ S′h by

uh : 〈fh, vh〉 := 〈Ahuh, vh〉 = ah(uh, vh) + εJσh (uh, vh)− �h(vh)∀vh ∈ Sh. (7.271)

In the next step we determine uh := Ehuh ∈ U from Auh = fh. We start with uh, v ∈Ue ⊃ Uo

e , such that the ah(uh, v), εJσh (uh, v), �h(v), fh, are well defined. By uh ∈ Uo

e ,the boundary condition and v ∈ Vb ∩ Uo

e , the terms with jumps, [u] = [v] = 0, andalong the boundary vanish, and we end up with uh ∈ U , and v ∈ Vb. So we determine

Ehuh = uh ∈ U with boundary conditions (uh − uD)|∂Ω = 0 implying (7.272)

〈Auh, v〉 = 〈fh, v〉 = ah(uh, v) + εJσh (uh, v)− �h(v) ∀v ∈ Vb

= ε∑

K∈Th

∫K

(∇uh, ∇v)n dx

− ε∑

e∈∂Ω

∫e

(∇uh, ν)nv + (∇v, ν)nuh

)dS + ε

∑e∈∂Ω

∫e

σuh v dS

−〈g, v〉 − ε∑

e∈∂Ω

∫e

(σuD v − (∇v, ν)n uD

)dS

= ε∑

K∈Th

∫K

(∇uh, ∇v)n dx− 〈g, v〉 = a(uh, v)− �(v) ∀v ∈ Vb.

For this uh ∈ Ue we determine the discrete solution uhh ∈ Sh solving, cf. (7.270),⟨

Auhh, v

h⟩

= ah(uh

h, vh)

+ εJσh

(uh

h, vh)− �h(vh) ∀vh ∈ Sh.

This is Equation (7.271) with uhh replaced by uh. Then the coercivity of ah

(uh

h, vh)

+εJσ

h

(uh

h, vh)

implies uh = uhh and, with the consistency, the convergence limh→0 ‖uh −

uhh‖ = 0 to the chosen discrete uh ∈ Uh. �


This allows the strategy, starting with Lemma 5.77, for proving the stability of Ah

for boundedly invertible linear A as in Theorem 3.29. We reformulate this theorem forour DCGMs.

Theorem 7.38. Stability criterion for DCGMs: Choose Uh = Vh �⊂ U as in(7.65),(7.95). Choose Ph = Ih and Q

′h as in Theorem 7.7, Corollary 7.8, and in(7.113), (7.115), and one of the DCGMs as formulated above. Let A : Ub → V ′ = V ′ beboundedly invertible and be one of the linear(ized) operators for equations or systems,studied in this chapter. These DCGMs for A, the Ah, are consistent with A in a smoothenough u. Then

A−1 ∈ L(V ′,Ub) =⇒ Ah is stable in Phu. (7.273)

Under these conditions the linear Ah is stable.By Theorem 3.23 this stability is inherited for boundedly invertible G′(u0) to ΦhG.

With our above consistency results, Theorem 4.53 implies the unique existence of thediscrete approximation uh

0 and its convergence to the exact solution u0.We summarize, for the different problems and their discrete operators.

Condition 7.39. Conditions for stability and consistency for DCGMs; summary ofthese results for the different previous problems:

1. Choose Uh = Vh = Sh �⊂ U as in (7.63), (7.95) with the penalty norm ‖uh‖Vh =‖uh‖ in (7.153) and the parameter cw as in (7.39), (7.164). For b(·, ·) we imposeAssumption (H), cf. (7.51)–(7.53), and for our DCG elements Sh of local degreed− 1 the conditions (A0)–(A2), (7.124)–(7.128), (7.146). In particular, the Sh areapproximating spaces for the function spaces considered here. We always denotethe Sh-interpolant for u from Theorem 7.7 as Phu.

2. We consider the nonsymmetric, the symmetric or the incomplete or generalinterior penalty Galerkin method, the NIPG, SIPG, IIPG or GIPG method,respectively, with θ = −1, 1, 0 or θ ∈ [−1, 1].

3. For all problems, the operator G and its linearization in u0, the G′(u0)u = f ∈ U ′,has to be combined with the boundary condition (7.14), so u |∂ΩD

= uD on ∂ΩD,and ∂u/∂ν |∂ΩN

= φ on ∂ΩN . In particular, we require for all cases the boundedinvertibility of G′(u0) with the Dirichlet and Neumann or only Dirichlet boundaryconditions, e.g. (7.14), cf. Theorem 7.14.

4. If in the consistency results a |u|Hmin{d,s}(Ω) is shown, we always requireu ∈ Hmin{d,s}(Ω); for the hp-methods in Section 7.14 we replace this by|u|Hmin{dK ,s}(K) for u ∈ Hmin{dK ,s}(K), respectively Throughout this section weuse the abbreviation

C(h, u) := hmin{d,s}−1|u|Hmin{d,s}(Ω). (7.274)

5. For the special convection–diffusion problem (7.13), (7.14) we impose (7.16), andchoose a numerical flux satisfying Assumption (H): (7.51)–(7.53). Define itsstandard DCGM as in (7.70). Then coercivity and stability is guaranteed byTheorem 7.16, (7.165), (7.166). The following consistency estimates are valid, cf.


Theorems 7.10, 7.21, 7.22, 7.23, 7.26,

|ch(u, vh)− ch(Phu, vh)| ≤ C10C(h, u)‖vh‖ for Dirichlet boundary cd.,(7.275)

|ch(u, vh)− ch(Phu, vh)| ≤ C10C(h, u)(1 + ‖vh‖) for Dir. and Neumann b.cs.

|ah(u, vh)− ah(Phu, vh)| ≤ ε C17C(h, u)‖vh‖,∣∣ε Jσh (u, vh)− ε Jσ

g (Phu, vh)∣∣ ≤ ε C15C(h, u)‖vh‖

and �h(vh)− �h(vh) = 0 ∀vh ∈ Vh = Sh.

6. The nonstandard DCGM for this problem (7.13), (7.14), (7.16) is defined in(7.71). The coercivity and consistency estimates for the ah, bh, �h are the sameas for the previous ah, ch, �h.

7. For the linear operator (7.74), (7.75) with its conditions we define its DCGMin (7.77). Then coercivity (and stability) and the consistency estimates areguaranteed by obvious modifications of Theorems 7.18, and 7.10, 7.23, 7.31.Modifying the previous consistency estimates requires changing the constant in(7.275), (7.276).

8. For the semilinear convection–diffusion operator (7.78), (7.79) we impose (7.80),and define its DCGM as in (7.89).Then coercivity (and stability) is guaranteedby Theorems 7.10, 7.18, (7.181), (7.182) for cw in (7.180). For u ∈W 2,∞(T h), weobtain the following consistency estimates, cf. Theorems 7.10, 7.21, 7.24, 7.31,here (7.204), (7.211), (7.245),

|ch(u, vh)− ch(Phu, vh)| ≤ C10C(h, u)‖vh‖ for Dirichlet boundary cd.,(7.276)

|ch(u, vh)− ch(Phu, vh)| ≤ C10C(h, u)(1 + ‖vh‖) for Dir. and Neumann b.cs.

|ah(u, vh)− ah(Phu, vh)| ≤ εC(h, u)(C∗

1 + 2C∗2‖u‖W 1,∞(T h)

)‖vh‖, (7.277)∣∣ε Jσ

h (u, vh)− ε Jσg (Phu, vh)

∣∣ ≤ ε C15C(h, u)‖vh‖

|�h(Phu, vh)− �h(u, vh)| ≤ C13C(h, u)(1 + ‖u‖L∞(∂Ω)

)‖vh‖H1(T h)∀vh ∈ Vh.

9. For the quasilinear system (7.93), we impose (7.94), and define its DCGM as in(7.101). Then coercivity (and stability) is guaranteed by Theorem 7.19, (7.195),(7.196) for cw in (7.194). For u ∈W 2,∞(T h,Rq), we get the following consistencyestimates, cf. Theorems 7.10, 7.23, 7.33, here (7.257),

|ah(�u,�vh)− ah(Ph�u,�vh)| ≤ C16C(h, u) (7.278)(1 + ‖�u‖W 1,∞(T h,Rq) + ‖�u‖2W 1,∞(T h,Rq)

)‖�vh‖,∣∣Jσ

h (u, vh)− Jσg (Phu, vh)

∣∣ ≤ C15C(h, u)‖vh‖

�h(vh)− �h(vh) = 0 ∀vh ∈ Vh = Sh.


10. For the special quasilinear equation in (7.102), Houston et al. [407] formulatetheir DCGM according to (7.109); for systems they provide all necessary tools,but do not formulate (7.110). Under the conditions (7.103)–(7.105), the coercivity(and stability) is guaranteed by Theorem 7.19, (7.195), (7.196) for cw in (7.194).The following consistency estimates are valid, cf. Theorem 7.10, 7.23, 7.36. Soinstead of ah(·, ·), . . . , in (7.278) we now formulate for the Gh, analogous to theGh in (7.109), (7.110), cf. (7.268),

|Gh(u, vh)−Gh(Phu, vh)| ≤ C∗1C(h, u)‖vh‖ ∀θ ∈ [−1, 1]. (7.279)

We summarize the convergence results for all DCGMs in Theorem 7.40. In par-ticular, we have the same order of convergence for the DCGMs as for the classicalFEMs. It is straightforward along the lines of the above proofs, to prove consistentdifferentiability as well. This is the condition for the mesh independence principle inTheorem 3.40.

Theorem 7.40. Convergence for the DCGMs for the above cases 5–10: Under theprevious conditions and notation (1)–(4) and for inhomogeneous Dirichlet and/orNeumann boundary conditions choose, for the problems in (5)–(10), the quotedDCGMs with the above coercivity. Hence stability results for a boundedly invertibleG′(u0), and the consistency estimates (7.274)–(7.279) are granted.

Then for the G(u0) = 0, the application of these DCGMs define the Gh(uh

0

)= 0.

Then for sufficiently small h, a unique solution uh0 ∈ Sh exists near u0 and∥∥uh

0 − u0

∥∥J

H1(T h)≤ CC(h, u0), with C(h, u0) := hmin{d,s}−1|u0|Hmin{d,s}(Ω) (7.280)

with C depending upon u0 for problems (8)–(10). There, we have to impose additionallyu0 ∈W 2,∞(T h).

For the special quasilinear equation (system) in [407] we formulate:

Theorem 7.41. Convergence via monotony for the DCGMs for case 10: We assumethe previous conditions and notation (1), (2), (4) and (7.103), consequently (7.105).Then for inhomogeneous Dirichlet and/or Neumann boundary conditions choosethe DCGMs in (7.109) and (7.110) for equations and systems, respectively, with theconsistency estimates (7.279). Then there exist unique solutions u0 and uh

0 ∈ Sh forG(u0) = f and Gh

(uh

0

)= Qhf , respectively These uh

0 converge to u0 according to∥∥uh0 − u0

∥∥J

H1(T h)≤ CC(h, u0), with C(h, u0) in (7.280). (7.281)

Proof. Under the above conditions, Lemmas 2.2 and 2.3 in [407] show that Gh

is Lipschitz-continuous and monotone with respect to uh in W 1,p(T h). Therefore,Theorem 2.68 implies the unique existence of the solutions, and Theorem 4.67 (5),and Theorems 7.5 ff., and Remark 7.4, show the convergence according to (7.281). �

Theorem 7.42. Discontinuous Galerkin methods for monotone operators, eigenvalueproblems, nonlinear boundary operators, quadrature approximations: Under theconditions of Theorem 7.40, discontinuous Galerkin methods applied to the following


problems yield unique converging solutions for variational methods for eigenvalueproblems, cf. Section 4.7, methods for nonlinear boundary operators, cf. Section5.3, quadrature approximate methods, cf. Section 5.4. For all these problems (7.280)remains valid.

Remark 7.43. By the appropriate modifications, cf. Section 7.14, these three theo-rems remain valid for hp-methods.

7.13 Solving nonlinear equations in DCGMs

7.13.1 Introduction

For solving the nonlinear DCG equations, we restrict ourselves to presenting a discreteNewton method based upon the mesh independence principle (MIP). It is formulatedand proved only for quasilinear systems. Except for the modified Lipschitz conditionfor the derivatives, the results are the same for semilinear equations as well.

For this MIP we have to determine the linearized form of the quasilinear systemand its discretization, based upon the same Sh as before. This derivative has to satisfythe following property, cf. (3.83). A general discretization method is differentiablyconsistent and of order p, if it is classically consistent and if for a Frechet differentiablenonlinear operator G,

‖(Gh)′(Phu)Phv −Q′h(G′(u)v)‖V′h → 0 and (7.282)

‖(Gh)′(Phu)Phv −Q′h(G′(u)v)‖V′h ≤ Chp(1 + ‖u‖Us

)‖v‖Usfor h→ 0.

These ‖v‖Usare usually ‖v‖W k,q(Ω) and p, k, q appropriately chosen. For our most

complicated DCGM, essentially including the previous problems as special cases, weprove this property.

7.13.2 Discretized linearized quasilinear system and differentiableconsistency

Hence we have to determine the Frechet derivatives of the ah(·, ·) in (7.93), (7.285).With the Ak(·, �u1,∇�u1) in (7.93), we define these weak forms, cf.(7.94), (7.93),

〈G′(�u1)�u,�v〉 :=∑

K∈T h

∫K

n∑k=0

(∂Ak

∂�u(·, �u1,∇�u1)�u +

∂Ak

∂�u(·, �u1,∇�u1)∇�u, ∂k�v

)q

dx

=∑

K∈T h

∫K

n∑k=0

(A′

k(·, �u1,∇�u1)∇0�u, ∂k�v)qdx (7.283)

=∑K∈T

∫K

(B′

A(�u1,∇�u1)∇0�u,∇0�v)0qdx,

7.13. Solving nonlinear equations in DCGMs 533

where we used and will use the notation

A′k(·, �u1,∇�u1)∇0�u :=

∂Ak

∂�u(·, �u1(·),∇�u1(·))�u(·)

+∂Ak

∂�u(·, �u1,∇�u1)∇�u ∈ Rq inx ∈ Ω

(B′

A(�u1,∇�u1)∇0�u,∇0�v)0q

:=n∑

k=0

(A′

k(·, �u1,∇�u1)∇0�u, ∂k�v)q∈ R in x ∈ Ω, (7.284)

and (B′

A(�u1,∇�u1)∇0�u, ν)n

:=n∑

k=1

A′k(·, �u1,∇�u1)∇0�uνk ∈ Rq in x ∈ Ω,

Again we omitted, in the nonlinear and consequently in the linearized form of theDCGM, the

( ([B′

A(�u1,∇�u1)∇0�u], ν)n, {�v}

)q, vanishing for �u = �u0 and only consider

the IIPG for θ = 0. We collect the∑

e∈T h\∂Ω and∑

e∈∂Ω into∑

e∈T h . Compared to(7.76), we replace the (Ba∇0u, ν)n by

(B′

A(�u1,∇�u1)∇0�u, ν)n

etc. With this notation,we introduce, similarly to (7.57)–(7.61), the linearized quasilinear, and linear forms,a′h(�u1; �u,�v), �h(�v), and use Jσ

h (�u,�v),

a′h(�u1; �u,�v) :=∑

K∈T h

∫K

(B′

A(�u1,∇�u1)∇0�u,∇0�v)0qdx (7.285)

−∑

e∈T h

∫e

( ({B′

A(�u1,∇�u1)∇0�u}, ν)n, [�v])qdS

�h(�v) := 〈g,�v〉+∑

e∈∂Ω

∫e

σ(�uD, �v)q dS.

For this discrete IIPG method (θ = 0) for the linearized quasilinear problem (7.93),(7.91) we determine the discrete approximate solution �uh

2 such that

(a) �uh2 ∈ Sh, cf. (7.95), (7.286)

(b)⟨(Gh)′

(�uh

1

)�uh

2 , �vh⟩

:= a′h(�uh

1 ; �uh2 , �v

h)

+ Jσh

(�uh

2 , �vh)− �h(�vh) = 0 ∀�vh ∈ Sh.

For quadratic convergence of Newton’s method, Lipschitz-continuous derivatives ofthe operator are necessary. The same holds for its discrete counterpart. We assumefor semilinear problems, in addition to (7.80),

(a) For (b) –(d) we assume |u| ≤ R0 and an L′ = L′(R0) such that (7.287)

(b) fs ∈ C1L′(R) are Lipschitz-continuous with the constant L′,

(c) (∂aij/∂u)(·, u) are Lipschitz-continuous with respect to u with the constant L′,

(d) (∂g/∂u)(·, u) are Lipschitz-continuous with respect to u with the constant L′.


For quasilinear problems, we avoid a formulation analogous to (7.94), by imposingconditions, similar to (7.287), in the form

(a) We assume the |�u| ≤ R0 and an L′ = L′(R0) such that (7.288)

(b) (∂Ak/∂�u)(�u, �u), (∂Ak/∂�u)(�u, �u), (∂A0/∂�u)(�u, �u), (∂A0/∂�u)(�u, �u)

are Lipschitz-continuous with respect to �u, �u, with the constant L′∀|�u| ≤ R0.

We turn to our main goal, the consistency of the terms in (7.286). This has beenproved for the Jσ

h (�uh, �vh) and �h(�vh) in the previous sections; only a′h(�uh

1 ; �uh, �vh)

ismissing. We proceed as above in two steps.

Proposition 7.44. Under the condition (7.288), and for all ‖�u1‖H1(T h) ≤ C ′, thea′h(�u1; �u,�v) are nonlinear and bounded with respect to �u1, and bilinear and boundedwith respect to �u,�v and satisfy the estimate for all �u,�v, �u, �u1 ∈ Ue, �u ∈W 2,∞(T h),and with C0 := L(1 + (C ′)2), L′ cf. (7.94), (7.288), H1(T h) = H1(T h,Rq), . . . ,

∣∣a′h(�u1; �u,�v)− a′h(�u1; �u,�v)∣∣ ≤ C0‖�u− �u‖H1(T h)‖�v‖H1(T h) (7.289)

+ 2nC0

(1cw

)1/2

(Jσh (�v,�v))1/2

×

⎛⎝ ∑K∈T h

hK |�u− �u|2H1(∂K) + hK‖�u− �u‖2L2(∂K)

⎞⎠1/2

+nL′‖�u1 − �u1‖H1(T h)‖�u‖W 1,∞(Ω)‖�v‖H1(T h)

+ 2nL′/(cw)1/2‖�u‖W 1,∞(T h) (Jσh (�v,�v))1/2

×

⎛⎝ ∑K∈T h

hK‖�u− �u‖2L2(∂K)

⎞⎠1/2

.

Proof. The boundedness follows from (7.289) for �u = 0 and use, cf. (7.94), of thebounds for |Ak, A

′k| ≤ C0. We consider the arguments of a′h(�u1; �u,�v) as �u, �u, �u1, �u1, �v ∈

Ue. Then

|a′h(�u1; �u,�v)− a′h(�u1; �u,�v)| ≤∣∣a′h(�u1; �u,�v)− a′h(�u1; �u,�v)

∣∣ (7.290)

+∣∣a′h(�u1; �u,�v)− ah(�u1; �u,�v)

∣∣ .


The first term is estimate via the σ19, σ20 as

|a′h(�u1; �u,�v)− a′h(�u1; �u,�v)| ≤

∣∣∣∣∣∣∑

K∈T h

∫K

(B′

A(�u1,∇�u1)∇0(�u− �u),∇0�v)0qdx

∣∣∣∣∣∣︸︷︷︸:=σ19

+

∣∣∣∣∣∣∑

e∈T h

∫e

( ({B′

A(�u1,∇�u1)∇0(�u− �u)}, ν)n

[�v])qdS

∣∣∣∣∣∣︸︷︷︸:=σ20

.

σ19 is estimated with both Cauchy–Schwarz inequalities (1.44), for p = q = 2, (1.45),

|σ19| ≤ C0

∑K∈T h

‖�u− �u‖H1(K)‖�v‖H1(K) ≤ C0‖�u− �u‖H1(T h)‖�v‖H1(T h).


|σ20| ≤ nC0

∑e∈T h

∫e

|[�v]|q | ∇0�u−∇0�u|q dS

≤ nC0

⎛⎜⎝⎛⎝∑

e∈T h

∫e

([�v], [�v])q

d(e)/cwdS

⎞⎠1/2⎛⎝∑e∈T h

d(e)cw

∫e

| ∇0�u−∇0�u|2q dS

⎞⎠1/2⎞⎟⎠

≤ 2nC0

(1cw

)1/2

× (Jσh (�v,�v))1/2

⎛⎝ ∑K∈T h

(hK |�u− �u|2H1(∂K) + hK‖�u− �u‖2L2(∂K)

)⎞⎠1/2

.

For the second term in (7.290) we introduce σ21, σ22 as

|a′h(�u1; �u,�v) − a′h(�u1; �u,�v)|

≤

∣∣∣∣∣∣∑

K∈T h

∫K

((B′

A(�u1,∇�u1)−B′A(�u1,∇�u1)

)∇0�u,∇0�v

)0

qdx

∣∣∣∣∣∣︸︷︷︸:=σ21

+

∣∣∣∣∣∣∑

e∈T h

∫e

( ({(B′

A(�u1,∇�u1)−B′A(�u1,∇�u1)

)∇0�u

}, ν)n

[�v])

qdS

∣∣∣∣∣∣︸︷︷︸:=σ22

.


Obviously we obtain with the Lipschitz constant, L′, for the BA with (1.43),

|σ21| ≤ nL′‖�u1 − �u1‖H1(T h)‖�u‖W 1,∞(Ω)‖�v‖H1(T h).

A combination with the ideas in (7.212) ff., allows the estimation of σ22. Here we need‖∇0�u‖L∞(e) ≤ ‖�u‖W 1,∞(K), valid for �u ∈W 2,∞(T h) by Theorem 1.38.

|σ22| ≤ 2nL′

⎛⎝∑e∈T h

‖∇0�u‖2L∞(e)

∫e

([�v], [�v])q

d(e)/cwdS

⎞⎠1/2

⎛⎝∑e∈T h

d(e)cw

∫e

n∑i=1

(�u− �u, �u− �u)q dS

⎞⎠1/2

≤ 2nL′/(cw)1/2‖�u‖W 1,∞(T h) (Jσh (�v,�v))1/2

⎛⎝ ∑K∈T h

hK‖�u− �u‖2L2(∂K)

⎞⎠1/2

.

Adding the estimates for σ19–σ22 proves (7.289). �

Proposition 7.44 implies with �u = Ph�u, as in Theorem 7.21

Theorem 7.45. Differentiable consistency for quasilinear DCGMs: We assume�u, �u1 ∈ Ue ⊂ U , �u ∈W 2,∞(T h), �u1 ∈ D(G), with

∥∥�uh1

∥∥H1(T h)

≤ C ′,∥∥�uh

1

∥∥L∞(T h)

≤ R0

and C0 := L(1 + (C ′)2), L′ = L′(R0), cf. (7.94), (7.288) and these latter conditions.Then (Gh)′(Phu1)Phu are consistent with G′(u1)u. More precisely, a′h

(�uh

1 ; �uh, �vh)

are bounded, nonlinear and bilinear with respect to �uh1 , and �uh, �vh. A constant,

C18 := C0ce(2 + nL′‖�u‖W 1,∞(Ω)) max{C7, 2n√

2C5C7/cw} exists, such that for theSh-interpolants Ph�u, Ph�u1 of �u, �u1 in Theorem 7.7,

|a′h(�u1; �u,�vh) − a′h(Ph�u1;Ph�u,�vh)| (7.291)

≤ C18

⎛⎝ ∑K∈T h

(h

min{d,s}−1k |�u|Hmin{d,s}(K)

)2

⎞⎠1/2

‖�vh‖.

With the classical consistency in Theorems 7.10, 7.33, this DCGM for the quasilinearsystem is differentiably consistent.


Proof. The proof is nearly identical to that of Theorem 7.26. For (7.291) and by(7.289) we again use (7.222), however replacing the first two lines by

‖Ph�u− �u‖H1(T h) ≤ 2C7

⎛⎝ ∑K∈T h

(h


)2

⎞⎠1/2

, by (7.139),

×

⎛⎝ ∑K∈T h

hK‖�u− Ph�u‖2H1(∂K)

⎞⎠1/2

(7.292)

≤ 2√

2C5C7

⎛⎝ ∑K∈T h

(h


)2

⎞⎠1/2

,

(7.222), (7.292) are applied to the different error terms �u− Ph�u in (7.289):∣∣a′h(�u1; �u,�vh)− a′h(Ph�u1;Ph�u,�vh)∣∣

≤ C0

⎛⎝ ∑K∈T h

(h


)2

⎞⎠1/2

×(2C7‖�vh‖H1(T h) + 4n

√2C5C7/cw

(Jσ

h (�vh, �vh))1/2

+ nL′C7‖�u‖W 1,∞(Ω)‖�vh‖H1(T h)

+ 2nL′√2C5C7/cwh‖�u‖W 1,∞(T h)

(Jσ

h (�vh, �vh))1/2

)≤ C0ce(2 + nL′‖�u‖W 1,∞(Ω))max

{C7, 2n

√2C5C7/cw

}⎛⎝ ∑

K∈T h

(h


)2

⎞⎠1/2

‖�vh‖.


We modify Theorem 7.45 for the semilinear, in particular, the model problem. Weonly have to replace (7.94), (7.288) by (7.80), (7.287), and C0 by L′.

Theorem 7.46. Differentiable consistency for semilinear and the special quasilinearDCGMs: We assume u, u1 ∈ Ue ⊂ U , u ∈W 2,∞(T h), u1 ∈ D(G), and, the model prob-lem, (7.13), the conditions (7.80), (7.287) (a), (b), for the semilinear problem, (7.78),the conditions (7.80), (7.287), and finally for the special quasilinear problem, (7.102),the conditions (7.288). Then the (Gh)′(Phu1)Phu are consistent with the G′(u1)u.


A C ′18 := ceL

′(2 + nL′ ‖u‖W 1,∞(Ω)) max{C7, 2n

√2C5C7/cw

}exists, such that

|a′h(u1;u, vh) − a′h(Phu1;Phu, vh)| (7.293)

≤ C18

⎛⎝ ∑K∈T h

(h


)2

⎞⎠1/2

‖vh‖.

With the classical consistency in Theorems 7.21–7.31, this DCGM for the semilin-ear systems, (7.13), (7.78), are differentiably consistent. For the special quasilinearproblem, (7.102), slight modifications of the constant C ′

18 are necessary.

We summarize the results of Theorem 3.40 and Corollary 3.41 for our DCGMs

Theorem 7.47. Newton’s method for DCGMs Gh: Let u0 ∈ Hs(Ω) be an isolatedsolution of G(u0) = 0, such that G′(u0) : U → V ′ is boundedly invertible and let Gh

indicate one of the DCGMs for nonlinear problems in Theorems 7.45 and 7.46 with thecorresponding conditions. Start the Newton process with u1 ∈ Hs(Ω), and the discreteNewton process for i = 1, . . . , with uh

1 := Phu1, for small enough ‖u0 − u1‖U andh. That yields

uhi+1 := uh

i −(Gh(uh

i

)′)−1

Gh(uh

i

), (7.294)

cf. (3.84). The uhi+1 in (7.294) uniquely exist and converge quadratically to uh

0 and toui+1 of order min{d, s} − 1, such that∥∥uh

i+1 − uhi

∥∥Uh ≤ C

∥∥uhi − uh

i−1


i − Phui

∥∥Uh ≤ Chmin{d,s}−1‖ui‖Us

, i = 1, . . . .

The following relation between the numbers of iterations of the exact and the discretemethod is the reason for calling this a “mesh independence principle”, MIP. For smallenough η :∣∣min{i ≥ 1 :

∥∥uhi − Phu0

∥∥Uh < η} −min{i ≥ 1 : ‖ui − u0‖U < η}

∣∣ ≤ 1. (7.295)

7.14 hp-variants of DCGM

For many important problems the local smoothness of the solution strongly varies inΩ. It is appropriate to choose the local degrees according to this smoothness. DCGMsare based upon a discontinuous piecewise polynomial approximation. So there is noprincipal obstacle for using these different degrees of polynomial approximation ondifferent elements.

The corners of the domain and singularities of the data are particularly interesting.The real power of hp-methods, e.g. the exponential convergence, strongly depends upona careful combination of the analytic structure of the singularities of the solution withmesh refinement strategies. In Chapter 6 we have discussed these problems for FEMs.Here we only formulate the necessary tools for hp-methods.

7.14. hp-variants of DCGM 539

We list a few from the many papers published for hp DCGMs, by B. Cockburn,P. Houston, M. Luskin, I. Perugia, D. Schotzau, Ch. Schwab, C-W. Shu, E. Suli, andT.P. Wihler [193,405,408,410–412].

7.14.1 hp-finite element spaces

To each K ∈ T h, we assign a positive integer sK (local Sobolev index) and a positiveinteger pK (local polynomial degree). Then we define a new type of vector

s ≡ {sK ,K ∈ T h}, p ≡ {pK ,K ∈ T h}. (7.296)

We generalize the above broken Sobolev space to the vector s and triangulation T h

Hs(T h) ≡ {v : v|K ∈ HsK (K) ∀K ∈ T h} (7.297)

with the norm

‖v‖Hs(T h) ≡

⎛⎝ ∑K∈T h

‖v‖2HsK (K)

⎞⎠1/2

(7.298)

and the seminorm

|v|Hs(T h) ≡

⎛⎝ ∑K∈T h

|v|2HsK (K)

⎞⎠1/2

(7.299)

for the Sobolev space HsK (K) ≡W sK ,2(K). If, as in (7.63), (7.95), sK = d− 1 ∀K ∈T h, d− 1 ∈ N then we use the notation Hd−1(T h) = Hs(T h). Obviously,

Hs(T h) ⊂ Hs(T h) ⊂ Hs(T h), (7.300)

where s = max{sK , sK ∈ s} and s = min{sK , sK ∈ s}.Furthermore, we define the space of discontinuous piecewise polynomial functions

associated with the vector p by

Shp ≡ {v : v ∈ L2(Ω), v|K ∈ PpK−1(K) ∀K ∈ T h}, (7.301)

where PpK−1(K) denotes the space of all polynomials on K of degree ≤ pK − 1, K ∈T h. For deriving a priori hp error estimates we assume that there exists a constantCP ≥ 1 such that

pK

pK′≤ CP ∀K,K ′ ∈ T h such that K,K ′ have a common face. (7.302)

The assumption (7.302) may seem to be rather restrictive. However, we suppose thatan application of hp-methods to practical problems is efficient and accurate when thepolynomial degrees of approximation on neighboring elements do not differ too much.Moreover, to each K ∈ T h we define the parameter

dhp(K) :=hK

p2K

, K ∈ T h. (7.303)

Another approach was presented by Dryja et al. [304] where the harmonic average ofhK and pK of neighboring elemenents was used. Let e ∈ T h be an inner face shared


by two elements Kel and Ke

r of T h. Then we put

dhp(e) := min (dhp (Kel ) , dhp (Ke

r )) . (7.304)

For a boundary face e ∈ ∂Ω ∩Kel we put

dhp(e) := dhp (Kel ) . (7.305)

7.14.2 hp-DCGMs

Since the essential differences for the hp-DCGMs are well demonstrated for the modelproblem (7.13)–(7.15), we restrict the discussion to this case. Then many of the ideasfor adaptive FEMs in Chapter 6 can be applied to DCGMs as well. Similarly as in(7.70) we define the standard hp-DCGM and its discrete approximate solution as afunction uh

0 : T h → R satisfying the conditions

(a) uh0 ∈ Shp, (7.306)

(b)(Gh(uh

0

), vh)

:= ah(uh

0 , vh)+ ch

(uh

0 , vh)+ εJσ

h

(uh

0 , vh)− �h(vh) = 0 ∀ vh ∈ Shp,

with ah(uh, vh) := ah(uh, vh)|Shp×Shp , ch(uh, vh) := ch(uh, vh)|Shp×Shp ,

and �h(vh) := �h(vh)|Shp

and the forms ah, ch, �h and Jσh are defined by (7.57)–(7.61). The only difference

occurs in the definition of the penalty parameter σ: here (7.39) is replaced by

σ|e =cw

dhp(e)e ∈ T h (7.307)

where dhp(e) is given by (7.304) or (7.305) and cw > 0.Similarly, the hp-DCGMs are defined by a formal exchange of Sh by Shp for the

general linear (7.77), semilinear (7.89) and quasilinear (7.286) problems consideredwithin this chapter. This is a great advantage for DCGM.

7.14.3 hp-inverse and approximation error estimates

For analysing the hp-variant of DCGM we have to restrict the discussion to elements,K, which are simplices and/or parallelograms. We allow (n− 1)- dimensional facese to be shared by more than two elements, cf. (7.304). We still do not require theconforming properties, usually imposed upon FEMs, but the elements cannot be asgeneral as in (7.27) ff. In particular, a triangulation, T h = {K}K∈T h (h > 0), is apartition of Ω into a finite number of open n-dimensional mutually disjoint simplexesand/or parallelograms K, hence star-shaped elements. We assume that T h is

� locally quasiuniform, i.e. there exists a constant CQ > 0 such that

hK ≤ CQ hK′ ∀K,K ′ ∈ T h sharing a face e ∈ T h \ ∂Ω; (7.308)� shape-regular, i.e. there exists a constant CS > 0 such that, cf. (7.126),

hK ≤ CS ρK ∀K ∈ T h, (7.309)

where ρK is the radius of the largest n-dimensional ball inscribed into K.


The assumptions (7.308) and (7.309) mean that the diameters of two neighboringelements do not differ too much and that elements do not degenerate, respectively.

The definitions (7.304)–(7.305) and the assumptions (7.302) and (7.308) imply that

dhp(e) ≤ dhp(K) and1

dhp(e)≤ CQC

2P

1dhp(K)

∀e ⊂ ∂K, K ∈ T h. (7.310)

The multiplicative trace inequality (7.129) is still valid for this section. On the otherhand, we need hp-variants of the inverse inequality and approximation error estimates.

Theorem 7.48. Inverse inequality: There exists a constant CI > 0 independent of v,h and K such that

|v|H1(K) ≤ CIp2

K

hK‖v‖L2(K), v ∈ PpK

(K), K ∈ T h, h ∈ (0, h0). (7.311)

Proof. Let K be a reference triangle or square for K ∈ T h, and FK : K → K be anaffine mapping such that FK(K) = K. From Proposition 5.41 it follows that if thefunction v ∈ Hm(K), N � m ≥ 0, then v(x) = v(FK(x)) ∈ Hm(K) and

|v|Hm(K) ≤ cchn/2−mK |v|Hm(K), (7.312)

|v|Hm(K) ≤ cchm−n/2K |v|Hm(K), (7.313)

where cc > 0 depends on CS but not on K and v. From [575], Theorem 4.76 we have

|v|H1(K) ≤ csp2K‖v‖L2(K), v ∈ PpK

(K), (7.314)

where cs > 0 depends on n but not on v and pK . A simple combination of (7.312)–(7.314) proves (7.311) with CI = csc

2c . �

For FE interpolation, Condition 4.16 is essential: for the local results in Theorem4.17 we know that (i): our K are star-shaped, (ii) we choose τ = −1 and anyinterpolation condition according to (iii) or (iv). The above shape regularity condition(7.309) implies the nondegeneracity of the T h, cf. Condition 4.16 (v), and (4.30). Notethat the constants appearing in Theorem 4.17 are not indepenent of p. Then the localDCG interpolation operator, Ih = Ph yields:

Theorem 7.49. Local approximation properties: There exists a constant CA > 0,independent of v and h, and a mapping πK

p : Hs(K) → Pp(K), s ≥ 1, e.g. the previousIh = Ph, such that ∀v ∈ Hs(K), K ∈ T h, h ∈ (0, h0) the following inequality is valid

∥∥πKp v − v

∥∥Hj(K)

≤ CAh

min(p,s)−jK

ps−j‖v‖Hmin(p,s)(K), (7.315)

where 0 ≤ j ≤ s.

Proof. See Lemma 4.5 in [53] for the case n = 2. The n = 3 arguments are completelyanalogous. �


Definition 7.50. Let s and p be the vectors introduced by (7.296). We define themapping Πhp : Hs(T h) → Shp by(

Πhpu)|K := πK

pK(u|K) ∀K ∈ T h, (7.316)

where πKpK

: HsK → PpK(K) is the mapping introduced in Theorem 7.49.

Theorem 7.51. Global approximation properties: Let Πhp : Hs(T h) → Shp be themapping introduced by (7.316), and v ∈ Hs(T h). Then

∥∥Πhpv − v∥∥2

Hq(T h)≤ C2

A

∑K∈T h

h2 min(pK ,sK)−2qK

p2sK−2qK

‖v‖2Hmin(pK ,sK )(K)

, (7.317)

where 0 ≤ q ≤ minsK∈s sK .

Proof. Using definition (7.316), the Cauchy inequality, and the approximation prop-erties (7.315) we obtain (7.317) and CA is given by Theorem 7.49. �

7.14.4 Consistency and convergence of hp-DCGMs

Here we analyze the hp–DCGM (7.306) for our model problem. We consider forsimplicity ∂ΩD = ∂Ω. More general linear, semilinear, and quasilinear cases can beanalyzed analogously. The proofs are very similar to h-variant (but not identical),since we have to use hp-inverse inequality or hp-interpolation error estimates. Thecoercivity as well as boundedness of the form Ah(u, v) := ah(u, v) + εJσ

h (u, v), provedin Theorem 7.16, remain valid for u, v ∈ Shp. Let us analyze the consistency of formsbh, ah and Jσ

h . �h is independent of u and therefore its consistency immediately followsfrom Theorem 7.23.

Theorem 7.52. Consistency for the ch and bh: For ∂Ω = ∂ΩD, there exists a constantC10 > 0 such that for ch and bh

|ch(u, vh)− ch(Πhpu, vh)| ≤ C10‖vh‖

⎛⎝ ∑K∈T h

h2 min(pK ,sK)K

p2sK

K

‖u‖2Hmin(pK ,sK )(K)

⎞⎠1/2

,

u ∈ Hs(T h), vh ∈ Sh, h ∈ (0, h0), (7.318)

where Πhpu is the Shp-interpolant of u from Theorem 7.51. This estimate remainsvalid for |bh(u, vh)− bh(Πhpu, vh)|, as well.

Proof. The first part of the proof of Proposition 7.20 remains unchanged, exceptreplacing d(e), hK in (7.202) by dhp(e), dhp(K) here. After (7.202) the proof contin-ues as:

Now, we insert u := Πhpu and v := vh ∈ Shp into (7.200)–(7.202) in the modifiedform. Using the multiplicative trace inequality (7.129), the discrete Cauchy inequality


and the approximating properties (7.317), we obtain∑K∈T h

dhp(K)‖u−Πhpu‖2L2(∂K) (7.319)

≤∑

K∈T h

C5dhp(K)‖u−Πhpu‖L2(K)‖u−Πhpu‖H1(K)

≤ C5C2A

∑K∈T h

hK

p2K

hmin(pK ,sK)K

psK

K

hmin(pK ,sK)−1K

psK−1K


≤ C5C2A

∑K∈T h

h2 min(pK ,sK)K

p2sK+1K


.

This estimate, (7.317) and (7.198), yield

|ch(u, v)− ch(Πhpu, v)| (7.320)

≤ C8‖v‖

⎛⎜⎝‖u−Πhpu‖L2(Ω) +

⎛⎝ ∑K∈T h

hK‖u−Πhpu‖2L2(∂K)

⎞⎠1/2⎞⎟⎠

≤ C8CA‖v‖

⎛⎜⎝⎛⎝ ∑

K∈T h

h2 min(pK ,sK)K

p2sK

K


⎞⎠1/2

+

⎛⎝C5

∑K∈T h

h2 min(pK ,sK)K

p2sK+1K


⎞⎠1/2⎞⎟⎠ ,

which proves (7.318) with C10 = CAC8(√C5 + 1) since 1/pK ≤ 1. �

The consistency of the form ah is given by

Theorem 7.53. Consistency for the ah: The ah(Πhpu, vh) in (7.57) are consis-tent with a(u, v) in (7.19) ∀ u ∈ Ue, ∀vh ∈ Sh = Vh, so a constant, C�

14 := CA +2nCA

√C5(1/c

−1/2w +

√C5(1 + C6)) exists, such that for the Shp-interpolant Πhpu of

u in Theorem 7.51,

|ah(u, vh)− ah(Πhpu, vh)| (7.321)

≤ ε C�14‖vh‖

⎛⎝ ∑K∈T h

h2 min(pK ,sK)−2K

p2sK−3K


⎞⎠1/2

.

Proof. In Proposition 7.25, we consider u, u, v ∈ Ue as arguments of ah(·, ·). Again,for the generalization to hp-methods, the proof remains unchanged, except replacingd(e), hK there by dhp(e), dhp(K) here. After that the proof of Theorem 7.26 ismodified as:


Putting u := Πhpu and v := vh we have from (7.217)–(7.220)

|ah(u, vh)− ah(Πhpu, vh)| ≤ ε |Πhpu− u|H1(T h)|vh|H1(T h) (7.322)

+ ε 2n

⎛⎜⎝( 1cw

)1/2 (Jσ

h (vh, vh))1/2

⎛⎝ ∑K∈T h

dhp(K)|u−Πhpu|2H1(∂K)

⎞⎠1/2

+

⎛⎝ ∑K∈T h

dhp(K)‖∇vh‖2L2(∂K)

⎞⎠1/2 ⎛⎝ ∑K∈T h

dhp(K)−1‖(u−Πhpu)‖2L2(∂K)

⎞⎠1/2⎞⎟⎠ .

Similarly as in (7.319), it is possible to show that

∑K∈T h

dhp(K)−1|u−Πhpu|2L2(∂K) ≤ C5C2A

∑K∈T h

h2 min(pK ,sK)−2K

p2sK−3K


.

(7.323)

and∑K∈T h

dhp(K)|u−Πhpu|2H1(∂K) ≤ C5C2A

∑K∈T h

h2 min(pK ,sK)−2K

p2sK−3K


(7.324)

These formulas yield, with (7.64), (7.171), (7.317), (7.319), (7.322), (7.323) and(7.324),

|ah(u, vh)− ah(Πhpu, vh)| (7.325)

≤ εCA‖vh‖

⎛⎝ ∑K∈T h

h2 min(pK ,sK)−2K

p2sK−2K

‖v‖2Hmin(pK ,sK )(K)

⎞⎠1/2

+2nε√cw‖vh‖

⎛⎝C5C2A

∑K∈T h

h2 min(pK ,sK)−2K

p2sK−3K


⎞⎠1/2

+ 2nε√C5(1 + C6)‖vh‖

⎛⎝C5C2A

∑K∈T h

h2 min(pK ,sK)−2K

p2sK−3K


⎞⎠1/2

≤ εC�14‖vh‖

⎛⎝ ∑K∈T h

h2 min(pK ,sK)−2K

p2sK−3K


⎞⎠1/2

,

C�14 := CA + 2nCA

√C5(1/c

−1/2w +

√C5(1 + C6)). �

Finally, we show the consistency of the penalty form Jσh .


Theorem 7.54. Consistency for the Jσh : There exists a constant C�

15 > 0 such that

∣∣Jσh (u, vh)− Jσ

h (Πhpu, vh)∣∣ ≤ C�

15‖vh‖

⎛⎝ ∑K∈T h

h2 min(pK ,sK)−2K

p2sK−1K


⎞⎠ 12

,

u ∈ Hs(T h), vh ∈ Sh, h ∈ (0, h0), (7.326)

where Πhpu is the Shp-interpolant of u from Theorem 7.51.

Proof. From (7.61), the Cauchy inequality, (7.149), and the multiplicative traceinequality (7.129) and (7.317) we have∣∣Jσ

h (u, vh)− Jσh (Πhpu, vh)

∣∣ (7.327)

≤∑

e∈T h

∫e

σ|[u−Πhpu]| |[vh]| dS

≤ Jσh (vh, vh)1/2

⎛⎝∑e∈T h

∫e

σ[u−Πhpu]2 dS

⎞⎠1/2

≤ ‖vh‖∑

K∈T h

(cwdhp(K)−1‖u−Πhpu‖2L2(∂K)

)1/2

≤√C5cw‖vh‖

( ∑K∈T h

dhp(K)−1(‖u−Πhpu‖L2(K)|u−Πhpu|H1(K)

+ dhp(K)−1‖u−Πhpu‖2L2(K)

))1/2

≤√

2C5cw‖vh‖C2A

⎛⎝ ∑K∈T h

(1hK

hmin(pK ,sK)K

psK

K

hmin(pK ,sK)−1K

psK−1K

+1h2

K

h2 min(pK ,sK)K

(psK

K )2

)‖u‖2

Hmin(pK ,sK )(K)

)1/2

≤ 2√C5cw‖vh‖C2

A

⎛⎝ ∑K∈T h

h2 min(pK ,sK)−2K

p2sK−1K


⎞⎠1/2

,

which proves Theorem 7.54 with C�15 := 2

√C5cwC

2A. �

For our previous problems we impose the conditions listed in Section 7.12 for thestandard DCGM and the new conditions in Subsections 7.14.1–7.14.3. We define the


hp-DCGMs by replacing the previous Sh by the Shp and (7.274) by

C(hs, u) :=

⎛⎝ ∑K∈T h

h2 min(pK ,sK)−2K

p2sK−1K


⎞⎠ 12

∀u ∈ Hs(T h), h ∈ (0, h0).

(7.328)

Previously we have proved the consistency for our model problem. For the other casesthe proofs follow similar modifications. The necessary stability with respect to the‖ · ‖ is available via the U- and Uh-coercivity as in Section 7.9. Then our hp-updatedconsistency results can be applied for proving the general stability and convergence asin Section 7.12. We summarize

Theorem 7.55. Convergence for the hp- DCGMs for the problem 5.–10. in Condition7.39. Under these conditions 1.–4. and for inhomogeneous Dirichlet boundary condi-tions choose, for the problems in 5.–10. the quoted hp-DCGMs. This implies coercivity,hence stability results for a boundedly invertible G′(u0), and the necessary consistencyestimates in Theorems 7.53,7.54 and the corresponding modifications for the othermethods.

Then for G(u0) = 0, the application of these hp-DCGMs define the Gh(uh

0

)= 0

with a uniquely existing solution uh0 near u0 converging according to

‖uh − u‖JHs(T h) ≤ C(hs, u). (7.329)

Again a numerical solution is efficiently computed via Newton’s method under theconditions of and as in Theorem 7.47. This again implies quadratic convergenceessentially independent of h.

7.15 Numerical experiences

7.15.1 Scalar quasilinear equation

In this section we verify theoretical results presented in the previous sections bynumerical experiments. Similarly as in [407], we consider the nonlinear diffusionequation

−2∑

s=1

∂

∂xs

(ν(|∇u|) ∂

∂xsu

)= g, u : Ω → R, (7.330)

where ν(w) : (0,∞) → R is chosen in the form

ν(w) = ν∞ +ν0 − ν∞(1 + w)γ

, γ > 0. (7.331)

7.15. Numerical experiences 547

The corresponding quasilinear weak form is

〈G(u), v〉 =∫

Ω

n∑k=0

(Ak(x, u,∇u), ∂kv

)dx (7.332)

:=∫

Ω

2∑k=1

(ν(|∇u|) ∂

∂xku

)∂

∂xkvdx−

∫Ω

gvdx.

These Ak(x, u,∇u), k = 1, . . . , n = 2, satisfy the conditions (7.94) (a)–(c) and A0 ≡ 0,cf. (2.284), (2.285), (2.286), (2.293).

We set ν0 = 0.15, ν∞ = 0.1, γ = 1/2, Ω = (0, 1)2, and define the function g and theboundary conditions in such a way that the exact solution has the form

u(x1, x2) = 2rαx1x2(1− x1)(1− x2) (7.333)

= rα+2 sin(2ϕ)(1− x1)(1− x2),

where (r, ϕ)(r := (x21 + x2

2)1/2) are the polar coordinates and α ∈ R is a constant. The

function u is equal to zero on ∂Ω and its regularity depends on the value of α, namely(cf. [52])

u ∈ Hβ(Ω) ∀β ∈ (0, α + 3), (7.334)

where Hβ(Ω) denotes (in general) the Sobolev–Slobodetskii space of functions with“noninteger derivatives”.

In the presented numerical tests we use the values α = 2 and α = −3/2. The valueα = 2 gives a function u sufficiently regular (∈ Hβ(Ω) for β < 5), whereas the valueα = −3/2 gives u ∈ Hβ(Ω), β < 3/2. Figure 7.5 shows functions u for both valuesof α.

The above problem is discretized by the IIPG method (7.286). The resulting systemof nonlinear algebraic equations by a pseudo-time discretization, i.e. we solve a timedependent problem(

∂wh0 (t)∂t

, vh

)+(Gh(wh

0 (t)), vh)

= 0 ∀ vh ∈ Sh, (7.335)

where wh0 (t, x) : (0, T )× Sh → R, T > 0 and Gh is an operator introduced by (7.286).

We seek a steady state solution, i.e.

∂wh0 (t, x)∂t

= 0 for T →∞

and put

uh0 (x) := lim

T→∞wh

0 (T, x), x ∈ Ω.

The system of ordinary differential equations (7.335) is solved by a three step backwarddifferential formula, which is formally of third order, see [293]. The following numerical


0 0.4 0.2 0.6 0.8 1 0 0.2 0.4

0.6 0.8

10

0.025

0.05

0.075

0.1

0 0.2

0.4 0.6

0.81 0

0.2 0.4

0.6 0.8

1

0

0.1

0.2

0.3

0.4

Figure 7.5 The exact solution (7.333) for α = 2 (above) and α = −3/2 (below).

flux is used for the discretization of the convective term

H(u1, u2,n) =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

2∑s=1

fs(u1)ns if A > 0,

2∑s=1

fs(u2)ns if A ≤ 0,

(7.336)

where

A =2∑

s=1

f ′s(u)ns, u =12(u1 + u2) and n = (n1, n2). (7.337)

Numerical experiments are carried out with the use of piecewise linear (P1), qua-dratic (P2), and cubic (P3) elements on six triangular meshes Thl

, l = 1, . . . , 6 having128, 288, 512, 1152, 2048 and 4608 elements. Figure 7.6 shows the coarsest mesh andthe finest one. We investigate the experimental orders of convergence (EOC) for grid


0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Figure 7.6 The coarsest and the finest grids used for the computations.

pairs Thland Thl−1 , l = 2, . . . 6, defined by

EOCl =log(ehl

/ehl−1)log(hl/hl−1)

, l = 2, . . . , 6, (7.338)

where ehlis the computational error obtained on mesh Thl

.Tables 7.1–7.4 show computational errors in the L2(Ω) norm and in the H1(Ω)

seminorm and the corresponding experimental orders of convergence (EOC) for α = 2and α = −3/2. These values together with the marked EOC are shown in Figures 7.7and 7.8. We observe the following:


Table 7.1: Computational errors and the corresponding experimental orders of conver-gence (EOC) of the P1, P2 and P3 approximations for α = 2 in the L2 norm, purelydiffusive case; notice the definition of α in (7.333) ff.

Grid h P1 P2 P3

‖eh‖L2(Ω) EOC ‖eh‖L2(Ω) EOC ‖eh‖L2(Ω) EOC

1 0.177E+00 0.1809E−02 — 0.9906E−04 — 0.4470E−05 —2 0.118E+00 0.8373E−03 1.90 0.3102E−04 2.86 0.9031E−06 3.943 0.884E−01 0.4791E−03 1.94 0.1354E−04 2.88 0.2895E−06 3.954 0.589E−01 0.2162E−03 1.96 0.4202E−05 2.89 0.5811E−07 3.965 0.442E−01 0.1224E−03 1.98 0.1839E−05 2.87 0.1856E−07 3.976 0.295E−01 0.5477E−04 1.98 0.5832E−06 2.83 0.3703E−08 3.98

Table 7.2: Computational errors and the corresponding experimental orders of conver-gence (EOC) of the P1, P2 and P3 approximations for α = 2 in the H1 seminorm, purelydiffusive case.

Grid h P1 P2 P3

|eh|H1(Ω) EOC |eh|H1(Ω) EOC |eh|H1(Ω) EOC


Table 7.3: Computational errors and the corresponding experimental orders of conver-gence (EOC) of the P1, P2 and P3 approximations for α = −3/2 in the L2 norm, purelydiffusive case.

Grid h P1 P2 P3




Table 7.4: Computational errors and the corresponding experimental orders of conver-gence (EOC) of the P1, P2 and P3 approximations for α = −3/2 in the H1 seminorm,purely diffusive case.

Grid h P1 P2 P3


1 0.177E+00 0.4512E+00 — 0.2098E+00 — 0.1548E+00 —2 0.118E+00 0.3730E+00 0.47 0.1662E+00 0.57 0.1129E+00 0.783 0.884E−01 0.3251E+00 0.48 0.1415E+00 0.56 0.9082E−01 0.764 0.589E−01 0.2668E+00 0.49 0.1133E+00 0.55 0.6779E−01 0.725 0.442E−01 0.2314E+00 0.49 0.9707E−01 0.54 0.5576E−01 0.686 0.295E−01 0.1890E+00 0.50 0.7822E−01 0.53 0.4305E−01 0.64

1e-09

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

P_1

P_2

P_3

1

2

1

3

1

4

0.0001

0.001

0.01

0.1

P_1

P_2

P_3

1

3/2

Figure 7.7 Computational errors and the corresponding experimental orders of convergence

of the P1, P2 and P3 approximations for α = 2 (left) and α = −3/2 (right), purely diffusive

case.

(i) For a sufficiently regular exact solution (case α = 2) we observe the optimalorder of convergence O(hp+1) for p = 1, 2, 3 in the L2 norm (a small decrease ofthe experimental order of convergence for the P3 approximation on the finestgrid is caused by rounding off errors, so that we obtain a better EOC thanthe theoretical results). It is a rather surprising fact that the IIPG variantgives (almost) optimal order of convergence also for an even degree polynomialapproximation, i.e. P2, cf. Table 7.1. On the other hand, Guzman and Riviere[380] show that the NIPG method gives only suboptimal EOC also for oddpolynomial degrees when special nonuniform grids are used.


1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

0.1

P_1

P_2

P_3

11

1

2

1

3

0.1

1

0.1

P_1

P_2

P_3

1

1/2


of the P1, P2 and P3 approximations for α = 2 (left) and α = −1/2 (right), purely diffusive

case.

(ii) For a sufficiently regular exact solution (case α = 2) we observe the optimalorder of convergence O(hp) for p = 1, 2, 3 in the H1 seminorm which is in goodagreement with the theoretical results, cf. Table 7.2.

(iii) For the case α = −3/2, we obtain an EOC equal to 3/2 in the L2 norm andan EOC equal to 1/2 in the H1 seminorm for p = 1, 2, 3. Using the resultfrom [312], for any β ∈ (1, 3/2) we get, cf. Table 7.3,

‖v − Ihv‖L2(Ω) ≤ C(β)hμ‖v‖Hβ(Ω), v ∈ Hβ(Ω), (7.339)

where Ihv is a piecewise polynomial Lagrange interpolation to v of degree≤ p, μ = min(p + 1, β) and C(β) is a constant independent of h and v. Theexact approximation corresponding precisely to our experimental results canbe obtained with the use of the interpolation in so-called Bessov spaces. See [51],Section 3.3 and the references therein.

Moreover, we consider a generalization of (7.330) given by

2∑s=1

u∂u

∂xs−

2∑s=1

∂

∂xs

(ν(|∇u|) ∂

∂xsu

)= g, u : Ω → R, (7.340)

where ν(w) : (0,∞) → R is given by (7.331). This problem represents the convection–diffusion equation with nonlinear convective as well as diffusive terms. The corre-sponding quasilinear weak form is given by (7.332) but with A0 �≡ 0. Although thisform violates condition (7.94)(d) it is possible to solve it numerically. The convectiveterm is discretized in the same way as in (7.89) where the discrete convective form ch


is defined by (7.59). There the following numerical flux is used

H(u1, u2,n) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

2∑s=1

fs(u1)ns if A > 0,

2∑s=1

fs(u2)ns if A ≤ 0,

(7.341)

where

A =2∑

s=1

f ′s(u)ns, u =

12(u1 + u2) and n = (n1, n2). (7.342)

Tables 7.5–7.8 show computational errors in the L2(Ω) norm and in the H1(Ω)seminorm and the corresponding experimental orders of convergence (EOC) for α = 2and α = −3/2. These values together with the marked EOC are shown in Figures 7.9and 7.10. We observe the completely same results as for problem (7.330).

Table 7.5: Computational errors and the corresponding experimental orders of conver-gence (EOC) of the P1, P2 and P3 approximations for α = 2 in the L2 norm, convection–diffusion case.

Grid h P1 P2 P3



Table 7.6: Computational errors and the corresponding experimental orders of conver-gence (EOC) of the P1, P2 and P3 approximations for α = 2 in the H1 seminorm,convection–diffusion case.

Grid h P1 P2 P3




Table 7.7: Computational errors and the corresponding experimental orders of convergence(EOC) of the P1, P2 and P3 approximations for α = −3/2 in the L2 norm, convection–diffusion case.

P1 P2 P3

Grid h ‖eh‖L2(Ω) EOC ‖eh‖L2(Ω) EOC ‖eh‖L2(Ω) EOC


Table 7.8: Computational errors and the corresponding experimental orders of conver-gence (EOC) of the P1, P2 and P3 approximations for α = −3/2 in the H1 seminorm,convection–diffusion case.

P1 P2 P3

Grid h |eh|H1(Ω) EOC |eh|H1(Ω) EOC |eh|H1(Ω) EOC

1 0.177E+00 0.2068E+00 — 0.7954E−01 — 0.3507E−01 —2 0.118E+00 0.1699E+00 0.48 0.6488E−01 0.50 0.2988E−01 0.403 0.884E−01 0.1475E+00 0.49 0.5622E−01 0.50 0.2650E−01 0.424 0.589E−01 0.1206E+00 0.50 0.4598E−01 0.50 0.2220E−01 0.445 0.442E−01 0.1045E+00 0.50 0.3987E−01 0.50 0.1947E−01 0.466 0.295E−01 0.8534E−01 0.50 0.3262E−01 0.50 0.1607E−01 0.47

7.15.2 System of the steady compressible Navier–Stokes equations

Let Ω ⊂ Rn, n = 2, 3, be a bounded domain, and by ∂Ω we denote the boundary ofΩ which consists of several disjoint parts. We distinguish inlet ∂Ωi, outlet ∂Ωo andimpermeable walls ∂Ωw, i.e. ∂Ω = ∂Ωi ∪ ∂Ωo ∪ ∂Ωw. We use the following notation: ρ– density, p – pressure, e – total energy, v = (v1, . . . , vn) – velocity, θ – temperature,γ – Poisson adiabatic constant, Re – Reynolds number, Pr – Prandtl number.

The system of Navier-Stokes equations describing steady state viscous compressiblefluids can be written in dimensionless form

∇ · �f(w)−∇ · �R(w,∇w) = 0 in Ω, (7.343)

where

w = (w1, . . . , wn+2)T = (ρ, ρv1, . . . , ρvn, e)T (7.344)


1e-10

1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0.001

0.01

0.1

P_1

P_2

P_3

1

1

1

2

3

4

1e-04

0.001

0.01

0.1

P_1

P_2

P_3

1

3/2


of the P1, P2 and P3 approximations for α = 2 (left) and α = −3/2 (right), convection–

diffusion case.

1e-07

1e-06

1e-05

1e-04

0.001

0.01

0.1

0.1

P_1

P_2

P_3

1

1

1

1

2

3

0.01

0.1

0.1

P_1

P_2

P_3

1

1/2

Figure 7.10 Computational errors and the corresponding experimental orders of conver-

gence of the P1, P2 and P3 approximations for α = 2 (left) and α = −3/2 (right), convection–

diffusion case.


is the so-called state vector, and

�f(w) = (f1(w), . . . ,fn(w)), (7.345)

with fs(w) =(f (1)

s (w), . . . , f (n+2)s (w)

)T

= (ρvs, ρvsv1 + δs1p, . . . , ρvsvn + δnp, (e + p) vs)T, s = 1, . . . , n

the so-called inviscid (Euler) fluxes. Finally

�R(w,∇w) = (R1(w,∇w), . . . ,Rn(w,∇w)), (7.346)

with Rs(w,∇w) =(R(1)

s (w,∇w), . . . , R(n+2)s (w,∇w)

)T

=

(0, τs1, . . . , τsn,

n∑k=1

τskvk +γ

Re Pr∂θ

∂xs

)T

, s = 1, . . . , n

are the so-called viscous fluxes. The symbols ∇ and ∇· mean the gradient anddivergence operators, i.e.

∇w :=(∂w

∂x1, . . . ,

∂w

∂xn

)∈ Rn+2 × · · · × Rn+2 (7.347)

and

∇ · �f(w) :=n∑

s=1

∂fs(w)∂xs

∈ Rn+2, (7.348)

respectively.Again, we can, but do not explicitly, formulate this as the quasilinear weak form

〈G(�u), �v〉 =∫

Ω

n∑k=0

(Ak(x, �u,∇�u), ∂k�v

)dx. (7.349)

These Ak(x, u,∇u), k = 1, . . . , n, satisfy the conditions (7.94) (a)–(c); A0(x, u,∇u),violates (d). If we omit A0(x, u,∇u), we would fully satisfy (7.94) (a)–(c), the (d) wouldbe dropped, and would have an example similar to [407]. There probably exist somephysical models whose mathematical description completely satisfies our assumptionfor a quasilinear model, but is not the case of this viscous compressible flow. In ourcontext it cerainly would not make any physical sense avoiding this term, e.g.

A0(x, u,∇u) = ∇ · �f(w),

where �f is the convective flux. This would mean considering a flow with only withdiffusion and not convection.


We consider a Newtonian type of fluid, i.e. the viscous part of the stress tensor hasthe form

τsk =1

Re

[(∂vs

∂xk+

∂vk

∂xs

)− 2

3

n∑i=1

∂vi

∂xiδsk

], s, k = 1, . . . , n. (7.350)

In order to close the system, we consider the state equation for a perfect gas andthe definition of the total energy

p = (γ − 1) (e− ρ|v|2/2), e = cV ρθ + ρ|v|2/2, (7.351)

Figure 7.11 NACA 0012 profile, computational grid around the NACA0012 profiles (top),

details of the leading (middle) and trailing edges (bottom).


where cV is the specific heat at constant volume which we assume to be equal to 1 inthe dimensionless case. The system (7.343)–(7.351) is equipped with the following setof boundary conditions on appropriate parts of the boundary:

(a) ρ = ρD, v = vD,

n∑k=1

(n∑

l=1

τlknl

)vk +

γ

Re Pr∂θ

∂n= 0 on ∂Ωi, (7.352)

(b)n∑

k=1

τsknk = 0, s = 1, . . . , n,∂θ

∂n= 0 on ∂Ωo,

(c) v = 0,∂θ

∂n= 0 on ∂Ωw,

where ρD and vD are given functions and n = (n1, . . . , nn) is a unit outer normal to∂Ω. Another possibility is to replace the adiabatic boundary condition (7.352) (c) by

(c′) v = 0, θ = θD on ∂Ωw. (7.353)

Although the existence of the solution of system (7.343)–(7.352) is still an openproblem we solve it numerically. The question of the existence and convergence of

P1-approximation

P2-approximation

P3-approximation

Figure 7.12 Flow around NACA0012 profiles, isolines of the Mach number (left) with

details around the leading (middle) and trailing edges (right) computed by P1, P2 and P3

approximations.


discrete solutions is equally open. However, the formulation (7.343) of the problemis pretty close to the quasilinear problem (7.90). We employ the IIPG variant ofthe DCGM (7.286) to the solution of the system of the compressible Navier-Stokesequations. We consider a flow around the NACA0012 profile characterized by theinlet Mach number Min = 0.5 (= |v|/

√γp/ρ at infinity), angle of attack α = 0 (=

arctg(v2/v1) at infinity) and Reynolds number Re = 5000. The Reynolds numberis near to the upper limit for steady laminar flow. A characteristic feature of thisflow problem is the separation of the flow occurring near to the trailing edge. Thecomputations were carried out on a relatively coarse triangular grid having 1 360elements which was obtained by the anisotropic mesh adaptation method (cf. [292,298]), see Figure 7.11. We employ P1, P2 and P3 polynomial approximation and thecorresponding isolines of the Mach number are shown in Figure 7.12. We observea significantly smooth resolution for the P3 polynomial approximation although thecorresponding grid is coarse.

Other interesting papers, applying DCGMs to the Stokes problem and to two-dimensional incompressible and to compressible Navier-Stokes equations, are due toHartmann et al. [390–392,409,505].

8

Finite difference methods

8.1 Introduction

In our synopsis it is impossible to present all methods in the same generalityas the dominant FEMs. Our choice for the results in finite difference methods ismotivated by the following aspects: The most popular approach between 1960 and1990, based upon M matrices, allows error estimates, e.g. for the Laplacian, of theform supx∈Ωh

{|u0(x)− uh

0 (x)|}≤ Ch2‖u0‖C4(Ω). Here u0 and uh

0 are the exact andthe discrete solutions of the strong equation Δu0 = f ∈ C(Ω) and the correspondingstrong symmetric difference equations on a grid Ωh ⊂ Ω with step size h. We apply ourgeneral discretization theory in Chapter 3, thus avoiding the “external approximationapproach”. This motivates considering finite difference methods as variational methodssimilar to FEMs, cf. earlier approaches, discussed below. The above |u0(x)− uh

0 (x)| canthen be estimated by a stronger discrete Sobolev energy norm on Ωh. More generally,convergence results and error estimates can be proved here for similarly generalproblems as for FEMs. However, handling boundary conditions and adaptivity isstrongly restricted. These difficulties are discussed by Heinrich [395] with appropriatecombinations of finite difference methods and finite volume techniques, on irregulargrids, Ωh. For intended applications to bifurcation, finite difference methods are oftenused on cuboidal domains Ω = Πn

i=1(ai, .bi). So this case will play an essential role.Finally we only discuss equidistant grids Ωh. All our results are based upon theanalytical results for general elliptic problems, summarized in Chapter 2.

Section 8.2 is devoted to simple examples and basic tools. We will discuss thechanges from unsymmetric to symmetric difference equations. Symmetric differenceequations remain unchanged, if h is replaced by −h, and thus they automatically are oforder two, cf. Lemmas 8.1–8.2 and Grossmann and Roos [374–376]. The correspondingconvergence theory is nearly the same as for unsymmetric forms. Section 8.3 presentsthe facts for discrete Sobolev spaces.

Section 8.4 elaborates general elliptic problems with Dirichlet boundary conditions,and their difference methods, Section 8.5 the unique existence of the discrete solutionsand their convergence.

For general problems, we consider unsymmetric difference methods first and gen-eralize them to curved boundaries. This yields methods converging of order 1. Amodification to symmetric difference methods converges of order 2. We consider forall cases the quartets of strong and weak differential and difference equations. Theseweak difference forms go back to the Russian literature, cf. Samarskii [564, 565], and


are used, e.g. by Temam [621] and Grossmann and Roos [375]. By combining the ∂j−

and ∂i+u

h, we define an unsymmetric divergence form as, e.g. in (8.44). We proceedfrom unsymmetric to symmetric forms in Subsection 8.4.2. The transformation ofthe strong forms Ah

suh = fh, essentially the classical difference equation, into the

weak forms Ahuh0 = fh, and the discussion of solving Ahuh

0 = fh, proceeds along thesame lines for all cases. So we give the details only for second order linear equationsin Subsections 8.4.2, 8.4.3. For the other cases we restrict ourselves to listing theequations in Subsections 8.4.4 ff. For nonlinear operators the domains are subsets ofthe corresponding Sobolev spaces in (8.43), (8.47). Conditions for the coefficients arelisted in Summary 8.6, and Remark 8.7. The above transformation from strong toweak forms becomes more complicated for natural boundary conditions in Section8.6, and is not applicable in Section 8.7.

With one exception, the proof for stability and convergence follows “the same generalprocedure as” for all other discretization methods: We combine discrete coercivity, cf.(8.120), compact perturbation of a boundedly invertible linear operator and consis-tency. This will yield the existence of a unique solution for the difference equation,and its convergence to the exact solution u0, and the corresponding generalizations.For equations of higher order and general domains violated boundary conditions aretreatable as for FEMs. We do not reformulate these rather technical considerations.Convergence for monotone and quasilinear operators follows the lines of Section 4.5and again yields Theorem 4.67.

Except for Section 8.6, we restrict the discussion to homogeneous Dirichlet condi-tions. In Lemma 2.26 we have formulated conditions allowing the transformation of ageneral Dirichlet boundary value problem into a homogeneous form. For differentialand difference equations the transformation of strong into weak forms is based uponpartial integration and summation. Here the vanishing Dirichlet boundary conditionsat least for the test functions, often for the solutions as well, are crucial, cf. e.g.Proposition 8.8. For nonlinear discrete problems the mesh independence principle, cf.Theorem 8.33, implies the quadratic convergence of Newton’s method for solving thedifference equations, essentially independent of the step size.

In Section 8.6 we formulate, for problems with natural boundary conditions, dif-ference methods and prove existence and convergence of the discrete solutions bymodifying the previous technique.

In Section 8.7, we discuss different methods of orders ≥ 1 on curved domains forspecial differential equations. This will require distinguishing points more or less nearthe boundary. These results are only summarized, but not proved.

We have confined the discussion to difference methods of first or second orderaccuracy. By our convergence results with respect to discrete Sobolev norms, theextrapolation or defect correction methods, based upon asymptotic expansions, pre-sented in Section 8.8, allow a very efficient formulation of higher order methods anyway.In particular, our symmetric forms yield second order accuracy and the advantageousasymptotic expansions in powers of h2.

The last Section 8.9 applies Richardson extrapolation in an academic example tothe von Karman equations.

562 8. Finite difference methods

Notice that the following problems are only discussed for FEMs. General convergencetheory for monotone operators, cf. Section 4.5, variational methods for eigenvalueproblems, cf. Section 4.7, methods for fully nonlinear elliptic problems, cf. Section 5.2,methods for nonlinear boundary conditions, cf. Section 5.3, and quadrature approxi-mate methods, cf. Section 5.4. All the results remain correct for difference methods.For appropriate updates see Remark 8.27.

Many difference methods for many problems are considered in the literature. Fromthe huge number of publications, we only list a very small choice of books and surveys.Papers are cited in the corresponding context. Ansorge [31], Cheng [171], Ciarletand Lions (ed.) [177] Collatz [206], Cuvelier et al. [231], Forsythe and Wasow [319],Grossmann et al. [375,376], Hackbusch [387], Heinrich [395], Isaacson and Keller [414],Marchuk [483], Quarteroni [536], Samarskii [564, 565], Sloan et al. [592], Strikwerda[604, 606], Temam [624], Thomas [626], Vainberg [643], Vainikko [644]. Xu [670] andZeidler [677,678].

8.2 Difference methods for simple examples, notation

We start with the simple boundary value problem

−u′′(x) = f(x) ∀ x ∈ Ω = (0, 1), u(0) = ua, u(1) = ue, (8.1)

hence −u′′ = −Δu in R1 = R. To determine an approximate solution we replace thesecond derivative u′′ by divided differences in the grid points. We choose a step sizeh, and introduce an equidistant one-dimensional grid as, cf. Figure 8.1,

Gh1 := Gh := {xi ∈ R : xi = ih, h := 1/k, k, i ∈ Z, k > 0}, (8.2)

Ωh0 := {xi ∈ R : xi = ih, 0 < i < k} ⊂ Gh, ∂Ωh := {xi = ih, i = 0, k},

Ωh := Ωh0 ∪ ∂Ωh = {xi ∈ R : xi = ih, 0 ≤ i ≤ k} ⊂ Gh.

We define for a function v: R → R. For v : Ωh → R, we restrict x to x ∈ Ωh orx ∈ Gh :

∂+v(x) := ∂h+v(x) := ∂

(1)+ v(x) := [v(x + h)− v(x)]/h, 0 ≤ x ≤ 1− h, (8.3)

∂−v(x) := ∂−h+ v(x) := ∂

(1)− v(x) := [v(x)− v(x− h)]/h, h ≤ x ≤ 1, recursively,

∂(j)+ v(x) := ∂

(j)+,hv(x) := ∂+

(∂

(j−1)+ v(x)

), ∂

(j)− v(x) := ∂

(j)−,hv(x) := ∂−

(∂

(j−1)− v(x)

),

e.g. ∂(2)+ v(x) = [v(x + 2h)− 2v(x + h) + v(x)]/h2, 0 ≤ x ≤ 1− 2h.

Ω

Ωh

Ωh ∪ ∂Ωh

Figure 8.1 One-dimensional grid.

8.2. Difference methods for simple examples, notation 563

The recursive definition of symmetric formulas is a bit more complicated than for(8.3), cf. Collatz [206], pp. 538 ff. In difference equations, these (8.4) can be directlyused for constant coefficients, but will be modified below for nonconstant coefficients:

∂(0)h v(x) := [v(x + h) + v(x− h)]/2, (8.4)

∂(1)h v(x) := [v(x + h)− v(x− h)]/(2h) =

(∂h+ + ∂h

−v(x))/2,

∂(2)h v(x) := ∂h

+

(∂h−v(x)

)= [v(x + h)− 2v(x) + v(x− h)]/h2

=(∂h−(∂h+v(x)

)+ ∂h

+

(∂h−v(x)

))/2

∂(3)h v(x) := ∂

(1)h

(∂

(2)h v(x)

)= [v(x + 2h)− 2v(x + h) + 2v(x− h)− v(x− 2h)]/(2h3)

with ∂(1)h

(∂

(2)h v(x)

)= ∂

(2)h

(∂

(1)h v(x)

),

∂(4)h v(x) := ∂

(2)h

(∂

(2)h v(x)

)= [v(x + 2h)− 4v(x + h) + 6v(x)− 4v(x− h) + v(x− 2h)]/(h4), etc.

The formulas in (8.4) have the invariance property with respect to replacing h by−h, so ∂

(i)h v(x) = ∂

(i)−hv(x).

Compared to the nonsymmetric divided differences in (8.3), the correspondingsymmetric formulas in (8.4) approximate the function values and derivatives in xof order 2, instead of the previous order 1. This implies an asymptotic expansionin powers of h2 instead of h, cf. the following Lemma 8.1, cf. [387], Lemma 4.1.1,and [103]. These advantages are inherited by the difference methods.

Lemma 8.1. The mean value and difference operators ∂(0)h , and ∂+, ∂−, ∂

(i)± , ∂

(i)h

satisfy the following relations, estimates, and asymptotic expansions. For i = 0, 1, . . . ,and with ci,j ∈ R, independent of u, h, we obtain

∂(0)h = (∂+ + ∂−)/2, ∂(2)

h = (∂+ − ∂−)/h = ∂+ ◦ ∂− = ∂− ◦ ∂+, and (8.5)∣∣∂(i)± u(x)− u(i)(x)

∣∣ ≤ Lh for u(i) ∈ CL(Ω), and for symmetric forms∣∣∂(i)h u(x)− u(i)(x)

∣∣ ≤ Lh2

i + 1for u(i+1) ∈ CL(Ω) or asymptotic expansions∣∣∣∣∣∂+u(x)− u′(x)−

m∑i=1

hiu(i+1)(x)(i + 1)!

∣∣∣∣∣ ≤ Lhm+1

(m + 1)!u(m+1) ∈ CL(Ω), (8.6)

∣∣∂(i)h u(x)− u(i)(x)−

q∑j=1

h2jci,ju(2j+i)(x)(2j + i)!

∣∣ ≤ Lh2q+1

(2q)!, for u(2q+i) ∈ CL(Ω).

These results remain valid, with obvious modifications, for the multidimensionalgeneralizations in (8.12) ff., cf. Lemma 8.2.


Replacing the u′′(x) in (8.1) by the ∂(2)h u(x) for x = xi ∈ Ωh

0 in (8.2) yields

−∂(2)h u(xi) = f(xi) +O(h2), 0 < i < k, h :=

1k, xi = ih, u(x0) = ua, u(xk) = ub.

Omitting the O(h2) term we end up with the (symmetric) difference equation for adiscrete function uh : Ωh → R of the form

− ∂(2)h uh(xi) = −[uh(xi − h)− 2uh(xi) + uh(xi + h)]/h2 = f(xi),

0 < i < k, uh0 = uh(x0) = ua, u

hk = uh(xk) = ub. (8.7)

As for all the previous discretization methods, the deciding question is: Does anapproximate solution uh

0 (xi), 0 ≤ i ≤ k, uniquely exist, and do these values convergeto the values of the exact solution u0(xi)? Do small perturbations in the equation onlyslightly perturb the solution uh

0 (xi)? This has to be generalized, and discussed in thischapter.

For motivating difference methods for PDEs, we generalize our above model problem(8.1) as

Au = −Δu = f ∈ C(Ω), (u− φ)|∂Ω = 0. (8.8)

We only consider finite difference methods with equal step sizes, h, in all directions.Different hi can be treated with essentially the same techniques.

For cuboidal Ω, Dirichlet boundary conditions can be exactly discretized:

Ω :=n∏

i=1

(ai, bi) ⊂ Rn s.t. ∃h > 0 with ai/h, bi/h ∈ Z, i = 1, · · · , n. (8.9)

The following approach works as well for unions of cuboids in (8.9), connectedalong (n− 1)-dimensional faces, perpendicular to the coordinate axes. We extend itto curved boundaries below.

For Ω ⊂ Rn in (8.9) or in (8.18) below we introduce step sizes, h, and grids:

H := {h as in (8.9) or h ∈ R+ for (8.18) below} : (8.10)

Gh := GhD := {x = (x1, . . . , xn)T ∈ Rn : xi/h ∈ Z},

Ghl := {∀j = 1, . . . , n : x = (x1, . . . , xn)T ∈ Rn : ∀i �= j : xi/h ∈ Z, xj/h ∈ R}.

For Ω in (8.9) we introduce

Ωh0 := Ω∩Gh,Ωh := Ω∩Gh, ∂Ωh := ∂Ω∩Gh = Ωh \ Ωh

0 (8.11)

but we have to modify it for Ω in (8.18).We consider grid functions vh, defined on Ωh, and define their multidimen-

sional forward, backward, and normal (divided) differences for x ∈ Gh, generalizing

8.2. Difference methods for simple examples, notation 565

: pointsin ∂Ωh

: pointsin Ωoh


(8.3), (8.4), as, cf. Figure 8.2,

∂i+v

h(x) := ∂i+,hv(x) := [vh(x + hei)− vh(x)]/h, ei = i-th unit vector (8.12)

∂i−v

h(x) := [vh(x)− vh(x− hei)]/h = ∂i−,hv(x), i = 1 . . . n,∀x ∈ Gh,

∂α±v

h(x) := Πni=1

(∂i±)αi

vh(x), and ∂ν±v

h(x) := [vh(x± hν)− vh(x)]/h,

with the normal differences, where, for x ∈ ∂Ω, Ω ⊂ Rn in (8.9), ei is replaced by theouter normal ν or by −ν. The symmetric divided differences are

∂ihv

h(x) := ∂i−hv(x) := [vh(x + hei)− vh(x− hei)]/(2h), (8.13)

∂(2,0)h vh(x) := ∂1

−(∂1+v

h(x))

= ∂1+

(∂1−)vh(x)), ∂(0,2)

h vh(x) := ∂2−(∂2+v

h(x)), and

∂(1,1)h vh(x) := ∂1

h

(∂2

hvh(x)

)or ∂(1,1)

h vh(x) :=(∂1−∂

2+ + ∂1

+∂2−)vh(x),

∂(3,0)h := ∂1

h∂(2,0)h , ∂

(2,1)h := ∂2

h∂(2,0)h or ∂1

h∂(1,1)h , ∂

(1,2)h := ∂2

h∂(1,1)h or ∂1

h∂(0,2)h ,

∂(4,0)h := ∂

(2,0)h ∂

(2,0)h , ∂

(3,1)h := ∂1

h∂2h∂

(2,0)h , . . . , and ∂

(j)ν,hv

h(x) := (∂νh)j

vh(x).

These symmetric formulas are invariant with respect to replacing h by −h. Theproblematic

∂(2,0)h,u vh(x) :=

(∂1

h

(∂1

hvh(x)

)= [vh(x + 2he1)− 2v(x) + vh(x− 2he1)]/(2h)2 (8.14)

is avoided. It would yield in (8.16) a five-point star of double step size 2h. Therefore,the previous higher order ∂

(i,j)h vh(x) in (8.13) are chosen to include only points as


Table 8.1: Stencil for the five-point star around x ∈ Ωh ⊂ R2 for −h2Δh.

0 −uh(x + he2) 0 0 −1 0−uh(x− he1) 4uh(x) −uh(x + he1) abbreviated as −1 4 −1

0 −uh(x− he2) 0 0 −1 0

close as possible to x. So we allow only points x± νhe1 + μhe2 in ∂(i,j)h vh(x) show

that in

∂(i,j)h vh(x) with2(m− 1) ≤ i + j ≤ 2m, only x± νhe1 + μhe2with|ν|, |μ| ≤ m. (8.15)

For the two-dimensional Laplacian in (8.8) and Ω = (0, 1)× (0, 1) we formulateits difference approximation analogously to (8.1) and (8.7). We replace the partial−Δu(x) = −(∂(2,0) + ∂(0,2))u(x) in (8.8) by the difference approximations −(∂(2,0)

h +∂

(0,2)h )uh(x), and only admit x = (xi, yj) ∈ Ωh

0 . Omitting the O(h2) term, the differ-ence equation for uh

0 : Ωh → R has the form of a five-star approximation

f(x) = Ahuh0 (x) := −Δhu

h0 (x) := −

(∂

(2,0)h + ∂

(0,2)h

)uh

0 (x) = −[uh

0 (x + he1) (8.16)

+ uh0 (x− he1) + uh

0 (x + he2) + uh0 (x− he2)− 4uh

0 (x)]/h2,∀x = (xi, yj) ∈ Ωh

0 ,

h = 1/k,∀0 < i, j < k, uh0 (x) = φ(x)∀x ∈ ∂Ωh ⇔ ∀0 ≤ i, j ≤ k, i · j = 0.

Sometimes we approximate the f(x) as in Proposition 8.5. Equation (8.16) isvisualized in Table 8.1. We will use the same standard abbreviation, showing onlythe factors of the uh(x± hei), uh(x) in the appropriate positions, in the later tables.

Equation (8.16), and the following symmetric, and unsymmetric finite differencemethods, essentially maintain the order 2 and 1, and the asymptotic expansion inpowers of h2 and h, in Lemma 8.2 for the difference between the exact solution, u0,e.g. of the two-dimensional Laplacian and its discrete approximation, uh

0 , e.g. of (8.16),evaluated in the grid points, cf. Theorem 8.31 in Section 8.5, and Theorem 8.43 inSection 8.8.

8.3 Discrete Sobolev spaces

Essential for our stability and consistency results is the modification of continuousto discrete Sobolev spaces. These are spaces of grid functions only defined on Ωh orGh. Correspondingly, we introduce these concepts, based upon partial differences. Forthe intended convergence results, we have to consider sequences of discrete Sobolevspaces, defined on increasingly finer grids, indicated by h ∈ H, cf. (8.10).

8.3.1 Notation and definitions

Similarly to the above, we define the higher partial differences with the usual multi-indices α := (α1, . . . , αn) ∈ Nn

0 , and the corresponding reals, and vectors in (8.17).

8.3. Discrete Sobolev spaces 567

They are based upon the above ∂(i)+ uh, ∂

(i)− uh, ∂

(i)h uh in (8.3), (8.4), generalized in

(8.12)–(8.15). We formulate the ∂α+, version, and indicate the ∂α

−, ∂αh forms. The

notation ∂α± indicates ∂α

+ or ∂α− or their combination, applied recursively.

With ∂i+u, ∂i

−u, ∂ihu as in (8.12), (8.13) let (8.17)

∂α+u := ∂α1

+ . . . ∂αn+ u :=

(∂1+

)α1 · · ·(∂n+

)αnu,∇k

+u :=(∂α+u)|α|=k

,

∇≤k+ u :=

(∂α+u)|α|≤k

, ∇0+u := u, ∂+u := ∇+u := ∇1

+u,

and correspondingly ∂i−u, ∂

α−u,∇k

−u, ∂−u, and ∂ihu, ∂

αhu,∇hu,∇k

hu, ∂hu,

with, e.g., u(x), ∂i+u(x), ∂α

+u(x) ∈ R, ∂+u(x) ∈ Rn, nk, Nk s.t. ∇k+u(x) ∈ Rnk ,

∇≤k+ u(x) ∈ RNk , ϑi, ϑα := (ϑ1)α1 · · · (ϑn)αn ∈ R, ϑ := (ϑ1, . . . , ϑn) ∈ Rn,

Θk := (ϑα)|α|=k ∈ Rnk , ϑα := ϑi := 1 for |α| = i = 0,Θ0 ∈ R,Θ := Θ1.

For the elliptic equations below, the boundary conditions, cf. (8.8), are transformedinto trivial Dirichlet conditions, see (8.62). For the unsymmetric and symmetricdifference methods, this is translated into vanishing uh(x) for grid points x near theboundary, cf. (8.19), (8.20). Then the ∂α

±uh in the difference equations are defined in

the standard way for all points, in particular for those near to, and at the boundary.The symmetric difference methods, cf. Subsection 8.4.2, (8.60), ff., and Section 8.7,are the motivation for choosing different forms for the relevant grid points, cf. (8.20),compared to (8.19), e.g. [678].

For relating the strong forms in (8.7), (8.16), and (8.44) ff. below to the corre-sponding generalized weak forms, we split the set of grid points into all relevant,interior, boundary and near boundary grid points, Ωh,Ωh

0 , ∂Ωh, . . . , ∂Ωhm or Ωh \ Ωh

0 ,generalizing (8.11). We allow curved boundaries, such that

Ω ⊂ Rn is open, bounded, nonempty, and ∂Ω is Lipschitz-continuous. (8.18)

We distinguish two cases for defining the relevant grid points. Often u0 ∈ Hk(Ω), k <m, so it cannot be boundedly extended to an Ecu0 ∈ Hm(Rn). Or the solutionu0 ∈ Hk(Ω), k ≥ m with (8.18) allows the bounded extension Ecu0 ∈ Hk(Rn), cf.Theorem 4.37.

The Ω-relevant grid points x ∈ Gh, cf. (8.10), are defined via half open cubes ch(x).In both cases, (8.19) and (8.20), the strong and weak difference equations will beformulated on Ωh

0 and Ωh, respectively, and the boundary conditions on Ωh \ Ωh0 .

For both cases and for the first case, u0 ∈ Hk(Ω), k < m, we define

ch(x) :=n∏

i=1

[x− hei/2, x + hei/2), ch(Ωh) :=⋃

x∈Ωh

ch(x), and for k < m (8.19)

Ωh := {x ∈ Gh : ch(x) ⊂ Ω}, ∂Ωh1 := {x ∈ Gh \ Ωh : ch(x) ∩ ∂Ω �= ∅},

Ωh0 :=

{x ∈ Ωh : dist

(x, ∂Ωh

1

)≥ mh

}, and ∂Ωh := ∂Ωh

m := Ωh \ Ωh0 ,


of interior and boundary knots. In the second case, we define the interior knots as Ωh0

and ∂Ωh, respectively.

Ωh0 := {x ∈ Gh : ch(x) ⊂ Ω}, and define boundary knots recursively: For (8.20)

m = 1 : ∂Ωh := ∂Ωh1 :=

{x ∈ Gh \ Ωh

0 : ch(x) ∩ ∂Ω �= ∅}, Ωh := Ωh

1 := Ωh0 ∪ ∂Ωh

1 ,

m ≥ 2 : Ωh := Ωhm := Ωh

0 ∪ ∂Ωh1 ∪ . . . ∂Ωh

m, ∂Ωh := Ωh \ Ωh0 = ∂Ωh

1 ∪ . . . ∂Ωhm with

∂Ωhm :=

{x ∈ Gh \

(Ωh

0 ∪ ∂Ωh1 ∪ . . . ∂Ωh

m−1

): ch(x) ∩

(∪x′∈Ωh

0 ..∪∂Ωhm−1

ch(x′))�= ∅}.

The half open cubes ch(x) yield a disjoint covering of Rn, and Ω :

Rn =⋃

x∈Gh

ch(x), Ω ⊂⋃

x∈Ωh∪∂Ωh1

ch(x) for (8.19)), and (8.21)

Ω ⊂⋃

x∈Ωh0∪∂Ωh

1

ch(x) for (8.20)), with ch(x) ∩ ch(x′) = ∅∀x′ �= x, x′ ∈ Gh.

For Ω in (8.18), the above ∂Ωh1 in (8.19) or (8.20) is equivalently defined by

∂Ωh1 =

{x ∈ Gh \ Ωh or \ Ωh

0 : ch(x) ∩ Ω �= ∅}. We have defined the Ωh

0 ,Ωh, ∂Ωh

m, in(8.20) differently from other authors, e.g. Temam [624]. This allows the presentationof symmetric difference methods of order 2 instead of the nonsymmetric order 1.

For the consistency of these methods we need

Lemma 8.2. The difference operators ∂(α)± , ∂

(α)h satisfy the following estimates, and

asymptotic expansions: With 1 := (1, . . . , 1), cα,j , c′α,j ∈ R, independent of h, and

cα,j = cα,j

(u(α+β)(x), |β| = j

), c′α,j = c′α,j

(u(α+β)(x), |β| = 2j, β even

), we get∣∣∂(α)

± u(x)− u(α)(x)∣∣ ≤ cα,1h for u(α) ∈ CL(Ω) for unsymmetric forms (8.22)∣∣∂(α)

h u(x)− u(α)(x)∣∣ ≤ c′α,1h

2 for u(α+1) ∈ CL(Ω) for symmetric forms,

with ∂(0,i)h u(x) = ∂

(α=0ei)h u(x) = (u(x + ei) + u(x− ei))/2, and asymptotic expansions∣∣∣∣∂(α)

± u(x)− u(α)(x)−m∑

j=1

hjcα,j

∣∣∣∣ ≤ Lhm+1cα,m+1 for u ∈ Cα+mL (Ω), (8.23)

∣∣∣∣∂(α)h u(x)− u(α)(x)−

m∑j=1

h2jc′α,j

∣∣∣∣ ≤ Lh2m+1c′α+2m+1, for u ∈ C(α+2m)L (Ω);

u ∈ Cα+mL (Ω) indicates derivatives of u only in directions ei with αi �= 0 or 0ei.


8.3.2 Discrete Sobolev spaces

Convergent difference methods require regularity assumptions for the boundary. Forour chosen Ω ∈ CL and the definition of Ωh

m the following, usually imposed conditionsare satisfied by Theorem 1.26 and by construction. In different papers the followingconditions are imposed:

1. Assume ∂Ω smooth enough, such that the embedding W 1,p(Ω) ↪→ Lp(∂Ω) iscontinuous, cf. Necas [510].

2. The boundary strip Ωhm := ∪x∈Ωhch(x) \ Ω is uniformly small. Hence there exist

0 < d, h0 ∈ R such that for all 0 < h < h0, h ∈ H, for all x ∈ Ωhm, there exists an

α with |α| ≤ m such that ∃x + dαh or ∃x− dαh �∈ Ω, cf. Raviart [545], p. 22.

The continuous embedding W 1,p(Ω) ↪→ Lp(∂Ω) is a consequence of (8.9), (8.18), theuniformly small boundary strip of our construction.

We consider the following grid functions, uh, and test functions vh : Gh → R or Rq,always vanishing outside of Ωh. The imposed vanishing Dirichlet boundary conditions(of higher order) imply test functions, vh(x), vanishing on Ωh \ Ωh

0 ,

uh, vh : Gh → R or → Rq, with uh(x), vh(x) = 0 ∀x ∈ Gh \ Ωh, and (8.24)

for test functions vh ∈Wm,p0,+ (Ωh) : vh(x) := 0 ∀x ∈ Ωh \ Ωh

0 , cf. (8.19), (8.20).

In Section 8.6, we will discuss natural boundary conditions, and modify the previous,and following definitions.

We define, for the grid functions, uh, and their differences in (8.17)–(8.20), thediscrete integral, the (scaled) discrete Lebesgue, and Sobolev scalar products, theirnorms, seminorms, and spaces, (of grid functions of finite norms ‖uh‖L2

h(Ωh) or‖uh‖W m,p

+ (Ωh)), cf. e.g. Hackbusch [387] and Zeidler [678]:∫Ωh

uhdxh := hn∑

x∈Ωh

uh(x),∫

Ωh0

uhdxh := hn∑

x∈Ωh0

uh(x), (8.25)

(vh, wh)L2h(Ωh) :=

∫Ωh

vhwhdxh, ‖vh‖L2h(Ωh) := (vh, vh)1/2

L2h(Ωh)

and

1 ≤ p ≤ ∞ : ‖vh‖Lph(Ωh) :=

(∫Ωh

|vh(x)|pdxh

)1/p

=

⎛⎝hn∑

x∈Ωh

|vh(x)|p⎞⎠1/p

, (8.26)

with ‖vh‖L∞h (Ωh) := max

x∈Ωh{|vh(x)|}, and Lp

h(Ωh) :={vh : ‖vh‖Lp

h(Ωh) <∞}.

We turn to discrete Sobolev spaces and the dual norms for bounded linear forms.Depending upon the different types of divided differences, mainly the ∂+, we introducethe higher order Sobolev spaces of grid functions, denoted as Wm,p

+ (Ωh) or Wm,p− (Ωh),

Wm,p± (Ωh), Wm,p

h (Ωh). Compared to Wm,p+ (Ωh) with uh of finite norms, we replace


in Wm,p− (Ωh), or the symmetric forms, Wm,p

h (Ωh), the ∂α+v

h by the corresponding∂α−v

h, and appropriate symmetric ∂αh v

h, cf. (8.12)–(8.17). The notation Lph(Ωh),

with h indicating the discretization parameter, is not appropriate for the followingWm,p

+ (Ωh), . . . , since the + , − , h have to indicate the chosen type of divideddifferences, assuming, e.g. well-defined ∂α

+uh(x)∀x ∈ Ωh, uh : Gh → R.

(uh, vh)Hm+ (Ωh) :=

∑|α|≤m

(∂α+u

h, ∂α+v

h)L2

h(Ωh), and (8.27)

|vh|W k,p+ (Ωh) :=

⎛⎝∑|α|=k

‖∂α+v

h‖pLp

h(Ωh)

⎞⎠1/p

, ‖vh‖W m,p+ (Ωh) :=

⎛⎝ ∑0≤k≤m

|vh|pW k,p

+ (Ωh)

⎞⎠1/p

,

Wm,p0,+ (Ωh) :=

{uh ∈Wm,p

+ (Ωh) : uh(x) = 0 ∀x �∈ Ωh0

}=: Wm,p

0,− (Ωh) =: Wm,p0,h , (8.28)

Hm+ (Ωh) := Wm,2

+ (Ωh), . . . ,W−m,p′

+ (Ωh) := L(Wm,p

+ (Ωh),R), 1/p + 1/p′ = 1,

(8.29)

〈fh, vh〉W−m,p′

+ (Ωh)×W m,p+ (Ωh)

=∑

|α|≤m

∫Ωh

fhα∂

α+v

hdxh, 1 < p <∞, cf. (2.107), and

‖fh‖W−m,p′

+ (Ωh):= sup

‖vh‖W

m,p+ (Ωh)=1

{|〈fh, vh〉|} =

⎛⎝ ∑|α|≤m

‖fα‖p′

Lp′h (Ωh)

⎞⎠1/p′

. (8.30)

By (8.24), (8.29), ‖ · ‖W m,p+ (Ωh), and ‖ · ‖W m,p

h (Ωh) are equivalent. For systems wemodify these definitions, corresponding to (8.69), as

(�uh, �vh)Hm+ (Ωh,Rq) :=

q∑i=1

∑|α|≤m

(∂α+u

hi , ∂

α+v

hi

)L2

h(Ωh), �uh =

(uh

1 , . . . , uhq

), (8.31)

|�uh|W m,p+ (Ωh,Rq), ‖�uh‖W m,p

+ (Ωh,Rq) :=

⎛⎝ q∑i=1

∑|α|≤m

∥∥∂α+u

hi

∥∥p

Lph(Ωh)

⎞⎠1/p

, cf. (2.107), with

‖�fh‖W−m,p′

+ (Ωh,Rq):=

(q∑

i=1

∥∥fhi

∥∥p′

W−m,p′+ (Ωh)

)1/p′

, for �fh =(fh1 , · · · , fh

q

).

For strong difference equations, and in Proposition 8.4, we additionally needW k,p

+ (Ωh) with uh(x) = 0∀x �∈ Ωh for k = 2m or 0 ≤ k < m.

Proposition 8.3. All these spaces of discrete functions with any, but fixed choicesof ∂α

+ or ∂α− or ∂α

h , are discrete Lebesgue and Sobolev spaces, in particular Banachspaces, with respect to the scalar products, and norms defined in (8.25)–(8.31).


As for standard Sobolev spaces, discrete Sobolev spaces satisfy (8.32), sometimescalled the Sobolev inequality, cf. Yulin [673], Theorems 4, 9. The norm and seminorm inW k,p

0,+(Ωh,Rq) in (8.33) are equivalent, cf. Ladyzenskaja [462], Schumann and Zeidler,[574], Lemma 4, Temam [624], a consequence of his Proposition 3.3. Its generalizationto m > 1 is straightforward.

Proposition 8.4. For any discrete function �uh ∈Wm,p+ (Ωh), and for any given ε > 0

there exists a constant K(ε,m, q) ∀0 ≤ k < m, 2 ≤ p ≤ ∞, independent of h, and uh,

such that the following inequality is valid ∀uh ∈W k,p+ (Ωh):

|�uh|W k,p+ (Ωh) ≤ ε|�uh|Hm

+ (Ωh) + K(ε,m, q)|�uh|L2h(Ωh). (8.32)

‖�uh‖W k,p+ (Ωh) and |�uh|W k,p

+ (Ωh) are equivalent norms in W k,p0,+(Ωh), hence there exists

C = C(m, q) ∈ R+, independent of h, such that

‖�uh‖W k,p+ (Ωh) ≤ C|�uh|W k,p

+ (Ωh) ≤ C‖�uh‖W k,p+ (Ωh) ∀uh ∈W k,p

0,+(Ωh). (8.33)

This allows a modification of (8.32) with ‖�uh‖W k,p+ (Ωh), ‖�uh‖W m,p

+ (Ωh) in W k,p0,+(Ωh).

In the following difference equations we sometimes employ the following mean valuefunction. We will come back to it in our later approximations. For (8.20), we assumeu or f to be extended to an Ω1 ⊃ Ωh or to Rn and define:

fh(x) :=1hn

∫ch(x)

f(x)dx. (8.34)

Proposition 8.5. This integral mean value fh(x) approximates f(x) ∈ C1L

(Ωh

e

):

fh(x) :=1hn

∫ch(x)

f(x)dx, |fh(x)− f(x)| ≤ Lh2. (8.35)

For functions only defined in Ω ⊂ Ωhe , extensions beyond Ω as (8.24) or in Gilbarg and

Tru-dinger [346], Lemma 6.37 have to be used.With

∑2m′,ev|α|=2 indicating even multi-indices and α + 1, 1 := (1, . . . , 1), we obtain

the asymptotic expansion for f ∈ C2m′L

(Ωh

e

)∣∣∣∣∣∣fh(x)−

⎛⎝f(x) +2m′,ev∑|α|=2

(h/2)α

(α + 1)!∂αf(x)

⎞⎠∣∣∣∣∣∣ ≤ Ch2m′+1. (8.36)

The analogous result holds for f ∈W 2m′+1,p(Ωhe ) and

∑2m′,ev|α|=2 · · · and |f |C2m′

L (Ωhe )

replaced by the averaged Taylor polynomial Q2m′+1,evf in (4.33) and by|f |W 2m′+1,p(Ωh

e ).


Proof. We use the Taylor expansion:

f(x + tej) = f(x) +m∑

i=1

(t)i∂i

xjf(x)i!

+ R1 with ∂ixjf = ∂if/

(∂xj

i)

and

Rm1 := R1 : =

∫ t

0

(t− σ)m−1

(m− 1)!

(∂m

xjf(x + σ)− ∂m

xjf(x)

)dσ with (8.37)

|R1| ≤ L(t)m+1

m!for ∂m

xjf ∈ CL(Ω).

This is inserted into∫ch(x)

f(x)dx =∫ h/2

−h/2

· · ·∫ h/2

−h/2

f

⎛⎝x +m∑

j=1

tjej

⎞⎠ dt1 · · · dtn

=∫ h/2

−h/2

· · ·

⎛⎝∫ h/2

−h/2

f

⎛⎝x + t1e1 +m∑

j=2

tjej

⎞⎠ dt1

⎞⎠ · · · dtn and with (8.37)

=∫ h/2

−h/2

· · ·

⎛⎝∫ h/2

−h/2

⎛⎝f

⎛⎝x +m∑

j=2

tjej

⎞⎠+

m,ev∑i=1

(t1)i∂i

x1f(x +

∑mj=2 tjej

)i!

+ R1

⎞⎠ dt1

⎞⎠ · · · dtnand, with m = 2m′ and x+ := x +

∑m,evj=2 tjej

=∫ h/2

−h/2

· · ·h∫ h/2

−h/2

(f(x+) +

m,ev∑i=2

(h/2)i ∂ix1f(x+)

(i + 1)!+ R1

)dt2 · · · dtn.

By induction we obtain the claim. �

8.4 General elliptic problems with Dirichlet conditions, and theirdifference methods

8.4.1 General elliptic problems

In Section 8.1 we have already described the unsymmetric and the symmetric types ofdifference methods. The previous definitions allow a generalization of these differenceequations to all types of elliptic differential equations studied in Chapter 2. To avoidobscuring the basic ideas we start, similarly to the above two examples in (8.1) and(8.8) with the case that Ω has the form (8.9). In contrast to (8.1) and (8.8) we considerunsymmetric difference methods first and generalize them to curved boundaries. Thisyields methods converging of order 1. We list for all cases the quartets of strong and

8.4. General elliptic equations with Dirichlet boundary conditions 573

weak differential and difference equations, e.g. As, Ahs , A,A

h. We have seen in (8.12)–(8.15) that for difference equations, replacing ∂+u

h, ∂−uh by ∂huh does not make

sense, cf. (8.14). Nevertheless, we will formulate symmetric divergence forms for allour problems. This modification to symmetric difference methods converges of order 2.

We start, summarizing for the different elliptic problems studied here, with thedefinitions of ellipticity, and the conditions for the coefficients, implying the bound-edness, and Hm

0 (Ω,Rq)-or Wm,p0 (Ω,Rq)-coercivity of the weak forms of the principal

parts for m, q ≥ 1. All equations have to satisfy a strong Legendre, or in the caseof higher order systems, a strong Legendre–Hadamard condition. We use the unifiednotation aα,β for equations, q = 1, and Aα,β(x) =

(aij

α,β(x))qi,j=1

∈ Rq×q for systems,q > 1, with aα,β = aij , and Aα,β = Akl for m = 1, and ∀|α|, |β| ≤ 1. We include theconditions for nonlinear operators G implying a bounded derivative G′(u0) with acoercive principal part. Difference equations will require appropriate modifications.

Summary 8.6. Let the bilinear form, a(·, ·), be induced by the (weak) linear operatorA or the linearized G′(u0). We assume the following conditions 1.–4. for boundedness,and (strong, and uniform) ellipticity with ϑ = (ϑ1, . . . ϑn) ∈ Rn:

1. For linear equations of second order, cf. (8.46), (8.47), (8.60), we require ∀0 ≤i, j ≤ n, for almost every (a.e.) x ∈ Ω, cf. Chapter 2, (2.136),

aij ∈ L∞(Ω), ∃λ > 0, a.e. ∀x ∈ Ω : λ|ϑ|2n ≤n∑

i,j=1

aij(x)ϑiϑj ∀ϑ ∈ Rn, (8.38)

where aij ∈ L∞(Ω) implies a.e.∑n

i,j=1 aij(x)ϑiϑj < Λ|ϑ|2n for Λ > λ > 0.2. For linear equations of order 2m,m > 1, in (8.63), cf. Theorems 2.36, (2.79),

(2.106), let

∀|α|, |β| ≤ m : aα,β ∈ L∞(Ω), and for ∀|α|, |β| = m : aα,β ∈ C(Ω), (8.39)

∃λ > 0, a.e. ∀x ∈ Ω : λ|ϑ|2mn ≤

∑|α|=|β|=m

aα,β(x)ϑαϑβ ∀ϑ ∈ Rn, x ∈ Ω.

This implies a.e.∑

|α|=|β|=m aα,β(x)ϑαϑβ ≤ Λ|ϑ|2mn ∀ϑ ∈ Rn, x ∈ Ω.

3. For linear systems of order 2m,m ≥ 1, q > 1, in (8.69), (8.70), we combinefor m = 1 the strong Legendre condition (8.40), cf. Theorems 2.89, (2.343),and for m > 1, the strong Legendre–Hadamard condition (8.41), cf. Theorems2.104, (2.394). Assume Akl, Aα,β ∈ L∞(Ω,Rq×q) for m ≥ 1. For m > 1, and∀|α|, |β| = m let Aα,β ∈ C(Ω,Rq×q). With �ϑ = (�ϑ1, . . . , �ϑn) ∈ Rn×q, ϑ ∈ Cn, η ∈Cq, we assume for (8.70) that ∃λ > 0 such that ∀�ϑ, ϑ, η

for m = 1 : λ|�ϑ|2nq ≤n∑

k,l=1

(Akl(x)�ϑl, �ϑk

)q≤ Λ|�ϑ|2nq, a.e.x ∈ Ω, (8.40)

for m > 1 : λ|ϑ|2m|η|2 ≤i,j=1,...,q∑|α|=|β|=m

aijαβ(x)ϑβηjϑ

αηi ∀x ∈ Ω. (8.41)


For m > 1 this47 implies∑i,j=1,...,q

|α|=|β|=m aijαβ(x)ϑβηjϑ

αηi ≤ Λ|ϑ|2m|η|2 ∀x ∈ Ω.

4. A nonlinear problem, G, as in Subsections 8.4.5–8.4.7, is called elliptic if itsderivative, G′(u0), satisfies the appropriate condition 1.–3., for m, q ≥ 1.

5. Then a(·, ·), induced by the previous linear or linearized operators, is bounded,and its principal part, ap(·, ·), is Hm

0 (Ω,Rq)-coercive. This Hm0 -coercivity remains

correct for bilinear forms, induced by G′(u0) of nonlinear G, for m, q ≥ 1, and inWm,p(Ω,Rq) for 2 ≤ p ≤ ∞, cf. Section 2.7, and Theorems 2.118, 2.119, 2.122,2.124, 2.125, 2.126, 2.127.

6. For a constant a0 > 0, the extended principal part, aep(u, v) := ap(u, v) +

a0·(u, v)L2(Ω,Rq), is also Hm(Ω,Rq)- coercive, cf. Theorem 1.24.7. The strong differential operators, As or (G′(u0))s, are bounded if some coeffi-

cients aα,β, and Aα,β,respectively, additionally are boundedly differentiable, moreprecisely:

aα,β ∈W |β|,∞(Ω), and Aα,β ∈W |β|,∞(Ω,Rq×q) ∀|α|, |β| ≤ m. (8.42)

8. We consider difference equations in the weak forms (8.47), (8.63), (8.65), (8.67),(8.71)–(8.73), and in the corresponding strong forms, essentially the classicalFDEs, (8.44), (8.62), (8.64), (8.66), (8.76). These difference equations are welldefined, if the previous L∞(Ω),W |β|,∞(Ω), . . . , are replaced by C(Ω), C|β|(Ω), . . . ,or by bounds on Gh or aij ∈ H1(Ω), . . . , by aij(x), cf. (8.34), and Remark 8.7.

9. The discrete principal parts, ahp(·, ·), are defined as the sum of the highest order

difference terms, corresponding to the coefficients, e.g. in (8.38)–(8.41).

The discrete forms of these linear operators, the Ah, or the linearizations of thediscrete operator, Gh, for m, q ≥ 1, and the coercivity of the principal parts areessential for stability.

In Subsections 8.4.2 and 8.4.3, we give the details for second order linear equa-tions for unsymmetric and symmetric forms. The other cases we restrict to listingthe equations in Subsections 8.4.4 ff. The domains, and ranges, e.g. the D

(Ah

s

)=

H2+(Ωh) ∩H1

0,+(Ωh), R(Ah

s

)= L2

h

(Ωh

0

)of the As, A

hs in (8.43), and the Ah in (8.47)

have to be obviously modified. For nonlinear operators the domains are subsets ofthe corresponding Sobolev spaces in (8.43), (8.47). Conditions for the coefficients arelisted in Remark 8.7 and Summary 8.6.

For general domains as in (8.18) with boundary grid points in (8.19) not in theboundary of Ω, specific difference methods require a separate discretization of thedifferential equation, and the boundary conditions. This would be necessary for fullynonlinear problems in Subsection 8.4.7, if we were to discuss the details. We indicatesome other examples in Section 8.7.

8.4.2 Second order linear elliptic difference equations

We present the detailed relations between the strong and weak differential anddifference equations, As, A

hs , A,A

h. We start assuming Ω in the form (8.9), and47 For difference methods several definitions for regular elliptic systems are used in the literature.

They are all more or less equivalent, cf. [151]. Here we have enforced in (8.41), towards the needs for

difference methods, the condition ϑ ∈ Rn, η ∈ Cq , ϑα for elliptic systems into ϑ ∈ Cn, η ∈ Cq , ϑα

.


consider the strong unsymmetric form of difference equations in divergence form.Combining the ∂j

−, ∂i+u

h(x), yields one of the classical or natural difference equations.The unsymmetric methods of order 1 are modified in Subsection 8.4.3 on rectangulardomains to symmetric methods of order 2.

We apply partial summation (8.49) in Proposition 8.8 to difference equations in thestrong form to get the weak forms. The following As is well defined for u ∈ H2(Ω), aij ∈H1(Ω), cf. (2.16), and Ah

s for uh ∈ H2+(Ωh). We use (−1)j>0 = 1 for j = 0, else = −1,

and sometimes replace aij(x), f(x) by ahij(x), fh(x) in (8.34).

As : H2(Ω) ∩H10 (Ω) → L2(Ω), Ah

s : H2+(Ωh) ∩H1

0,+(Ωh) → L2h

(Ωh

0

),

Asu0 =n∑

i,j=0

(−1)j>0∂j(aij∂

i u0) = f in L2(Ω), cf. (8.42), (8.43)

Ahsu

h0 (x) =

n∑i,j=0

(−1)j>0∂j−(aij(x)∂i

+uh0 (x)

)= fh(x) ∀x ∈ Ωh

0 , with (8.44)

boundary conditions, cf.(8.24), (2.16), u0 ∈ H10 (Ω), uh

0 (x) ∈ H10,+(Ωh). (8.45)

Similarly to Table 8.1, we visualize the unsymmetric (8.44) for x ∈ Ωh0 ⊂ R2. We

use the same abbreviation as in Table 8.1, by listing the coefficients of, but omittingthe unknowns uh(x± hei). We include, here and below, only the highest order terms.The three-point stars in the direction of ei, i = 1, 2, for the −h2∂i

−aii∂i+u

h(x) arelisted in Table 8.2. They have to be combined with the seven-point stars for the−h2

(∂2−a12∂

1+ + ∂1

−a21∂2+

)uh(x) in Table 8.3. Summation of the numbers in Table

8.2, i = 1, 2, and in Table 8.3 yields the seven-point star for the full (8.44) in Table 8.4.Obviously Table 8.4 reduces to Table 8.1 for a11 = a22 = 1, a12 = a21 = 0.

The following dilemma applies to all our elliptic difference equations. For (8.43) weonly need aij ∈ H1(Ω). This is certainly problematic for (8.44).

Remark 8.7.

1. aij ∈ H1(Ω) in (8.43) is sufficient for analytical results: We require, for sim-plicity, and fitting to the standard situation in difference methods, aij ∈ C(Ω),with well-defined aij(x)∀x ∈ Ωh in (8.44). For interesting cases aij ∈ H1(Ω), . . . ,we might replace the aij(x)∀x ∈ Ωh by aij(x), as defined in (8.34). This isunproblematic for linear(ized) problems. With minor modifications all resultsremain correct. The proof for this claim is left as an exercise for the interestedreader.

2. For semi-, quasi-, and fully nonlinear problems below, this averaging, necessarilydefined with respect to x, but for fixed values of uh(x), e.g., (8.64),(8.65), wouldbe problematic.

3. The conditions for boundedness and ellipticity of linear(ized) operators inSummary 8.6 may be appropriately modified for difference equations, andsystems for higher order problems as well. In particular, generalizations to the


Table 8.2: Three-point star for (8.44), x ∈ Gh ⊂ R2

for −h2(∂i−aii∂

i+

)uh(x), i = 1, 2.

−aii(x− hei) aii(x) + aii(x− hei) −aii(x)

Table 8.3: Seven-point star for (8.44), x ∈ R2, and−h2

(∂2−a12∂

1+ + ∂1

−a21∂2+

)uh(x).

a21(x− he1) −a21(x) 0−a21(x− he1) (a12 + a21)(x) −a12(x)

0 −a12(x− he2) a12(x− he2)

Table 8.4: Seven-point star for the complete −h2× (8.43) and x ∈ R2.

a21(x− he1) −a22(x)− a21(x) 0−(a11 + a21)(x− he1) a11(x− he1) + a22(x− he2) −(a11 + a12)(x)

0 +(a11 + a22 + a12 + a21)(x) a12(x− he2)−(a22 + a12)(x− he2)

W 1,p0 (Ω),Wm,p

0 (Ω) situation and the appropriate conditions for this and thefollowing subsections are discussed there.

The usual multiplication of Asu with v ∈ H10 (Ω), integration over Ω, and partial

integration yields the (standard) weak bilinear form a(·, ·), and the induced A. Bothare well defined for u, v ∈ H1(Ω) and aij ∈ L∞(Ω), see (2.18). With

〈Au, v〉H−1(Ω)×H10 (Ω) = (Asu, v)L2(Ω)∀u ∈ H2(Ω), v ∈ H1

0 (Ω), (8.46)

aij ∈ H1(Ω)∀j > 0,

compute u0 ∈ H10 (Ω) : a(u0, v) :=

n∑i,j=0

∫Ω

aij∂iu0∂

jvdx = 〈Au0, v〉H−1(Ω)×H10 (Ω)

= 〈f, v〉H−10 (Ω)×H1

0 (Ω)∀v ∈ H10 (Ω) with f ∈ H−1

0 (Ω).

Similarly to the previous strong form, Ahs , we introduce the corresponding weak

operator, Ah, and weak bilinear form, ah(·, ·). Both are well defined for uh, uh0 , v

h ∈H1

+(Ωh) and aij ∈ C(Ω). Choosing uh, vh as in (8.24), we determine uh0 from

ah : H10,+(Ωh) × H1

0,+(Ωh) → R, Ah : Vhb = H1

0,+(Ωh) → H−10,+(Ωh) = V ′h

b , uh0 ∈ Vh

b

ah(uh

0 , vh)

:=n∑

i,j=0

∫Ωh

aij∂i+u

h0∂

j+v

hdxh =n∑

i,j=0

hn∑

x∈Ωh

(aij∂

i+u

h0∂

j+v

h)

(x)

=:⟨Ahuh

0 , vh⟩V′h

b ×Vhb

= 〈fh, vh〉V′hb ×Vh

b∀vh ∈ Vh

b = H10,+(Ωh). (8.47)


Depending upon the structure of f (and of Au) we employ different kinds of approx-imations. The (f, vh)L2

h(Ωh) or 〈f, vh〉V′hb ×Vh

bcan be evaluated exactly or appropriate

quadrature approximations may be used, cf. Section 5.4. For nonsmooth f, we mayreplace f(x) by fh(x) := fh(x) in (8.34). We denote all these approximations as〈fh, vh〉V′h

b ×Vhb. For other strategies, cf., e.g. Thomee and Westergren [627], Heinrich

[395], Bube et al. [151,605] and Grossmann et al. [374–376].Similarly to partial integration transforming differential equations in strong form

into the weak form, we apply partial summation. For avoiding unnecessary repetition,we formulate Proposition 8.8 for linear and quasilinear elliptic equations of orders2m,m ≥ 1, in (8.62), (8.66), with obvious extensions to systems. We require for vh ∈Hm

0,+(Ωh) the extension (8.24), but not for all uh considered here.We simplify the notation for the proof, and the following results by introducing the

grid points, and their function values in Ωh in the ej direction as, cf. (8.16).

xi := x0 + ihej , i ∈ Z, xi ∈ Ωh0 for i = 1, . . . , k − 1, xi ∈ ∂Ωh for i ≤ 0, i ≥ k,

ui := uh(xi), and the Ωj := {xi, i = 1, . . . , k − 1},Ωj:= {xi, i = 0, . . . , k},

Ωh,j

m := {xi, i = −m + 1, . . . , k + m− 1},Ωj:= Ω

h,j

1 , with sums, e.g.,∑Ω

j

vh∂j−u

h :=k∑

i=0

(vh∂j

−uh)

(xi), and obvious modifications for Ωj ,Ωh,j

m ,Ωh, . . . .

Proposition 8.8. For uh ∈ H2m+ (Ωh), vh ∈ Hm

0,+(Ωh),m ≥ 1, (8.48)–(8.52) are valid,with obvious (vh, · · · )L2

h(Ωh0 ) = (vh, · · · )L2

h(Ωh), . . . in (8.49)– (8.52), hence

for j > 0 : −∑Ωj

vh∂j−u

h = −∑Ω

j

vh∂j−u

h =∑Ω

j

uh∂j+v

h ⇒ for m = 1, (8.48)

⎛⎝ n∑i,j=0

vh, (−1)j>0∂j−(aij∂

i+u

h0

)⎞⎠L2

h(Ωh0 )=L2

h(Ωh)

=

⎛⎝ n∑i,j=0

aij∂i+u

h0 , ∂

j+v

h

⎞⎠L2

h(Ωh)

,(8.49)

for m > 1, |β| ≤ m : (−1)|β|(vh, ∂β

−uh)L2

h(Ωh0 )=L2

h(Ωh)=(uh, ∂β

+vh)L2

h(Ωh), (8.50)

(−1)|β|(∂β−(Aβ

(·, uh,∇m

+uh))

, vh)

L2h(Ωh

0 )=...=(Aβ

(·, uh,∇m

+uh), ∂β

+vh)L2

h(Ωh)⇒

⎛⎝ ∑|α|,|β|≤m

vh, (−1)|β|∂β−(aαβ∂

α+u

h0

)⎞⎠L2

h(Ωh0 )=..

=

⎛⎝ ∑|α|,|β|≤m

aαβ∂α+u

h0 , ∂

β+v

h

⎞⎠L2

h(Ωh)

(8.51)

578 8. Finite difference methods⎛⎝ ∑|β|≤m

vh, (−1)|β|∂β−Aβ

(·, uh

0 , · · · ,∇m+uh

0

)⎞⎠L2

h(Ωh0 )=L2

h(Ωh)

=

⎛⎝ ∑|β|≤m

Aβ

(·, uh

0 , · · · ,∇m+uh

0

), ∂β

+vh

⎞⎠L2

h(Ωh)

. (8.52)

Equations (8.48) ff. remain correct if ∂+, and ∂− are interchanged.

Remark 8.9.

1. According to (8.48)–(8.52), the following weak forms of the difference methods use(·, ·)L2

h(Ωh). The unequal previous condition uh ∈ Hm

+ (Ωh),m ≥ 1, vh ∈ Hm0,+(Ωh)

for uh, vh is important, since we apply (8.48)–(8.52) in the difference methods tovery different terms uh, cf. (8.44), Proposition 8.10, and (8.56),(8.65).

2. These results remain correct, if we allow uh ∈Wm,p+ (Ωh), 2 ≤ p <∞, and

for the linear case vh and the quasilinear case ∀uh : Aβ

(·, uh, · · · ,∇m

+uh)∈

W−m,p′

0,+ (Ωh), 1/p + 1/p′ = 1, respectively

Proof. We start with the first order result (8.48) for j > 0. Since the proofs for∂j+u

h, and ∂j−u

h are very similar, we only consider ∂j−u

h. We use H10,+(Ωh) � vh,

vh(x) = 0∀x ∈ Gh \ Ωh0 , only for vh, but not for uh.

j > 0 : −∑Ωj

vh∂j−u

h = −∑Ω

j

vh∂j−u

h (8.53)

= −k∑

i=0

vi(ui − ui−1)/h = −(

k∑i=0

viui −k−1∑

i′=−1

vi′+1ui′

)/h

v0=vk=vk+1=0=

(k∑

i=0

(vi+1 − vi)ui

)/h =

∑Ω

j

uh∂j+v

h.

Replacing the previous uh by wh :=(a�j∂

�+u

h)

we obtain with v0 = vk = vk+1 = 0

j > 0 : −∑Ωj

vh∂j−(a�j∂

�+u

h)

=∑Ω

j

(a�j∂

�+u

h)∂j+v

h. (8.54)

For (8.49), we multiply (8.54) with hn and sum over all 0 ≤ i = �, j ≤ n.The first line of (8.50) with 1 < m ≥ |β| ≥ 1 is proved inductively. Equation (8.54)

with m > 1 ≥ |α|, |β| = 1 represents the beginning of the induction. So we continuewith considering the components βj ≥ 2, and the sum over the lines Ωj . In (8.53), wereplace, for the next step of the induction, the vh, and Ωj , and Ω

jby, e.g. ∂�

+vh, and


Ωj, and Ω

h,j

2 , respectively. We obtain with 0 = ∂�+v−1 = ∂�

+vk+1 :

∂βj−1− ui := ∂

βj−1− u(xi) : −

∑Ω

j

∂�+v

h∂βj

− uh = −∑Ω

h,j2

∂�+v

h∂j−∂

βj−1− uh (8.55)

= −k+1∑

i=−1

∂�+vi∂

j−∂

βj−1− ui

= −(

k+1∑i=−1

∂�+vi∂

βj−1− ui −

k+1∑i=−1

∂�+vi∂

βj−1− ui−1

)/h

0=∂�+v−1=∂j

+vk+2=

(k+1∑

i=−1

(∂�+vi+1 − ∂�

+vi

)∂

βj−1− ui

)/h

=∑Ω

h,j2

(∂�+∂

j+

)vh∂

βj−1− uh.

This allows reducing inductively each component βj > 0 in (8.55) to 0, and proceedingas above. Then vh ∈ Hm

0,+ is sufficient for the first line of (8.50).For the second line of (8.50), the quasilinear equations of order 2m in (8.66) below,

we obtain with

βj := β − ej , ∂β− = ∂j

−∂βj

− , wh := (−1)|βj |∂βj

− Aβ

(·, uh(·), . . . ,∇m

+uh(·))

:∑Ω

j

vh((−1)|β|∂β

−Aβ

(·, uh(·), . . . ,∇m

+uh(·)))

= −∑Ω

j

vh∂j−

((−1)|β

j |∂βj

− Aβ

(·, uh(·), . . . ,∇m

+uh(·)))

= −∑Ωj

vh∂j−w

h = −∑Ω

j

vh∂j−w

h = −k∑

i=0

vi(wi − wi−1)/h

= −(

k∑i=0

viwi −k∑

i=0

viwi−1

)/h

0=v0=vk+1= =

(k∑

i=0

wi(vi+1 − vi)/h

)

=∑Ω

j

wh∂j+v

h =∑Ω

j

((−1)|β

j |∂βj

− Aβ

(·, uh(·), . . . ,∇m

+uh(·)))

∂j+v

h.

Again, we reduce |β| to 0. Now multiplying the first and second line of (8.50) with hn,and summation over all directions yields (8.51) and (8.52). �

We reformulate the results in Proposition 8.8, (8.49), (8.51) now for the weak andstrong difference operators, Ah

s , and Ah, and extend them to systems.


Proposition 8.10. For the difference operators Ahs , A

h in (8.44), (8.47) for m = 1and (8.62), (8.63), (8.71) for m > 1, q ≥ 1 we obtain ∀vh ∈ Vh

b = Hm0,+(Ωh,Rq) :⟨

Ahsu

h, vh⟩

L2(Ωh0 ,Rq) =

⟨Ah

suh, vh

⟩L2(Ωh,Rq)

= 〈Ahuh, vh〉V′hb ×Vh

b. (8.56)

Equation (8.56) remains correct for the quasilinear problems Ghs , and Gh in (8.65),

(8.67), and linear and quasilinear systems (8.71) and (8.72), (8.73), cf. Remark 8.9

Exchanging the two difference operators ∂h+, and ∂h

− in the previous (8.44) (8.47),and all the following cases, allows the same convergence results. In the symmetric formbelow we will use the mean value of these two equations.

Equation (8.44) is a classical system of difference equations for (8.43), and has to besolved. It is, possibly with slight modifications, included in the usual software packages.For the corresponding weak difference form (8.47) of (8.44), and its generalizationsbelow, a direct stability, and convergence proof with respect to the ‖ · ‖H1

+(Ωh) willbe given in Subsections 8.5.4 ff. This implies the results for (8.44) as well. In fact, astrong solution of (8.44) satisfies, by (8.56), the weak (8.47) as well, and vice versa:

uh0 ∈ H2

+(Ωh) ∩(Vh

b = H10,+(Ωh)

):(Ah

suh0 , v

h)L2

h

(Ωh

0

) = (fh, vh)L2

h

(Ωh

0

)∀vh ∈ Vhb

f ∈ L2(Ω) =⇒ uh0 ∈ Vh

b :⟨Ahuh

0 , vh⟩V′h

b ×Vhb

= (fh, vh)L2h(Ωh)∀vh ∈ Vh

b . (8.57)

Hence we solve (8.44) by standard software packages. This solution inherits theconvergence properties of the weak solution, uh

0 , of (8.47), cf. Theorem 8.11.If a discrete solution uh

0 (x)∀x ∈ Ωh for (8.44) or (8.47) with the boundary conditions(8.45) is computed, it is straightforward to determine its first and second divideddifferences, ∂i

+uh0 (x), ∂j

+∂i+u

h0 (x), yielding the desired uh

0 ∈ Vhb or uh

0 ∈ H2+(Ωh) ∩ Vh

b .This remains correct for order 2m, and the solutions uh

0 ∈ Vhb = Hm

0,+(Ωh,Rq), anduh

0 ∈ H2m+ (Ωh,Rq) ∩ Vh

b , m, q ≥ 1, of the weak, and strong difference problems in thefollowing subsections.

We use the same approach as in the previous chapters, related to Stetter [596]. Thiscontrasts to, e.g. Raviart [545], Temam [621] and Schumann and Zeidler [574, 678].They apply external approximations.

We want to outline the differences in alternative approaches. Let uh0 and uh

0 indicatethe solutions of the strong and the weak problem, respectively. For the strong Ah

s uh0 =

fh(= f) with special A, often stability results are obtained via M-matrices, see, e.g.Hackbusch [387], Sections 4.2 ff. This yields estimates of the form

Ahs u

h0 (x) = f(x)∀x ∈ Ωh

0 , uh0 (x) = 0∀x ∈ ∂Ωh (8.58)

=⇒ supx∈Ωh

∣∣uh0 (x)

∣∣ = ∥∥uh0

∥∥L∞

h (Ωh)≤ C‖f‖L∞

h (Ωh).

The approach, based upon the weak Au0 = f, starts with the coercive bilinear form,ap(·, ·), of the principal part, Ap, of the above general A. We will show that its discretecounterpart ah

p(·, ·), cf. Summary 8.6, is again coercive, so stable as well. Thus theconsistency, and Theorem 3.29, imply, for a boundedly invertible A, the stability of


Ah, hence the existence of a weak solution uh0 = uh

0 by (8.57), such that

for fh ∈ L∞h (Ωh), and for uh

0 (x) = 0 ∀ x ∈ ∂Ωh : Ahs u

h0 = fh ⇐⇒ (8.59)⟨

Ahuh0 , v

h⟩V′h

b ×Vhb

= ah(uh

0 , vh)


b∀vh ∈ Vh

b = H10,+(Ωh), and∥∥uh

0

∥∥H1

+(Ωh)=∥∥uh

0

∥∥Vh

b

≤ C‖fh‖V′hb

=⇒∥∥uh

0

∥∥Vh

b

≤ C ′‖fh‖L∞h (Ωh) ≤ C‖fh‖V′h

b,

since ‖fh‖L∞h (Ωh) is dominated by ‖fh‖V′h

b. This argument from (8.57) to (8.59)

remains correct for the following linear cases as well, if only the ‖uh0‖H1

+(Ωh) arereplaced by ‖uh

0‖Hm+ (Ωh,Rq),m, q ≥ 1. For quasilinear problems corresponding stability–

consistency arguments apply. Fully nonlinear problems require another treatment.

Theorem 8.11. Hm+ convergence for classical (strong) difference methods:

We assume a stability estimate (8.59) for the weak Ahuh0 = fh ∈ H−m

0,+

(Ωh,Rq),m, q ≥ 1, in the form ‖uh0‖Hm

+ (Ωh,Rq) ≤ C‖fh‖H−m0,+ (Ωh,Rq). Let A : Hm

0 (Ω,Rq)

→ H−m0 (Ω,Rq), and As : H2m(Ω,Rq) ∩Hm

0 (Ω,Rq) → L2(Ω,Rq) ⊃ L∞(Ω,Rq) beboundedly invertible. This implies ‖uh

0‖Hm+ (Ωh,Rq) ≤ C‖f‖L∞

h (Ωh,Rq) for the solutionof Ah

s uh0 = fh ∈ L∞

h (Ωh,Rq) as well, strongly improving (8.58).

This ‖uh0‖Hm

+ (Ωh,Rq) ≤ C‖f‖L∞h (Ωh,Rq) is a special case of regularity results for

the solutions of difference equations in the style of, e.g. Bube et al. [151, 605],Fossmeier [320], Geissert [339], Hackbusch [381, 383, 384, 387] and Thomee andWestergren [627].

Other concepts of generalized weak solutions for difference equations are studied byJovanovich et al. [416,430–432,613], and the viscosity solutions for nonlinear problemsby Crandall and Lions [217] ff., cf. Subsection 8.4.7

8.4.3 Symmetric difference methods

After our excursion to linear and quasilinear equations, and systems of orders 2 and2m, we return to linear equations of order 2, and their symmetric difference equations.The symmetric forms are particularly interesting, since the discretization error for thedifference equations is O(h2), compared to O(h) for unsymmetric forms. The abovediscussion has shown that replacing the ∂i

± by ∂ih does not yield usable symmetric

forms. However, for the previous unsymmetric forms a simple modification yields thedesired symmetric forms. This idea is identically applicable to all the following cases.Only the boundary conditions for higher order equations have to be modified, cf.Subsection 8.4.4. We define the strong and the corresponding weak symmetric discreteoperators Ah

ss, and Ahws, and the bilinear form ah

ws(·, ·) as the mean of two difference


equations, exchanging ∂i+, and ∂i

− :

Ahssu

h(x) :=12

n∑i,j=0

(−1)j>0

(∂j−(aij(x)∂i

+uh(x)) + ∂j

+(aij(x)∂i−u

h(x))), (8.60)

ahws(u

h, vh) :=12

n∑i,j=0

∫Ωh

aij(∂i+u

h∂j+v

h + ∂i−u

h∂j−v

h)dxh =:⟨Ah

wsuh, vh

⟩V′h

b ×Vhb

.

We turn to the boundary conditions. The error for the difference equations, ofO(h2), only makes sense if the boundary conditions are discretized equally accuratelyor better. For the cuboidal domains Ω in (8.9) this is straightforward: we chooseΩh,Ωh

0 , ∂Ωh as in (8.11). Then the boundary points x ∈ ∂Ωh satisfy x ∈ ∂Ω, hence,allow an exact discretization uh(x) = 0 ∀x ∈ ∂Ωh. For curved domains the techniquesin Section 8.7 apply.

So, based upon Proposition 8.10, and this boundary condition we determine

uh0 ∈ Vh

b : Ahssu

h0 (x) = f(x)∀x ∈ Ωh

0 ⇐⇒ (8.61)⟨Ah

wsuh0 , v

h⟩V′h

b ×Vhb

= 〈f, vh〉V′hb ×Vh

b∀vh ∈ Vh

b .

Only the highest order divided differences in (8.60), so i, j = 1, . . . , n, are requiredin this special form. For the lower order terms with i · j = 0 we may choose any ofthe symmetric formulas, e.g. as in (8.13). In fact, they do satisfy the conditions inProposition 8.4, thus the Vh

b -coercivity of the principal part is maintained.We visualize the symmetric Ah

ss in (8.60), starting with the three-point stars inTable 8.5, the seven-point stars in Tables 8.6, 8.7, and the final seven-point starin Table 8.8. Note that the factor 1/2 in front of every table represents the factor1/2 in (8.60). Furthermore we define, in Tables 8.5–8.7, several a±ij used in Table 8.8.

Table 8.5: Three-point star for (8.60) for −h2(∂i−aii∂

i+ + ∂i

+aii∂i−)uh(x).

1/2 −a−ii := −aii(x)− aii(x− hei) a−ii + a+ii −a+

ii := −aii(x)− aii(x + hei)

Table 8.6: Seven-point star for (8.60), and −h2(∂2−a12∂

1+ + ∂2

+a12∂1−)uh(x).

a12(x + he2) −a+12 := −a12(x + he2) 0

1/2 −a12(x) 2a12 := 2a12(x) −a12(x)0 −a−12 := −a12(x− he2) a12(x− he2)

Table 8.7: Seven-point star for (8.60), and −h2(∂1−a21∂

2+ + ∂1

+a21∂2−) uh(x).

a−21 := a21(x− he1) −a21(x) 01/2 −a21(x− he1) 2a21 := 2a21(x) −a21(x + he1)

0 −a21(x) a+21 := a21(x + he1)


Table 8.8: Seven-point star for the complete −h2× (8.60) and x ∈ Ωh0 ⊂ R2.

a+12 + a−21 −a+

22 − a+12 − a21 0

1/2 −a−11 − a12 − a−21 a+11 + a−11 + a+

22 + a−22 −a+11 − a12 − a+

21

+2a12 + 2a21

0 −a−22 − a−12 − a21 a−12 + a+21

8.4.4 Linear equations of order 2m

For these equations we need the higher order Dirichlet boundary conditions mentionedabove. So we assume the discrete boundary values in the form uh ∈ Vh

b = Hm0,+(Ωh)

as in (8.28), and the same for Hm0,−(Ωh). Symmetric formulas only are worthwhile for

cuboidal domains or for the special methods in Section 8.7.We formulate the equations of order 2m: for aαβ ∈W |β|,∞(Ω), f ∈ L2(Ω), and

aαβ ∈ C|β|(Ω), f ∈ C(Ω), cf. Summary 8.6, and Remark 8.7, we determine u0, anduh

0 , respectively:

u0 ∈ H2m(Ω) ∩ (Vb := Hm0 (Ω)), and uh

0 ∈ H2m+ (Ωh) ∩

(Vh

b := Hm0,+(Ωh)

)s.t.

Asu0 =∑

|α|,|β|≤m

(−1)|β|∂β(aαβ∂αu0) = f ∈ L2(Ω), and

Ahsu

h0 (x) =

∑|α|,|β|≤m

(−1)|β|∂β−(aαβ(x)∂α

+uh0 (x)

)= f(x) ∀x ∈ Ωh

0 . (8.62)

Similarly as above we formulate, for aαβ ∈ L∞(Ω), the weak forms a(u, v), Au, and,for aαβ ∈ C(Ω), the ah(uh, vh), Ahuh, cf. (8.46), (8.47), and compute u0 ∈ Vb, anduh

0 ∈ Vhb for f ∈ V ′

b from

a(u0, v) := 〈Au0, v〉V′b×Vb

:=∫

Ω

∑|α|,|β|≤m

aαβ∂αu0∂

βvdx = 〈f, v〉V′b×Vb

∀v ∈ Vb,

uh0 ∈ Vh

b : ah(uh

0 , vh)

:= hn∑

x∈Ωh

∑|α|,|β|≤m

(aαβ∂

α+u

h0∂

β+v

h)

(x) (8.63)

=:⟨Ahuh

0 , vh⟩V′h

b ×Vhb


b

= hn∑

x∈Ωh

∑|β|≤m

(fh

β ∂β+v

h)

(x)∀vh ∈ Vhb ,


cf (2.107). The form for f in (8.63) is unusual in classical difference meth-ods, cf. (2.110). However it makes sense for the weak form, and the later errorestimates.

Generalizations to Banach spaces with aαβ ∈W∞(Ω), u, u0, v ∈Wm,p(Ω), 2 ≤ p <∞ or aαβ ∈W p1(Ω), u, u0 ∈Wm,p2(Ω), v ∈Wm,p3(Ω), with 1/p1 + 1/p2 + 1/p3 = 1and the corresponding discrete spaces Wm,p

0,+ (Ωh) are possible, similarly to Section 2.3and Subsection 2.4.4, considering the corresponding coercivity problems. This allowsresults for the discrete Hm

+ (Ωh) norm.As in (8.60) symmetric difference equations can be defined, again only for cuboidal

domains Ω, cf. (8.9), and ∀x ∈ Ωh0 or the methods in Section 8.7.

8.4.5 Quasilinear elliptic equations of orders 2, and 2m

According to the theoretical results we assume homogeneous Dirichlet boundaryconditions of the form (8.45) or (8.62). Even more general Dirichlet systems, see(2.82), Lemma 2.26, (2.158), (2.159), and Theorem 2.50 can be transformed into thestandard homogeneous Dirichlet boundary conditions. Furthermore, the first typesof nonlinear elliptic equations in Section 2.5, Subsection 2.5.3, are either specialquasilinear equations, or simply the term a00u in (8.62) has to be replaced byf(u). So we only give the difference equations corresponding to a general quasilinearform, with the boundary conditions in (8.45) or (8.62), compare (2.270), (2.306).The inhomogeneous function f is included in the nonlinear Aj and Aα. Appropri-ate conditions for the unique existence of solutions are listed in Subsections 2.5.4and 2.5.6.

These strong quasilinear equations are defined for u, u0 ∈W 2,p(Ω) ∩W 1,p0 (Ω), and

uh, uh0 ∈W 2,p

+ (Ωh) ∩(Vh

b := W 1,p0,+(Ωh)

). Applying (8.49) to (8.64) we get the weak

form Gh, cf. (8.65), defined for uh, uh0 , v

h ∈ Vhb , and 1/p + 1/p′ = 1:

Gsu0 =n∑

j=0

(−1)j>0∂jAj(·, u0,∇u0) = 0 ∈ L1/p′

(Ω), (8.64)

u0 ∈W 2,p ∩W 1,p0 (Ω),

Ghsu

h0 (x) =

n∑j=0

(−1)j>0∂j−Aj

(·, uh

0 ,∇+uh0

)(x) = 0∀x ∈ Ωh

0 , and (8.65)

⟨Ghuh

0 , vh⟩V′h

b ×Vhb

:= hn∑

x∈Ωh

n∑j=0

Aj

(·, uh

0 ,∇+uh0

)(x)∂j

+vh(x)

:= ah(uh

0 , vh)

= 0 ∀ vh ∈ Vhb = W 1,p

0,+(Ωh) with 2 ≤ p <∞

uh0 ∈ Vh

b ,∀uh ∈ Vhb ∩ D(Aj) : Aj(·, uh,∇+u

h)

∈ V ′hb = W−1,p′

0,+ (Ωh)


for the weak form. Notice that ah(uh, vh) here, and for the following nonlinearproblems, is linear and bounded in vh, but not linear in uh.

For the analysis of linearized quasilinear problems we will need general Sobolevspaces W 2m,p(Ω) ∩Wm,p

0 (Ω) with 1 ≤ p ≤ ∞, and their discrete counterparts. Accord-ing to Theorem 2.122 this linearization essentially requires the Hm

0 (Ω)-coercivity ofthe principal part, cf. the end of Subsection 8.4.4. For our discretization approachvia linearization, we again restrict p from 1 < p <∞ to 2 ≤ p <∞ in vh ∈ Vh

b =W 1,p

0,+(Ωh) and Aj(·, uh,∇+uh) ∈ V ′h

b = W−1,p′

0,+ (Ωh) or vh ∈Wm,p0,+ (Ωh). If we employ

the monotone approach, 1 < p <∞ is still possible, cf. Section 4.5 and Theorem 8.32.The following strong equations Gh

s of order 2m are transformed, by iteratively apply-ing (8.49), into the weak form Gh. Again we consider trivial Dirichlet conditions, andrequire u, u0 ∈W 2m,p(Ω) ∩Wm,p

0 (Ω), and uh, uh0 ∈W 2m,p

+ (Ωh)∩(Vh

b := Wm,p0,+ (Ωh)

)for the strong, and uh, uh

0 , vh ∈ Vh

b for the weak form, cf. (8.17) for ∇m+u, then

Gsu0 =∑

|β|≤m

(−1)|β|∂βAβ(·, u0, · · · ,∇mu0) = 0 ∈ L2(Ω), (8.66)

Ghsu

h0 (x) =

∑|β|≤m

(−1)|β|∂β−Aβ

(·, uh

0 , · · · ,∇m+uh

0

)(x) = 0 ∀x ∈ Ωh

0 , and

⟨Ghuh

0 , vh⟩V′h

b ×Vhb

= hn∑

x∈Ωh

∑|β|≤m

Aβ

(·, uh

0 , · · · ,∇m+uh

0

)(x)∂β

+vh(x) (8.67)

:= ah(uh

0 , vh)

:= 0 ∀ vh ∈ Vhb = Wm,p

0,+ (Ωh) with 2 ≤ p <∞

uh0 ∈ Vh

b ,∀uh ∈ Vhb ∩ D(Aj) : Aj

(·, uh, . . . ,∇m

+uh)∈ V ′h

b = W−m,p′

0,+ (Ωh).

Schumann and Zeidler [574, 678] prove for these equations convergence of order 1via “external approximation schemes”, so totally differently from our approach.

We update Proposition 8.10 with Proposition 8.11, to the quasilinear case:

Proposition 8.12. For the difference operators Ghs , and Gh in (8.65)–(8.67) we

obtain ∀vh ∈ Vhb = Wm,p

0,+ (Ωh), hence vh ∈W 2m,p(Ωh) as well, cf. Remark 8.9:∫Ωh

0

vh(Gh

suh)dxh =

∫Ωh

vh

⎛⎝ ∑|β|≤m

(−1)|β|∂β−Aβ

(·, uh(·), . . . ,∇m

+uh(·))⎞⎠ dxh (8.68)

=∫

Ωh

∑|β|≤m

Aβ

(·, uh(·), . . . ,∇m

+uh(·))∂β+v

hdxh

=⟨Ghuh, vh

⟩V′h

b ×Vhb

.

For the weak form we require uh, vh ∈Wm,p+ (Ωh), 2 ≤ p <∞, and

∀uh : Aβ

(·, uh

0 , · · · ,∇m+uh

0

)∈W−m,p′

0,+ (Ωh), 1/p + 1/p′ = 1.

For the symmetric difference equations we proceed as in (8.68).


8.4.6 Systems of linear and quasilinear elliptic equations

We formulate the differential and difference systems only for order 2m, cf. (2.391),(2.306). The transition from order 2m back to second order is obvious from (8.65)–(8.67). The main reason is that for systems of order 2m the proof for coercivityrequires a modification, based upon the linearized forms. Preparing the quasilinearproblems, we formulate the linear problems in the Sobolev setting W 2m,p

+ (Ω,Rq). Wemaintain the notation in Chapter 2, e.g. (2.389), and formulate the necessary differencemodifications compared to (8.17), e.g. as

�uh =(uh

1 , · · · , uhq

), ∂l

+�uh :=

(∂l+u

h1 , · · · , ∂l

+uhq

), ∂α

+�uh :=

(∂α+u

h1 , · · · , ∂α

+uhq

), (8.69)

∇≤k+ �uh :=

{∂α+�u

h ∀|α| ≤ k},∇k

+�uh :=

{∂α+�u

h ∀|α| = k},

with ∇+�uh := ∇1

+�uh, ∇k

+�uh := ∂l

+�uh := ∂α

+�uh := �uh for k = l = |α| = 0, and

�uh(x), ∂l+�u

h(x) ∈ Rq,∇+�uh(x) ∈ Rq×n,∇k

+�uh(x) ∈ Rq×nk , and

�ϑj =(ϑ1

j , . . . , ϑnj

)∈ Rn, j = 1, . . . , q, �ϑl =

(ϑl

1, . . . , ϑlq

)∈ Rq, l = 1, . . . , n, with

�ϑ :=(�ϑ1, · · · , �ϑn

)= (�ϑ1, · · · , �ϑq) ∈ Rn×q, and|�ϑ| = |�ϑ|nq ∈ R,

�ϑα :=(ϑα

1 , · · · , ϑαq

)∈ Rq, and �Θk := (�ϑα)|α|=k ∈ Rq×nk , �Θ≤k := (�ϑα)|α|≤k.

Correspondingly we define, e.g. the∇k−�u

h,∇kh�u

h. Then for the strong linear differentialand difference systems, and the weak difference systems we determine the solu-tions �u0, and �uh

0 from (8.70), (8.71). We choose �u0 ∈W 2m,p(Ω,Rq) ∩Wm,p0 (Ω,Rq),

�uh0 ∈W 2m,p

+ (Ωh,Rq) ∩(Vh

b := Wm,p0,+ (Ωh,Rq)

), �vh ∈ Vh

b , for the strong, and delete theW 2m,p

+ (Ω,Rq) for the weak problems. The conditions for the Nemyckii operators, cf.Subsection 2.5.5, Aαβ in (8.70), and Aβ in (8.72), and their second order form, Akl,and Ak, are discussed in Subsections 2.6.3–2.6.6, and are summarized for differencemethods in Summary 8.6.

�f = As�u0 =∑

|β|,|α|≤m

(−1)|β|∂β(Aαβ ∂α�u0) ∈ Lp′(Ω,Rq), Aαβ(x) ∈ Rq×q, (8.70)

1/p + 1/p′ = 1, �uh0 ∈ Vh

b ∩W 2m,p+ (Ωh,Rq), �fh ∈ Lp′

+ (Ωh,Rq), and

�fh(x) = Ahs�u

h0 (x) =

∑|β|,|α|≤m

(−1)|β|∂β−(Aαβ(x) ∂α

+�uh0 (x)

)∀ x ∈ Ωh

0 (8.71)

〈�fh, �vh〉V′hb ×Vh

b= hn

∑x∈Ωh

∑|α|≤m

(�fhα , ∂

α+�v

h)

q(x) =

⟨Ah�uh

0 , �vh⟩V′h

b ×Vhb

:=

ah(�uh

0 , �vh)

:=∫

Ωh

∑|β|,|α|≤m

(Aαβ ∂α

+�uh0 , ∂

β+�v

h)

qdxh ∀�vh ∈ Vh

b ,�fh ∈W−m,p′

0,+ (Ωh,Rq).

The quasilinear versions with �f included in the Aβ are defined, and the solutions,�u0, �u

h0 , are computed in D(G),D(Gh), subsets of the same spaces as in (8.70), (8.71),


see (2.424), (2.425), so

Gs�u0 :=∑

|α|≤m

(−1)|β|∂βAβ(·,∇≤m�u0) = 0 ∈ Lp′(Ω,Rq), and (8.72)

Ghs�u

h0 (x) :=

∑|β|≤m

(−1)|β|∂β−Aβ

(·,∇≤m

+ �uh0

)(x) = 0, ∀x ∈ Ωh

0 ,

ah(�uh

0 , �vh)

:=⟨Gh�uh

0 , �vh⟩V′h

b ×Vhb

with Vhb := Wm,p

0,+ (Ωh,Rq) (8.73)

:=∫

Ωh

∑|β|≤m

(Aβ

(·,∇≤m

+ �uh0

), ∂β

+�vh)

qdxh = 0 ∀�vh ∈ Vh

b .

Again we proceed as in (8.60) for symmetric difference equations.

8.4.7 Fully nonlinear elliptic equations and systems

For the fully nonlinear equation (8.74) in general no weak formulations are possible,cf. Subsections 2.5.7, 2.6.8, and Section 5.2. So, as for FEMs in (5.2) ff., we formulateonly the strong form of the difference methods. As for FEMs, stability is proved viathe detour to the linearized operator A = G′(u0). It requires regularity results of theform Au = f ∈ L2(Ω) =⇒ u ∈ H2m(Ω) =: U . These results are available for equationsof order 2m for boundaries ∂Ω ∈ C2m or ∂Ω ∈ C∞, cf. Theorems 2.79 ff. For equationsand systems of order 2 this is known for convex polygonal boundaries, cf. Kozlov et al.[452]. We consider essentially unsymmetric difference methods for curved boundaries ofsecond order equations, cf. (8.76). For special cuboidal domains as in (8.9), symmetricmethods are formulated at the end of this subsection. For G in (8.76) we find itsderivative in the form (8.43). We restrict the discussion to order 2m = 2.

Crandall et al., e.g. [216, 217] discuss the concept of viscosity solutions for fullynonlinear uniformly elliptic, e.g. [216], and parabolic equations of second order usingspecific monotonicity properties of the operator. They extend these concepts todifference methods on equidistant grids and their convergence to the exact solutions.Kuo and Trudinger [456] consider discrete versions of maximum principles, Holderestimates and Harnack inequalities for linear elliptic difference operators of positivetype and generalize them to fully nonlinear problems on equidistant and moderatelynonequidistant grids. Relations between discrete and continuous elliptic equations, butno convergence results, are discussed.

We compute u0(x), uh0 (x) for (8.74), (8.76), cf. (8.45),

Gs : D(G) ⊂ Ub := H2m(Ω) ∩Hm0 (Ω) → V := L2(Ω), (8.74)

Gsu0 := G(·, u0,∇u0,∇2u0) = 0 ∈ L2(Ω), with u0(x) = 0 on ∂Ω, and (8.75)

Ghs :D(Gh

s )⊂Uhb := H2m

0,+(Ωh)→ Vh := L2(Ωh), Ghsu

h0 := G(., uh

0 ,∇+uh0 ,∇2

+uh0 )|Ωh

0= 0,

uh0 (x)|∂Ωh = 0 (8.76)

⇐⇒(Gh

suh0 , v

h)L2(Ωh)

= 0∀vh ∈ Vh, uh0 |∂Ωh = 0, hence u0(x) ∈ Ub, u

h0 (x) ∈ Uh

b .


We do not reformulate the highly technical proof for stability, consistency andconvergence for fully nonlinear problems for the finite difference form (8.76). We onlyindicate the necessary steps, strongly analogous to the FEM approach:

For special polyhedral, e.g. cuboidal, domains in (8.9) the boundary conditions aresatisfied exactly, otherwise they are violated. Correspondingly, (8.74) is discretized as(8.76) for (8.9). The general discretization approach in Subsection 5.2.4 is then reducedto elaborating the difference operator, our Gh, the Fh

1 = Gh in (5.41). Consequently,we only need projectors for the G component and no extension operators for theconsistency. This strongly simplifies the analogue of the proof for Theorem 5.4. For theother domains we again need the full machinery of discretization errors for differentialand boundary operators Gh, Bh as for FEMs in Section 5.2. In the proof of stabilityvia the linearized operator, G′(u0), we use the equality of weak and strong discretebilinear forms induced by G′(u0), cf. Proposition 8.10, (8.56). We obtain the regularityof the difference solutions, as in Lemma 5.10, and Hackbusch [387], Theorem 9.2.26.The corresponding convergence results are summarized in Theorem 8.31 below.

This difference method avoids the main disadvantage of the preceding FEMs, theneed of C1 FEs. These require necessarily piecewise polynomials of higher degree, e.g.≥ 5 for n = 2 and ≥ 9 for n = 3.

For symmetric difference equations, we slightly modify (8.60). The boundary condi-tions are formulated in (8.19), (8.20). So we require:

Ghssu

h0 (x) :=

12(G(x, uh

0 ,∇+uh0 ,∇2

+uh0

)+ G

(x, uh

0 ,∇−uh0 ,∇2

−uh0

))(x) = 0 ∀x ∈ Ωh

0 .

Extensions of these results to equations and systems of order 2m, and to curvedboundaries ∂Ω ∈ C2m can be formulated as in the preceding subsections.

A totally different approach is due to Crandall and Lions. They developed, in a seriesof papers, the theoretical tools and full convergence proofs for difference methods forquasi- and fully nonlinear elliptic and parabolic equations [217, 226, 227], and withKocan and Swiech in [216]. The basis is a concept of generalized weak solutions, theviscosity solutions; for a user’s guide see Crandall et al. [215].

For other types of results Kuo and Trudinger combine nonlinear elliptic and par-abolic difference equations of positive type with their concepts of weak solutions, local,and Schauder, Harnack and Holder estimates, and a discrete local maximum principle,see [456–460].

8.5 Convergence for difference methods

Here we apply our general discretization theory from Chapter 3 to the differencemethods formulated in the last chapter. As mentioned, we have only studied methodsof orders 1 and 2. We formulate the Richardson extrapolation and the defect correctionmethods, presented in Section 8.8, as an appropriate strategy for developing high ordermethods as well. The obtained convergence results with respect to the Hm

+ (Ωh) normsis a special case of the regularity results for difference methods mentioned at the endof Subsection 8.4.2. Important in this context are exact conditions for the linearizedoperators, collected in Summary 8.6 and in Remark 8.7. From the many papers for

8.5. Convergence for difference methods 589

general difference equations we only mention Borzı et al. [134] and Vainikko withTamme [646,646–649].

Although W−m,p′(Ω) �= W−m,p′

0 (Ω) and W−m,p′

+ (Ωh) �= W−m,p′

0,+ (Ωh), we often donot distinguish them. The restriction, e.g. of f ∈W−m,p′

(Ω) yields an f |W−m,p′

0 (Ω)∈

W−m,p′

0 (Ω).

8.5.1 Discretization concepts in discrete Sobolev spaces

We briefly summarize the general discretization theory in Chapter 3, cf. Summary 4.52as well. Our approach is much more similar to the FEM or DCGM form than to theearlier combination of strong forms of the difference equations with M -matrices. Incontrast, we discuss here stability and convergence for the weak forms, cf. Proposition8.11. These variational difference methods yield results that are usually stronger thanM -matrix results. These ideas have been applied to several problems, e.g. by Samarskii[564,565] and Temam [621].

However, we simplify the major part of the following proof for convergence. Werestrict the full presentation to the case of discretely exactly reproduced boundaryconditions. This is satisfied for the cuboidal domains Ω in (8.9), and second orderelliptic problems, hence m = 1, q ≥ 1. The extension to general Ω in (8.18), andm > 1, q ≥ 1, can be proved by modifying the FE approach for fully nonlinear problemsin Section 5.2, and Subsection 5.2.4. We would have to replace the differential operatorA or G by the two-component operator, F, including the boundary operator, B,thus F = (A,B) or F = (G,B), e.g. F = (A,B) : H1

0 (Ω) → H−10 (Ω)×H1/2(∂Ω). This

allows verifying the consistency of the “nonconforming finite difference method” as inSubsection 5.2.4, and the stability by combining the techniques in Subsections 8.5.5and 5.2.7.

This technique will be applied to the natural boundary conditions in Section 8.6as well. For the difference methods for curved domains in Section 8.7 totally differenttechniques are necessary. These proofs will be omitted there.

In the last section we have transformed the strong (classical) form of differenceequations into the corresponding weak form except for fully nonlinear problems. Forthe case of trivial Dirichlet conditions we need lines 1, 2 of (8.76), for the strong andweak form the next two lines for their weak discretization.

U = H2m(Ω), or W 2m,p(Ω), or H2m(Ω,Rq), and V ′ = Lp′(Ω), . . . , (8.77)

Ub = U ∩Hm0 (Ω),Wm,p

0 (Ω),V ′b = Lp′

(Ω), for m, q ≥ 1, 1 ≤ p ≤ ∞,

Uh = Vh = Hm+ (Ωh),Wm,p

+ (Ωh), V ′h = U ′h = W−m,p′

+ (Ωh),

Vhb = Uh

b = Hm0,+(Ωh),Wm,p

0,+ (Ωh), V ′hb = U ′h

b = W−m,p′

+,0 (Ωh).

or one can use the or the Wm,p− (Ωh), . . . , or the Wm,p

0,− (Ωh) forms.

For nonlinear G,Gh, the domains D(G),D(Gh) are subsets of U ,Uh. The A,G, willusually be defined on U → V ′, and the Ah, Gh on Uh → V ′h, cf. Figure 8.3. Uniqueness


Ub ⊂ U ⊃ D (G) A, G tested by←→ Vb

P h⏐ Φh⏐

Ubh ⊂ Uh ⊃ D(Gh ) Ah, Gh tested by

←→ VbhV ′h

Q′h

V ′

Figure 8.3 Difference methods: Spaces, and operators, usually with Au, Gu ∈ V ′b∀u ∈ Ub

and Ahuh, Ghuh ∈ V ′hb ∀uh ∈ Uh

b .

requires boundary conditions Ub,Vb and Uhb ,Vh

b , tested according to Au,Gu ∈ V ′b and

Ahuh, Ghuh ∈ V ′hb .

For applying the general discretization theory to A,G, we need restriction, andprojection operators, Ph, and Q

′h, relating the original and the discrete spaces.

Ph : U → Uh, Q′h : V ′

b → V ′hb , or only Ph : Ub → Uh

b . (8.78)

Then Ah = ΦhA and Gh = ΦhG are defined by the constructions in Section 8.4, cf.Figure 8.3. We formulate the W k,p

+ (Ωh) case. The modification to W k,p− (Ωh) is obvious.

For our difference method, applicable to the problem Au0 = f or Gu0 = 0, seeDefinition 3.12, we thus have the sequence of quintuples{Uh

b ,Vhb , P

h, Q′h, Gh

}h∈H

,with inf{0 < h ∈ H} = 0,dimUhb = dimVh

b <∞, (8.79)

and h < h0, thus limh→0 := limh∈H,h→0 always makes sense.According to Section 3.2, cf. (3.16), (3.24), we need bounded, not necessarily linear

Ph, Q′h in (8.78), such that Ph0 = 0, Q

′h0 = 0, and

limh→0

‖Phu‖Uh = ‖u‖U∀u ∈ U , limh→0

‖Q′hf‖hV′

b= ‖f‖V′

b∀f ∈ V ′

b. (8.80)

We employ the claim of Theorem 3.21 in Chapter 3: consistency and stability essen-tially imply uniquely existing discrete solutions, uh

0 , and their convergence to theexact solution u0. The (classical) consistency error, and consistency for differentialoperators in u, are defined as, cf. Definition 3.14,

(AhPhu− fh)−Q′h(Au− f) ∈ V ′h

b , or GhPhu−Q′hGu ∈ V ′h

b , and (8.81)

limh→0

||(AhPhu− fh)−Q′h(Au− f)||V′h

b= 0 or lim

h→0||GhPhu−Q

′hGu||V′hb

= 0.

Gh are called stable in uh, if ∃h0, r, S ∈ R+ fixed, such that for all h ∈ H,h < h0:

uh1 , u

h2 ∈ Br(uh) ∩ Ub ⇒

∥∥uh1 − uh

2

∥∥Uh ≤ S

∥∥Gh(uh

1

)−Gh

(uh

2

)∥∥V′h

b

. (8.82)

For Ah or Ah − fh this reduces to (for any r, and uh, but for h < h0)

(Ah)−1 ∈ L(V ′h

b ,Uhb

)exists and ||(Ah)−1||Uh

b ←V′hb≤ S.

Gh is stable if its derivative (G′(u0))h is stable, cf. Theorem 3.23.


So we have to introduce Ph, Q′h, and prove consistency and stability, via coercivity

or an inf-sup condition of its principal part. Alternatively the monotony arguments inSection 4.5 are applicable.

8.5.2 The operators Ph, Q′h

We do not repeat the basic proofs for well-defined, linear and bounded Ph and Q′h, cf.

e.g. Raviart [545], Schumann and Zeidler [574, 678] and Temam [621]. But we prove,in a slightly nonstandard way, consistency and stability.

Again we consider the two cases (8.19) and (8.20), with the two different definitionsof the Ωh

0 ⊂ Ωh, ∂Ωh in parallel. Either Ωh ⊂ Ω for the original u ∈W k,p(Ω) or weextend u to u ∈W k,p(Ω1) with Ωh ⊂ Ω1. This is possible for Ω in (8.9) by Theorem4.37. The same holds for functionals.

The definition of the Ph : U → Uh, and Q′h : V ′ → V ′h is nearly the same. But the

trivial boundary conditions in Ph : Ub → Uhb require modifications. This is achieved

by defining Phu(x) either ∀x ∈ Ωh or ∀x ∈ Ωh0 . We discuss two different possibilities.

Ph = Ph1 is studied, e.g. in Raviart [545], Schumann and Zeidler [574,678], and Ph =

Ph2 , e.g. in Temam [621]. We mainly consider Ph = Ph

2 , and only sometimes indicatePh

1 . Note that in (8.84) supp u is closed, and Ω is open.

Definition 8.13. For functions, u ∈ U , defined in Ω, or extended to Ω1 as Ecu, andwith ch(x) ⊂ Ωh ⊂ Ω or Ωh ⊂ Ω1, for Ec,Ωh,Ωh

0 cf. (8.19)–(8.20), define Ph = Ph1 by

the integral mean value uh(x) of u or Ecu as

Ph = Ph1 : U → Uh : ∀x ∈ Ωh : Ph

1 u(x) := uh(x) :=1hn

∫ch(x)

u(x)dx, and

Ph = Ph1 : Ub → Uh

b : ∀x ∈ Ωh0 : Ph

1 u(x) := uh(x) otherwise = 0. (8.83)

For Ph = Ph2 we modify (8.83) by replacing Ph

1 u(x) by Ph2 u(x)∀x ∈ Ωh or Ωh

0 : For

u ∈ C(Ω) := {v ∈ C(Ω) or C(Ω1,R) : carr v ⊂ Ω or ⊂ Ω1} : Ph2 u(x) := u(x) (8.84)

and uniquely extend Ph2 from C(Ω) to U = W k,p(Ω) defining Ph = Ph

2 : U → Uh.Obviously these Ph and the Q

′h below can be trivially extended as

Ph : U ∪ Uh → Uh : Ph|U = Ph in (8.83) and Ph|Uh = I, identity on Uh. (8.85)

Parallel to the standard grid functions, uh, we employ the step functions, uhe , with

χch(x), the characteristic functions of the ch(x), cf. (8.19),

Eh : Uh → Uhe :=

⎧⎨⎩uhe := Ehuh :=

∑x∈Ωh

uh(x)χch(x)

⎫⎬⎭ . (8.86)

We find, possibly except for a subset of measure 0 on the boundaries of the ch(x), that

∂α+E

huh = Eh∂α+u

h. (8.87)


This implies that

‖Uh‖Uh := ‖uh‖W k,p+ (Ωh) := [[Ehuh]]W k,p

+ (Ωh) :=

⎛⎝∑|α|≤k

‖∂α+E

huh‖pLp(Ωh)

⎞⎠1/p

. (8.88)

Remark 8.14. It is important to realize that the results in this and the next subsectionare measured with respect to this specific norm ‖uh‖Uh = ‖uh‖W k,p

+ (Ωh), unless statedotherwise. This holds, in particular, for the consistency estimates.

The following proposition is an obvious extension of Temam [621], Lemma 3.1 onp. 51 for Lp(Ω), to higher order derivatives, and to both our cases (8.19), (8.20). HisProposition 3.1, p. 42, shows that Proposition 8.15 had only to be proved for functionsu in a dense subspace of W k,p(Ω). This remains correct for Theorem 8.21, (8.104) aswell. We do not repeat the argument in [621], Proposition 3.1, since we use it in part(b) of the proof of the following Theorem 8.21. The Ec in Theorem 4.37 combinedwith this result yields, cf. (8.19)–(8.20):

Proposition 8.15. Ph, Eh, cf. (8.84), (8.86), satisfy ∀u ∈W k,p(Ω), |α| ≤ m

for (8.19): limh→0

‖∂α+E

hPhu− ∂αu‖Lph(ch(Ωh)) = 0 (8.89)

for (8.20): limh→0

‖∂α+E

hPhu− Ec∂αu‖Lp

h(ch(Ωh)) = 0, this implies

limh→0

‖Phu‖W k,p+ (Ωh) = lim

h→0[[EhPhu]]W k,p

+ (Ωh) = ‖u‖W k,p(Ω),

cf. (8.80). Similarly, we obtain

for (8.19): limh→0

‖∂α+P

hu− Ph(∂αu)‖Lph(Ωh) = 0 (8.90)

for (8.20): limh→0

‖∂α+P

hu− Ph(∂αEcu)‖Lph(Ωh) = 0.

Remark 8.16. A direct comparison between the original functions and grid functionsis not possible. So the concept of external approximation schemes was introduced, cf.Petryshyn [528–530, 532], Temam [621] and Zeidler [677]. To this end a mapping isdefined for V = Wm,p(Ω) as

ω : V → W := Π|α|≤mLp(Ω), ωu := (∂αu)|α|≤m, ‖ωu‖W := max|α|≤m

‖∂αu‖Lp(Ω).

ω is bounded, since all norms in Π|α|≤mR are equivalent. Then, e.g. the first line of(8.89) can be reformulated as

limh→0

‖(∂α+E

hPhu)|α|≤m

− ωu‖W = 0. (8.91)

In this generalized sense, grid functions approximate functions in Sobolev spaces.

By (8.30), cf. Proposition 2.34, (2.107), (2.108), for 1 < p <∞, 1/p + 1/p′ = 1, andfunctionals, f ∈ V ′ = W−m,p′

(Ω), unique fβ ∈ Lp′(Ω) exist, such that,


〈f, v〉V′×V :=∫

Ω

∑|β|≤m

fβ∂βvdx ∀ v ∈ V, with ‖f‖V′ =

⎛⎝ ∑|β|≤m

‖fβ‖p′

Lp′ (Ω)

⎞⎠1/p′

, (8.92)

cf. Proposition 2.34. For 1 = p or p =∞ the fβ ∈ Lp′(Ω) are no longer unique, and

we obtain ‖f‖V′ = minfβ{∑|β|≤m ‖fβ‖Lp′ (Ω)} over all possible fβ .

Two different possibilities for Q′h correspond to the above Ph = Ph

1 or = Ph2 .

Definition 8.17. For functionals, f ∈ V ′ = W−m,p′(Ω), tested by v ∈ V = Wm,p, we

define the Q′h : V ′ → V ′h componentwise for the fβ in (8.92). Using the first line of

(8.83) or (8.84), and testing ∀ vh ∈ Vh = Wm,p+ yields

Q′h : V ′ → V ′h : Q

′hf :=(fh

β (x) := Phfβ(x)∀x ∈ Ωh)|β|≤m

, fhβ ∈ Lp′

h (Ωh), (8.93)

=⇒ 〈Q′hf, vh〉V′h×Vh :=∫

Ωh

∑|β|≤m

fhβ ∂

β+v

hdxh =∑

x∈Ωh

hn∑

|β|≤m

∂β+v

h(x)(Phfβ)(x).

In contrast to Definition 8.13, usually Phfβ(x) �= 0 for x ∈ Ωh \ Ωh0 .

Proposition 8.18. The previous Ph : V → Vh in (8.83), (8.84) and Q′h : V ′ → V ′h

in (8.93), are well-defined linear bounded operators, satisfying (8.80), (8.89).

Proof. We start with the properties of the Ph. Obviously the Ph1 are well defined. By

Raviart [545], pp. 22–25, and Schumann and Zeidler [574, 678], Lemma 6, Ph = Ph1

are linear bounded operators. For Ph = Ph2 this is a consequence of Temam [621],

Proposition 3.1, and Lemma 3.1, cf. its proof. In particular, the extension of Ph2 from

C(Ω) to U = W k,p(Ω) is possible, and yields this result. Relation (8.80) is implied, andobtained for Ph

2 by combining [621], Proposition 3.1 with the proof of Lemma 3.1.Obviously Q

′h is well defined and linear. The boundedness follows from (8.93), andtwice the discrete Holder inequality (1.44) with 1/p + 1/p′ = 1:

〈Q′hf, vh〉V′h×Vh =∑

x∈Ωh

hn∑

|β|≤m

∂β+v

h(x)(Phfβ)(x) (8.94)

≤∑

|β|≤m

⎛⎝∑x∈Ωh

hn(∂β+v

h(x)(x))p

⎞⎠1/p⎛⎝∑x∈Ωh

hn(Phfβ(x))p′

⎞⎠1/p′

=∑

|β|≤m

‖∂β+v

h)‖Lph(Ωh)‖Phfβ‖Lp′

h (Ωh)

≤ ‖vh‖W m,ph (Ωh)‖Q

′hf‖W−m,p′

h (Ωh).

This shows that Q′h is bounded. A combination with (8.80) for the Ph, and (8.92),

(8.35) with the minimal fβ , implies (8.80). �


8.5.3 Consistency for difference equations

For the previous examples consistency does not require u ∈ Ub, vh ∈ Vh

b , but is validfor u ∈ U , vh ∈ Vh as well. In fact, the boundary conditions are exactly satisfied. Sincehere the norms in V and Vb are equivalent, we often do not distinguish them. Thepreviously introduced Ph, Q

′h, Ah, Gh allow the proof of this consistency.We only consider finite difference methods with equal step sizes, h, in all directions.

For special Ω, Dirichlet boundary conditions can be exactly reproduced:

Ω :=n∏

i=1

(ai, bi) ⊂ Rn s.t. ∃h > 0 with ai/h, bi/h ∈ Z, i = 1, · · · , n. (8.95)

For Ω ⊂ Rn in (8.95) or in (8.98) below we introduce step sizes, h, and grids:

H := {h as in (8.95) or h ∈ R+ for (8.98)} : (8.96)

Gh := GhD := {x = (x1, . . . , xn)T ∈ Rn : xi/h ∈ Z},

Ghl := {∀j = 1, . . . , n : x = (x1, . . . , xn)T ∈ Rn : ∀i �= j : xi/h ∈ Z, xj/h ∈ R}.

For Ω in (8.95) we introduce, cf. Figures 8.4,

Ωh0 := Ω∩Gh,Ωh := Ω∩Gh, ∂Ωh := ∂Ω∩Gh = Ωh \ Ωh

0 . (8.97)

This has to be modified as in (8.20) for

Ω ⊂ Rn is open, bounded, nonempty, and ∂Ω is Lipschitz-continuous. (8.98)

Let, with the norm for Uh in (8.87),

Ph = Ph1 : Ub → Uh

b : ∀x ∈ Ωh0 : Ph

1 u(x) := uh(x) otherwise = 0. (8.99)

: points in ∂Ωh

: points in Ω0h



For proving consistency of orders 1 or 2, for the difference methods, we have toapproximate the u ∈ V = Hm+μ(Ω). A very up-to-date choice, and very appropriatefor our bounded Ω, and equidistant Ωh, are radial basis functions. Since they are notquite as well known as other approximations, we give a very short introduction, andrefer for all the specific conditions to Buhmann [152] and Bohmer [120]. We start withφ, defined by a positive completely monotonic g

φ : R+ → R+, φ(r) := g(r2), g ∈ C(R+), e.g. g(r2) =√r2 + c2 (8.100)

and define ψ(x) :=∑

k∈Zn,|k|<N

ckφ(‖x− k‖)ck ∈ R, x ∈ Ω ⊂ Rn,

with just one ψ, a finite sum, and appropriate ck. We introduce

Sh :=

⎧⎨⎩sh(x) :=∑j∈Zn

cjφ(‖x/h− j‖)

⎫⎬⎭ , e.g. sh(x) :=∑j∈Zn

f(jh)ψ(‖x/h− j‖),

(8.101)

the latter interpolating f in the jh, if ψ is appropriately chosen. The followingproposition requires a slightly modified semi-inner product and norm in Hk(Ω):

(u, v)Hk∗ (Ω) :=

∑|α|=k

k!α!

(∂αu, ∂αv)L2h(Ωh), ‖u‖Hk

∗ (Ω) := ((u, u)Hk∗ (Ω))

1/2, (8.102)

and ‖p‖Hk∗ (Ω) = 0∀p ∈ Pn

k−1.

If Ω is not too narrow and h is small enough, Ωh contains a unisolvent subset Ωhu for

Pnk−1, the set of polynomials of degree ≤ k − 1, in Rn, such that

p ∈ Pnk−1 : ∀x ∈ Ωh

u : p(x) = 0 ⇒ p ≡ 0.

We have mentioned that the second sh in (8.101) interpolates f in Ωh, however notuniquely in Sh. This is possible in a modified way by, cf. [152], Proposition 5.2.

Proposition 8.19. Assume Ωh contains a unisolvent subset for Pnk−1, for u ∈ Hk(Ω),

the function values u(x)∀x ∈ Ωh are well defined, cf. Theorem 1.26 or [152]. Then thereexists a unique minimal norm interpolant sh ∈ Sh interpolating sh(x) = u(x)∀x ∈ Ωh.Compared to any other interpolating g ∈ Sh, this sh has minimal seminorm, hence‖sh‖Hk

∗ (Ω) ≤ ‖g‖Hk∗ (Ω).

For these interpolants Buhmann [152] proves, cf. Theorem 5.5 for m = 0 andWendland [662], for m ≥ 0:

Theorem 8.20. Convergence for radial basis functions: For u ∈ V = Wm,p(Ω), andsmall enough h, the conditions of Proposition 8.19 are satisfied. They guarantee aunique minimal norm interpolant sh

u ∈ Sh. For μ = 1, 2 we obtain convergence and


bounds for

u ∈ Z := Wm+μ,p(Ω), μ = 1, 2, =⇒ ‖u− shu‖V < Chμ‖u‖Z and (8.103)

=⇒ ‖ωu− ωshu‖W < 2hμ‖ω‖C‖u‖Z with ‖ωsh

u‖W < 2‖ω‖‖u‖V .Theorem 8.21. Consistency of linear operators with respect to ‖ · ‖Vh in (8.87):

1. With Ph, Q′h in (8.99), (8.84), (8.93), and the conditions for the coefficients

of A in Summary 8.6 and Remark 8.7, we obtain for Au = f ∈ V ′, u ∈ Ub =Wm,p

0 (Ω), A ∈ L(U ,V ′) and fh = Q′hf , consistent unsymmetric and symmetric

approximations Ah, Ahsw, cf. (8.60), for linear operators, e.g.

∀vh ∈ Vh, vh �= 0 : limh→0

〈AhPhu−Q′hAu, vh〉V′h×Vh/||vh||Vh = 0, or (8.104)

limh→0

||AhPhu−Q′hAu||V′h = lim

h→0||Ah

swPhu−Q

′hAu||V′h = 0 ∀u ∈Wm,p(Ω).

2. More precisely, we obtain, for u ∈Wm+μ,p(Ω), A ∈ L(Wm+μ,p(Ω), W−m+μ,p′

(Ω)) consitency of order μ = 1 for unsymmetric, and μ = 2 for symmetric dif-ferences:

||AhPhu−Q′hAu||V′h ≤ Ch‖u‖W m+1,p(Ω) ∀u ∈Wm+1,p(Ω), or (8.105)

||AhswP

hu−Q′hAu||V′h ≤ Ch2‖u‖W m+2,p(Ω)∀u ∈Wm+2,p(Ω). (8.106)

3. For possible choices of fh �= Q′hf , limh→0 ||fh −Q

′hf ||V′h = 0 has to be proved.

Proof. We prove the result only for m ≥ q = 1; the generalization to q ≥ 1 is straight-forward. The proof requires three steps:

(a) We estimate 〈AhPhu−Q′hAu, vh〉V′h×Vh for u ∈ Cm

L (Ω), and 〈AhswP

hu−Q

′hAu, vh〉V′h×Vh for u ∈ Cm+1L (Ω), thus (8.105), (8.106), cf. (8.108)

(b) The dense CmL (Ω) ⊂Wm,p(Ω) implies consisteny in (8.104) for u ∈Wm,p(Ω).

(c) For the stronger forms with order 1 in (8.105), and 2 in (8.106) we needinterpolation with radial basis functions with the error estimates in Theorem8.20 on the rectangular grids in (8.95), (8.98).(a) For aαβ �∈ C(Ω) the aαβ(x) have to be replaced by Phaαβ(x). We obtain, cf.

(8.47), for ∀u ∈ V = Wm,p(Ω), ∀vh ∈ Vh = Wm,p+ (Ωh), ah : Vh × Vh → R :

ah(Phu, vh)− 〈Q′hAu, vh〉V′h×Vh (8.107)

= 〈AhPhu−Q′hAu, vh〉V′h×Vh

=∫

Ωh

∑|α|,|β|≤m

(aαβ∂

α+P

hu− Ph(aαβ∂

αu))

∂β+v

hdxh

= hn∑

x∈Ωh

∑|α|,|β|≤m

((aαβ∂

α+P

hu)(x)− Ph(aαβ∂

αu)(x))∂β+v

h(x)

= hn∑

x∈Ωh

∑|α|,|β|≤m

aαβ(x)( (

∂α+P

hu)(x)− Ph

(∂αu)(x))∂β+v

h(x).


In (8.107), only the terms with |α| > 0 are different. We use Lemma 8.2.For the unsymmetric and symmetric divided difference, respectively, Thisyields with (8.22) for u ∈ Cm

L (Ω) and u ∈ Cm+1L (Ω), and with (8.99), (8.84),∣∣∂α

+Phu(x) − Ph

(∂αu)(x)∣∣ ≤ cα,1h ∀x ∈ Ωh for u ∈ Cm

L (Ω) and (8.108)∣∣∂αhP

hu(x) − Ph(∂αu)(x)∣∣ ≤ c′α,1h

2 ∀x ∈ Ωh for u ∈ Cm+1L (Ω).

(b) This part is an update of Temam’s proof of his Proposition 3.1 [621]. CmL (Ω)

is a dense subset of V = Wm,p(Ω). So we approximate the u ∈Wm,p(Ω) byappropriate un ∈ Cm

L (Ω), and employ the approximation results in (8.91),(8.108). For each n ∈ N a un ∈ Cm

L (Ω) exists such that

‖u− un‖V <1n

⇒ ‖ωu− ωun‖W <‖ω‖n

. (8.109)

We fix this un. Since ‖∂α+E

hPhu− ∂αu‖Lph(ch(Ωh)) → 0 by (8.89), (8.91),

there exists an ηn > 0, such that for 0 < h < ηn,

‖(∂α+E

hPhun

)|α|≤m

− ωun‖W <1n. (8.110)

We may additionally choose ηn < 1/n, thus {ηn} monotonely converge to0. For

Ph+u := Phun for ηn+1 < h < ηn

we obtain that

‖ωu−(∂α+E

hPh+u)|α|≤m

‖W

≤ ‖ωu− ωun‖W + ‖ωun−(∂α+E

hPhun

)|α|≤m

‖W

+ ‖(∂α+E

hPhun

)|α|≤m

−(∂α+E

hPh+u)|α|≤m

‖W

≤ 1 + ‖ω‖n

.

This ‖ωu−(∂α+E

hPh+u)|α|≤m

‖W ≤ (1 + ‖ω‖)/n for h < ηn with (8.91),(8.107), (8.108), proves the claimed consistency for u ∈Wm,p(Ω).

(c) A slight modification of the proof in (b) yields the stronger results (8.105),(8.106). Only for this case do we need the approximation properties in(8.103). It implies with Phu := Phsh

u that

‖ωeEcu−(∂α+E

hPhu)|α|≤m

‖We

≤ ‖ωeEcu− ωeshu‖We

+ ‖ωeshu −(∂α+E

hPhshu

)|α|≤m

‖We

+ ‖(∂α+E

hPhshu

)|α|≤m

−(∂α+E

hPhu)|α|≤m

‖We

≤ h(C ′ + ‖ω‖)‖u‖Z and

≤ h2(C ′ + ‖ω‖)‖u‖Z .


This ‖ωeEcu−(∂α+E

hPhu)|α|≤m

‖We≤ hμ(C ′ + ‖ωe‖)‖u‖Z proves the

claimed consistency of orders μ = 1, 2, for u ∈Wm+μ,p(Ω) for unsymmetricand symmetric differences.

�

In Subsections 8.4.5 and 8.4.6 we introduced the difference equations for the quasi-linear equations of orders 2 and 2m, cf. (8.65) and (8.67), and for linear and quasilinearsystems in (8.71) and (8.72). The symmetric forms only were described analogously tothe linear problems. For the different types of quasilinear problems different conditionsare discussed for unique existence of solutions and their regularity, cf. Subsections2.5.4, 2.5.6, 2.6.4, 2.6.6 and bounded Frechet derivatives, cf. Subsections 2.7.2, 2.7.3,2.7.5. To simplify the following consistency result for quasilinear problems, we assume aunifying condition. It is oriented towards that for bounded Frechet derivatives, neededanyway for the proof of the coercivity, stability and convergence in Subsections 8.5.4and 8.5.5. For minimal conditions the reader has to consult Subsections 2.5.4, 2.5.6,2.6.4, 2.6.6, 2.7.2, 2.7.3, 2.7.5. We do not fully formulate the next condition, and thetheorem for systems, since the generalization is obvious.

We assume all the Aj in (8.65) and Aα in (8.67) to be equicontinuous in a“neighborhood” of u0, more precisely, with Nm in (8.17),

Aj , Aα equicontinuous in Ω×N∇m ⊂ Ω× RNm with bounded N∇m , s.t. (8.111)

∀u ∈ Ub : ‖u− u0‖V ≤ ρ,∇≤mu(x) ∈ N∇m∀x ∈ Ω,∇≤m+ u(x) ∈ N∇m∀x ∈ Ωh

0 .

Theorem 8.22. Consistency of quasilinear equations and systems with ‖ · ‖Vh in(8.87)

1. Choose the Ph, Q′h in (8.84), (8.99), (8.93), and let Aj , Aα in (8.67) and u sat-

isfy (8.111). Choose Gh, similarly Ghsw, as in (8.65), (8.67), and (8.71), (8.72).

Then we obtain for Gu ∈W−m,p′(Ω) consistent unsymmetric and symmetric

approximations Gh, Ghsw, hence, e.g.

∀vh ∈ Vh, vh �= 0 : limh→0

〈GhPhu−Q′hGu, vh〉V′h×Vh/||vh||Vh = 0, (8.112)

thus

limh→0

||GhPhu−Q′hGu||V′h = lim

h→0||Gh

swPhu−Q

′hGu||V′h = 0 ∀u ∈Wm,p(Ω).

2. For u ∈Wm+μ,p(Ω), Aα : Wm+μ,p(Ω) →W−m+μ,p′(Ω)) with Lipschitz-conti-

nuous Aj , Aα with respect to the variables in N∇m , we obtain consitency of orderμ = 1 for unsymmetric, and μ = 2 for symmetric differences:

||GhPhu−Q′hGu||V′h ≤ Ch‖u‖W m+1,p(Ω) ∀u ∈Wm+1,p(Ω), (8.113)

or

||GhswP

hu−Q′hGu||V′h ≤ Ch2‖u‖W m+2,p(Ω)∀u ∈Wm+2,p(Ω). (8.114)


Proof. The proof is very similar to the proof of Theorem 8.21, here restricted tom = q = 1. The generalization to m, q ≥ 1 is straightforward. From the three steps,we only formulate (a) and only indicate (b) and (c)

(a) Again, we formulate this proof for u,∇u ∈ C(Ω), Aj ∈ C(Ω,R,Rn). Otherwisethese u,∇u,Aj have to be replaced by Phu, Ph∇u, PhAj . We obtain, cf. (8.65),for ∀u ∈ V = W 1,p(Ω), ∀vh ∈ Vh = W 1,p

+ (Ωh),

ah(Phu, vh)− 〈Q′hGu, vh〉V′h×Vh = 〈GhPhu−Q′hGu, vh〉V′h×Vh (8.115)

=∫

Ωh

n∑j=0

(Aj(·, Phu,∇+P

hu)

−PhAj(·, u,∇u))∂j+v

hdxh

= hn∑

x∈Ωh

n∑j=0

(Aj

(·, Phu,∇+(Phu)

)−Aj(·, u,∇u

))(x)∂j

+vh(x).

Lemma 8.2 yields with (8.22) for u ∈ C1L(Ω), u ∈ C2

L(Ω), but now for Aj ∈CL(Ω,R,Rn),∣∣(Aj

(·, (Phu), (D+(Phu))

)−Aj(·, u,∇u)

)(x)∣∣ ≤ c1,1h for u ∈ C1

L(Ω) (8.116)∣∣(Aj

(·, (Phu), (D+(Phu))

)−Aj(·, u,∇u)

)(x)∣∣ ≤ c′1,1h

2 for u ∈ C2L(Ω).

(b) We approximate the u ∈W 1,p(Ω) by uk ∈ C1L(Ω). Then a combination of the

equicontinuity in (8.111), (8.109), (8.110), implies the consistency (8.112).(c) The final consistency of orders 1 and 2 follows as in the proof of Theorem 8.21,

now with the Lipschitz-continuity of the Aj . �

Finally, we introduced in Subsection 8.4.7 the difference equations for the fullynonlinear equations restricted to order 2, cf. (8.76) and (8.67). The symmetric form islisted at the end of Subsection 8.4.7. Again we restrict the discussion to the Hilbertspace setting U := H2(Ω) → V := L2(Ω). For fully nonlinear equations different con-ditions for unique existence of solutions and their regularity are listed in Subsection2.5.7 and the bounded Frechet derivatives in Subsections 2.7.3 and 2.7.5. For systemsnearly nothing is known, cf. Section 5.2.8.

Similarly to (8.111), we assume the fully nonlinear G to be equicontinuous in a“neighborhood” of u0, more precisely, with Nm in (8.17),

G equicontinuous in Ω×N∇2 ⊂ Ω× RN2 with bounded N∇2 , s.t. (8.117)

∀u ∈ V : ‖u− u0‖V ≤ ρ,∀x ∈ Ω∇≤2u(x) ∈ N∇2∀x ∈ Ωh0 .∇≤2

+ u(x) ∈ N∇2 .

Theorem 8.23. Consistency for fully nonlinear problems with ‖ · ‖Vh = ‖ · ‖L2h(Ωh) :


1. Choose the Ph, Q′h in (8.99), (8.84), (8.93), and let G now in (8.74) satisfy

(8.117). Then we obtain for G : D(G) ⊂ U := H2m(Ω) → V := L2(Ω) and thegrid functions, Uh,Vh, a consistent strong approximation, Gh = Gh

s , and thestrong symmetric approximation, Gh

ss, hence we find for ‖u− u0‖U ≤ ρ,

∀vh ∈ Vh, vh �= 0 : limh→0

(GhPhu−Q′hGu, vh)Vh/||vh||Vh = 0, (8.118)

thus

limh→0

||GhPhu−Q′hGu||Vh = lim

h→0||Gh

ssPhu−Q

′hGu||Vh = 0 ∀u ∈ U .

2. For u ∈ H2+μ(Ω), G : D(G) ⊂ H2+μ(Ω) → Hμ(Ω)) with Lipschitz-continuous Gwith respect to the variables in N∇2 , we obtain consitency of order μ = 1 forunsymmetric, and μ = 2 for symmetric differences:

||GhPhu−Q′hGu||V′h ≤ Chμ‖u‖H2+μ(Ω) ∀u ∈ H2+μ(Ω). (8.119)

Proof. This is an obvious modification of the last proofs. �

8.5.4 Vhb -coercivity for linear(ized) elliptic difference equations

In contrast to the previous consistency results, the boundary conditions are crucial forcoercivity and the following stability of elliptic difference operators. Vh

b -coercivity is, byTheorems 3.23, 2.12 ff., and 8.29, the essential step towards stability of our differencemethods. In fact, let the principal part of a linear, or linearized, elliptic differenceoperator be Vh

b := Hm0,+(Ω,Rq)-coercive, see below, and the method be consistent,

cf. Subsection 8.5.3. Assume a bounded and boundedly invertible Frechet derivativeG′(u0) of the original problem in the exact solution, u0. Then the difference equation,Gh(uh) = 0, similarly Ah(uh) = 0, is stable, by Theorem 8.29. With its consistency,this implies its uniquely existing solution, uh

0 , converging to u0 with respect to‖ · ‖Hm

+ (Ω,Rq). We summarize these results in Subsection 8.5.5.We start with coercivity results for the original linear operators. In Chapter 2,

Theorems 2.43, 2.89, and 2.104, we have seen that coercive bilinear forms induceboundedly invertible linear operators. For the different cases, we have formulatedin Summary 8.6 the definitions of ellipticity and the conditions for the coefficients,implying Hm

0 (Ω,Rq)-or Wm,p0 (Ω,Rq)-coercivity, and boundedness of the weak (and

strong) forms of the principal parts of the original operators for m, q ≥ 1. As we havepreviously discussed, cf. Chapters 2–5, we will usually only have Hm

0 (Ω,Rq)-coercivity,even for (nonlinear) problems defined in Wm,p

0 (Ω,Rq).As indicated in Remark 8.7 with its pros and cons, the conditions in Summary 8.6 for

the coefficients of the linear difference equations, (8.38)–(8.41), could be considerablyreduced towards those of differential equations, by replacing the C(Ω,Rq), C|β|(Ω,Rq)by L∞(Ω,Rq),W |β|,∞(Ω,Rq). The nonlinear problems in Subsections 8.4.5 and 8.4.6are included via linearizations of their discrete operators, for m, q ≥ 1. They arebounded and have Hm

0 (Ω,Rq)-coercive principal parts, for 2 ≤ p <∞, under appro-priate conditions, cf. (8.38)–(8.41), by Theorems 2.124, 2.125, 2.126, 2.127. Thus


we obtain stability for 2 ≤ p <∞, with respect to the Hm+ (Ωh,Rq) norm. The case

1 ≤ p <∞, with respect to the Wm,p+ (Ωh,Rq) norm can be studied via monotone

operators.We memorize the discrete Vh

b -coercivity and ellipticity. We choose the discrete spacesVh

b = Hm0,+(Ωh,Rq), or Vh

b = Wm,p0,+ (Ωh,Rq), m, q ≥ 1, 1 ≤ p ≤ ∞ for the homogeneous

Dirichlet boundary value problem. Let ah(uh, vh), cf. e.g. (8.47), (8.63), (8.71) for m ≥1, be a bounded bilinear form, defined for Vh, and Ah : Vh → V ′h the induced linearoperator. Then ah(uh, vh) are called Vh

b -coercive, and Vhb -elliptic, if h-independent

constants α ∈ R+, Cc ∈ R exist such that

ah(uh, uh) > α‖uh‖2Vh , and ≥ α‖uh‖2Vh − Cc‖uh‖2Wh ∀uh ∈ Vhb ⊂ Wh, (8.120)

respectively, e.g. Vhb = Hm

0,+(Ωh,Rq) or Wm,p0,+ (Ωh,Rq) ⊂ Wh = L2

h(Ωh,Rq),

m, q ≥ 1.

If additionally C ∈ R+ exist such that

Ah : Vhb → V ′h

b is boundedly invertible with ‖(Ah)−1‖V′hb →Vh

b≤ C ∀h > 0, (8.121)

then Ah is called Vhb -stable or Vh

b -regular, e.g. by Thomee and Westergren [627],Hackbusch [381,383,387] and Bube and Strikwerda [151].

The proofs for the Vb-coercivity in Chapter 2, and the Vhb -coercivity here are very

similar. So we only indicate the proof for one equation of order 2, but present it fora system of order 2m. This case shows the strongest differences between the Vb-, andthe Vh

b -situation.48

48 We recapitulate the discrete Fourier transform, cf. Theorem 2.105, Thomee and Westergren[627], Stoer and Bulirsch [603] and Bube and Strikwerda [151], pp. 660 ff., and Bube et al. [605]. Instandard applications to periodic functions it yields (the coefficients of) the trigonometric polynomialsinterpolating discrete function values uh : Gh

n = Gh → R, see (8.96), e.g. via FFT, but not the Fourierseries. This method will be extensively studied in spectral methods, see [120]; see also any standardtextbook on numerical methods, cf. [151,603]. We only summarize the results needed for the followingcoercivity proof in this special case. We assume Ω in (8.95) or (8.98). For (8.98) we extend Ω to acuboidal Ωe with uh(x) = 0∀x �∈ Ωh

0 . Then we transform it, such that Ω = (−π, π)n, and

Ωh := {xν := hν} ⊆ [−π, π]n with h := π/N, N ∈ N, (8.122)

ν = (ν1, · · · , νn) ∈ Zn, ∀j = 1, · · · , n : |νj | ≤ N.

We periodically extend the grid functions, uh ∈ Hm0,+(Ωh), to Gh := {hν : ν = (ν1, · · · , νn) ∈ Zn},

since uh has identical, here even vanishing, values on opposite faces of [−π, π]n. We extend scalar

products, and norms, see (8.25), to uh, vh : Ωh → C with uh(x)∗ = uh(x) as

(uh, vh)L2h(Ωh) := hn

∑x∈Ωh

uh(x)∗vh(x), ‖vh‖Hm+ (Ωh) :=

⎡⎣ ∑|α|≤m

hn∑

x∈Ωh

|∂α+vh(x)|2

⎤⎦1/2

.

We define the discrete Fourier transform with the imaginary unit, ι, ι2 = −1, and

〈ξ, x〉n =

n∑k=1

ξkxk ∀ x ∈ Ωh, ∀ ξ ∈ Γh := {ξ = (ξ1, · · · , ξn) ∈ Zn : |ξj | ≤ N} (8.123)


For order 2m, discrete Fourier transforms are employed. They have been used in theliterature to prove regularity results for the solutions of difference equations, similarto those of differential equations, and the convergence to exact solutions, cf. Agmonet al. [2, 3], Thomee and Westergren [627], Bube and Strikwerda [151], Strikwerdaet al. [605] and Hackbusch [387]. But the author did not find any application towardsdiscrete coercivity as in this section.

Theorem 8.25. Vhb -coercive principal parts: Assume the conditions in Summary

8.6, and Remark 8.7, let Vhb := Hm

0,+(Ωh,Rq), Uhb := H2m

+ (Ωh,Rq) ∩ Vhb , Vh

b = V ′hb :=

L2h(Ωh,Rq), and let fh ∈ V ′h

b be bounded so p = 2 compared to Subsections 8.5.2, 8.5.3.

1. Then the weak bilinear forms ah(uh, vh) are bounded with respect to Vh, theprincipal parts ah

p(uh, vh), and ah(uh, vh) are Vhb -coercive, and Vh

b -elliptic, respec-tively. The ah

p(uh, vh) induce a stable Ahp , which is consistent with Ap.

2. The strong difference operators, Ahs , or (G′(u0))h

s , are also bounded with respectto Vh

s .

Proof.

(1) The claim that the strong operators, Ahs , (G′(u0))h

s and ah(uh, vh) are boundedwith respect to Vh

s , and Vh, respectively, is nearly obvious.(2) Since the ah(uh, vh) are compact perturbations of the principal parts

ahp(uh, vh) the Vh

b -ellipticity of the ah(uh, vh) can be shown, via the detour

as

(Fhuh)(ξ) := uh(ξ) := hn∑

x∈Ωh

e−ι〈ξ,x〉nuh(x) = (eι〈ξ,·〉n , uh(·))L2h(Ωh) ∀uh ∈ Hm

0,+(Ωh).

Obviously this uh(ξ) is 2N periodic in each ξj for ξ ∈ Zn. The fact that uh(x) and uh(ξ) are defined

on different grids Ωh and Γh or their extensions, is reflected in a certain asymmetry of the inversediscrete Fourier transform, in contrast to (2.397). The following properties are proved in standardtextbooks, and, e.g. Bube and Strikwerda [151].

Theorem 8.24. For the discrete Forier transform its inverse is defined as(F−1

h uh)

(x) :=∑

ξ∈Γh

eι〈ξ,x〉n uh(ξ), it yields uh(x) =(F−1

h uh)

(x), and (8.124)

(uh, vh)L2h(Ωh) = hn

∑x∈Ωh

uh(x)∗vh(x) =∑

ξ∈Γh

uh(ξ)∗vh(ξ) = (uh, vh)L2h(Γh)∀uh, vh ∈ L2(Ωh),

the so-called Parseval formula, and Fh

(∂β+uh

)(ξ) =

∂β+uh(ξ) = ξh,β uh(ξ), with

ξh,β :=

n∏=1

(exp(ιhξ) − 1)β�

hβ�, ξh,β =

n∏=1

2(exp(ιhξ/2)β� sin(hξ/2))β�

hβ�, |ξh,β | ≈ ±

n∏=1

2(ξ/2)β�

for small h. The squares of the norm and seminorm, ‖uh‖2Hk

+(Ωh), and |uh|2

Hk+(Ωh)

, are related as

∥∥∥uh∥∥∥2

Hk+(Ωh)

=∑

ξ∈Γh

∑|α|≤k

∣∣∣ξh,α∣∣∣2 uh(ξ)∗uh(ξ),

∣∣∣uh∣∣∣2Hk

+(Ωh)=∑

ξ∈Γh

∑|α|=k

∣∣∣ξh,α∣∣∣2 uh(ξ)∗uh(ξ).


ap(u, v), a(u, v) as in Chapter 2 by results similar to Theorem 1.26 andProposition 2.24.

(3) So we turn to the Vhb -coercivity of the ah

p(uh, vh), only indicating the proof form = q = 1. In fact for (8.38) we only have to replace ϑi by ∂i

+u(x), and sumover x ∈ Ωh to get the claim. This is analogous to (8.38) where ϑi = ∂iu(x) iscombined with integrating over Ω.

(4) For m, q > 1, so for systems (8.71) of order 2m, we impose the above (8.41).For equations and systems of order 2m, similarly to the situation in Theorem2.36, we cannot repeat the above argumentation in (3) based upon (8.38). Thereason is again the inequality

(∂i+u)αi �= ∂αi

+ u for αi > 1. So we apply insteadthe discrete Fourier transform to each component ∂β

+uj and ∂α+uj . So we prove:

(4) With Theorem 8.24 for the discrete Fourier transform, we prove the Vhb =

Hm0,+(Ωh)-coercivity of the principal part. We only discuss constant coefficients

Aαβ , |α|, |β| = m. The reduction of Aαβ ∈ Cm(Ω) to constant coefficients by thetechnique of frozen (constant) coefficients is presented in the proof of Theorem 2.43.It can be applied to difference methods as well.

We combine Theorem 8.24 with the Legendre–Hadamard condition (8.41). With thenotation in (8.123), (8.124) and

ξh,β =n∏

�=1

(exp(ιhξ�)− 1)β�

hβ�=

n∏�=1

2(exp(ιhξ�/2)β� sin(hξ�/2))β�

hβ�, (8.125)

we obtain with ξh,β =∏n

�=1

(ξh�

)β� , ϑ� := ξh� ,

ahp(�uh, �uh) =

⎛⎝ q∑j,k=1

∑|α|=|β|=m

hn∑

x∈Ωh

(ajk

αβ∂β+u

hk(x)

)∂α+u

hj (x)

⎞⎠=

⎛⎝ q∑j,k=1

∑|α|=|β|=m

ajkαβ

⎛⎝hn∑

x∈Ωh

∂β+u

hk(x)∂α

+uhj (x)

⎞⎠⎞⎠=

⎛⎝ q∑j,k=1

∑|α|,|β|=m

ajkαβ

(∂β+u

hk , ∂

α+u

hj

)L2(Ωh)

⎞⎠=

⎛⎝ q∑j,k=1

∑|α|,|β|=m

ajkαβ

(Fh

(∂β+u

hk

), Fh

(∂α+u

hj

))L2(Γh)

⎞⎠=

q∑j,k=1

∑|α|=|β|=m

ajkαβ

∑ξ∈Γh

([ξh,β uk(ξ)

][ξh,αuj(ξ)]

), by (8.41),(8.125)

≥ λ∑ξ∈Γh

(n∑

�=1

∣∣ξh�

∣∣2)m q∑j=1

∣∣uhj (ξ)

∣∣2 .


For a fixed m > 1 there exists an ε > 0, such that(∑n

�=1

∣∣ξh�

∣∣2 )m > ε∑

|α|=m |ξh,α|2.This is verified by the polynomial formula. So we find with (8.124)

ahp(�uh, �uh) > λε

∑ξ∈Γh

q∑i=1

⎛⎝ ∑|α|=m

(ξh,α)2

⎞⎠∣∣uhi (ξ)

∣∣2 = λε|�uh|2Hm+ (Ωh,Rq).

With the equivalent norms |�uh|Hm+ (Ωh,Rq) and ‖�uh‖Hm

+ (Ωh,Rq) on Vhb = Hm

0,+(Ωh,Rq),the Vh

b -coercivity of ahp(�uh, �vh) is proved for constant coefficients. The general case is

proved as in Theorem 2.42. �

8.5.5 Stability and convergence for general elliptic difference equations

In Subsection 8.5.3 we proved the consistency of our discrete operator with theoriginal operators. Subsection 8.5.4 showed the Vb-coercivity of the principal partsof the linear(ized) operators, and hence its stability. This implies, by Theorem 8.21,the convergence of the discrete to the exact solutions of these elliptic problems forthe principal part. The corresponding Lemma 8.28 is combined in Theorem 8.29 forproving the stability for boundedly invertible linear elliptic operators.

Condition 8.26. Choose Uhb = Vh

b and their duals as Hm+ (Ωh,Rq), V ′h = H−m

+

(Ωh,Rq), cf. (8.27)–(8.31), (8.77), and Ph = Ih and Q′h as in Subsection 8.5.2 for

p = 2. The original linear and nonlinear problems and the difference methods are listedin Section 8.4. For the consistency, and its order 1 or 2, we refer to Theorems 8.21,8.22, and 8.23; for the coercivity of the principal parts we refer to Theorem 8.25.

Remark 8.27. Notice that we have chosen the Vh = Hm+ (Ωh,Rq), in contrast to V =

Wm,p(Ω,Rq). This allows our linearization approach based upon coercive principalparts, as used in this section. The direct discrete counterpart Vh = Wm,p

+ (Ωh,Rq) forV = Wm,p(Ω,Rq) is possible for quasilinear problems and their difference methods aswell. Then we have to use the monotony technique in Section 4.5, and Theorem 4.67remains valid for 1 < p <∞ and e.g. cf. (4.197), (8.146)∥∥Phu0 − uh

0

∥∥Uh

b

≤(hμL‖u0‖W m+μ(Ω)

)1/(p−1), (8.126)

for μ = 1 and = 2 for unsymmetric Vh = Hm+ and symmetric difference methods.

Even better results are obtainable by linearization techniques. They require 2 ≤ p <∞ and yield stability and convergence with respect to the discrete Hm

+ (Ωh) norm, cf.the discussion in Subsection 8.4.5, following (8.65).

The following lemma enforces the consequences of Theorem 8.21. The different errorsare related with Ahuh = Q

′hAu as:



interp.error


class.cons.err.

; Ahuh = Q′hAu. (8.127)


Lemma 8.28. Under Condition 8.26, let Ap : Ub → V ′ be a coercive principal part, cf.(5.278) or (5.281). Then Apu = f ∈W−m,p′

(Ω), and for small enough h0 > h ∈ H,

Ahpu

h = Q′hf as well, have unique solutions, and uh converge to u as

‖uh − Phu‖Uh ≤ ε ‖u‖U , for h < h0, hence limh→0

‖uh − Phu‖Uh = 0. (8.128)

For general A, and u ∈ Ub, the preceding consistency errors satisfy, for h < h0,

‖Q′hAu−AhPhu‖Vh′ ≤ ε ‖u‖U , hence limh→0

‖Q′hAu−AhPhu‖Vh′ = 0. (8.129)

Proof. We can essentially dublicate the proof of Lemma 5.76, if we replace ‖u− uh‖Uh

there by ‖Phu− uh‖Uh here. �

Theorem 8.29. Stability inherited to invertible compact perturbations. UnderCondition 8.26, let A,B ∈ L (Ub,V ′

b) , Ub = Wm,p0 (Ω),V ′

b = W−m,p′

0 (Ω), be boundedly

invertible, and Bh ∈ L(Uh

b ,V′hb

)be stable, e.g. B = Ap. Let A := B + C, with C ∈

C (Ub,V ′b), the set of compact operators from Ub → V ′

b, and let Ah, Bh, hence Ch, beconsistent with A,B,C. Then, for small enough h ∈ H,

A−1 ∈ L(V ′,Ub) =⇒ Ah is stable in Phu hence

(Ah)−1 ∈ L(V ′h

b ,Uhb

), ‖(Ah)−1‖Uh

b ←↩V′hb≤ C.

Remark 8.30. We can choose B = Δm. Then, for Uhb = Wm,p

0,+ (Ωh) with 2 < p ≤ ∞only Hm

0,+(Ωh)-coercivity, stability and convergence is obtained, cf. Summary 8.6,Remarks 8.7, 8.14 and Theorems 8.31, 8.32.

Proof. This proof essentially modifies that for FEMs with variational crimes. Weconsider the appropriate extensions Eh, and Eh, cf. (8.86), (8.131).

(1) Again we define a kind of anticrime transformation from Uhb to Ub = Wm,p

0 (Ω) ⊂U , cf. Lemma 5.77. It allows a direct comparison between the elements in,e.g. Uh

b = Wm,p0,+

(T h

c

)and Ub = Wm,p

0 (Ω),V ′b = W−m,p′

0 (Ω). We again choosethe boundedly invertible B : Ub → V ′

b, here B = Δm. We have proved Bh : Uhb →

Vh′

b to be stable and consistent with B for every u ∈ Ub, with a consistency orderp > 0 for smoother u. In the following procedure we combine Proposition 8.18and Theorem 8.21 with the results in Theorems 8.25 and 3.21:

For any given uh ∈ Uhb define fh := Bhuh ∈ Vh′

, and fh := Ehfh ∈ V ′,

with ‖fh‖V′ ≤ ||Bh||Vh′←↩Uhb‖uh‖Uh(1 + o(1)), cf. (8.30). (8.130)


We define the bounded

Eh : Uhb → Ub : uh := u0,h := Ehu

h := (B)−1fh ∈ Ub = Wm,p0 (Ω) yielding

‖Ehuh‖U ≤ C1‖uh‖Uh , and ∀ε > 0 ∃h0 = h0

(∥∥uh∥∥

H1(T hc )

): ∀h < h0 :

(8.131)∥∥PhEhuh − uh

∥∥Uh

b

< ε‖uh‖Uhb, hence, lim

h→0‖PhEhu

h − uh‖Uhb

= 0.

(2) We derive several relations and inequalities. For an arbitrary u ∈ Ub, and v′ :=Cu ∈ V ′

b, we determine the exact, and discrete solutions, u, and uh, of

Bu = v′ and Bhuh = Q′hCu = Q

′hv′ ∈ V ′hb .

By assumption they uniquely exist and the uh converge to u in the sense that

∀ε > 0 ∃h0 = h0(‖u‖U ) : ∀h < h0 :∥∥Phu− uh

∥∥Uh < ε ‖u‖U . (8.132)

With the operators B−1, (Bh)−1Q′h, and the notation

T := B−1 ∈ L (V ′b,Ub) , and

Th := (Bh)−1Q′h ∈ L

(V ′

b ∪ V′hb ,Uh

b

), Q

′h|V′hb

= Id,

we relate this result to the stability and consistency of Bh, as

Bhuh −BhPhu = Q′hCu−Q

′hBu + Q′hBu−BhPhu = Q

′hBu−BhPhu⇒‖Phu− uh‖Uh ≤ ‖(Bh)−1‖Uh

b ←↩V′hb‖Q′hBu−BhPhu‖V′h

b→ 0 for h→ 0.

Applying Bu = Cu and the equibounded (Bh)−1 to the terms in the last norm,we find

‖(Bh)−1Q′hCu− PhB−1Cu‖Uh = ‖(Th − PhT )Cu‖Uh → 0

for h→ 0 ∀ u ∈ Ub.

Since C is compact, and ‖(Th − PhT )‖Uhb ←V′

bare equibounded, this implies∥∥(Th − PhT )C

∥∥Uh

b ←Ub→ 0 for h→ 0. (8.133)

We combine uh ∈ Uhb with Eh. With the boundedly invertible A we estimate

‖uh‖Uh ≤ 2∥∥Ehu

h∥∥U ≤ 2

∥∥A−1∥∥Ub←↩V′

∥∥AEhuh∥∥V′

= 2∥∥A−1

∥∥Ub←↩V′

∥∥B(I + TC)Ehuh∥∥V′ (8.134)

≤ 2∥∥A−1

∥∥Ub←↩V′ ‖B‖V′←↩Ub

∥∥(I + TC)Ehuh∥∥U ,

hence, ∥∥(I + TC)Ehuh∥∥U ≥

∥∥uh∥∥Uh /(2

∥∥A−1∥∥Ub←↩V′ ‖B‖V′←↩Ub

). (8.135)


With the stability of Bh we get∥∥(I + ThCh)uh∥∥Uh =

∥∥(Bh)−1Bh(I + ThCh)uh∥∥Uh

≤∥∥(Bh)−1

∥∥Uh

b ←↩Vh′∥∥Bh(I + ThCh)uh

∥∥Vh′ .

This implies∥∥Bh(I + ThCh)uh∥∥Vh′ ≥

∥∥(I + ThCh)uh∥∥Uh /

∥∥(Bh)−1∥∥Uh

b ←↩Vh′ . (8.136)

The linearity of difference methods implies

Φh(A) = Ah = Φh(B + C) = Bh + Ch (8.137)

= Bh(I + (Bh)−1Ch) = Bh(I + ThCh) : Uhb → Vh′

.

(3) For the final stability estimates we return to (8.136) with (Bh)−1Ahuh = (I +ThCh)uh, and use (8.137), cf. Remark 8.14. We extend Q

′h : V ′ → Vh′as Q

′h :V ′ ∪ Vh′ → Vh′

by defining for the new component Vh′, the Q

′h|Vh′ = IVh′ .

This is combined with Th|Vh′ = (Bh)−1Q′h|Vh′ = (Bh)−1, the equibounded-

ness of Ch and Th, the consistency of Ch, the indicated error terms, and thetriangle inequality estimating

‖(Bh)−1‖Uhb ←↩Vh′

b||Ahuh||Vh′ (8.138)

= ‖(Bh)−1‖Uhb ←↩Vh′

b||Bh(I+(Bh)−1Q

′hCh)uh||Vh′

≥ ‖(Bh)−1‖Uhb ←↩Vh′

b||Bh(I + ThCh)uh||Vh′

≥(||Ph(I + TC)Ehu

h||U − ||uh − PhEhuh||Uh − ||ThChuh − PhTCEhu

h||Uh

)≥(Ph||(I + TC)Ehu

h||U − ||uh − PhEhuh||Uh

− ||(Th − PhT )CEhuh||Uh − ||Th(Chuh − CEhu

h)||Uh

)≥(||(I + TC)Ehu

h||U/2− ||uh − PhEhuh||Uh

− ||(Th − PhT )CEhuh||Uh − ||Th(ChPhEhu

h − CEhuh)||Uh

− ||Th(Ch − ChPhEh)uh||Uh

)≥

⎛⎜⎝|| (I + TC)Ehuh︸︷︷︸

by (8.135)

||U/2− ||uh − PhEhuh︸︷︷︸

diff. err. (8.131)

||Uh

− 2|| (Th − PhT )C︸︷︷︸by (8.133)

||Uhb ←↩Ub

||uh||Uh

− ||Th (ChPhEhuh − CEhu

h)︸︷︷︸class.consist. C,(8.129)

||Uh − ||ThCh (uh − PhEhuh)︸︷︷︸

diff. error (8.131)

||Uh

⎞⎟⎠ .


We choose h small enough, such that all the preceding errors in (8.131), (8.133),(8.129), (8.89), (8.91) are < ε||uh||Uh with

ε < 1/(8(2 + ||Th||Uh

b ←↩Vh′b

+ ||ThCh||Uhb ←↩Uh

b

)||A−1||Ub←↩V′ ||B||V′←↩Ub

).

This implies

||Ahuh||Vh′ ≥(||uh||Uh/

(8||A−1||Ub←↩V′ ||B||V′←↩Ub

))/||Bh−1||Uh

b ←↩Vh′ (8.139)

≥ K ′||uh||Uh ,

i.e. (Ah)h∈H is stable. �

For difference methods, Theorem 8.29, similarly to Theorem 5.78 for FEMs, yields acriterion for the stability of discretizations of operators, which are compact perturba-tions of coercive operators. An important class are A ∈ L (Ub,U ′

b) satisfying a so-calledGarding inequality. Hence Theorem 3.32 can directly be generalized to our differencemethods.

We combine the consistency and coercivity/stability results in Theorems 8.21–8.23,8.29 with Condition 8.26 for obtaining the following theorem. We only formulatethe convergence results of orders μ for u ∈Wm+μ,p(Ω) and μ = 1, 2. For the changefrom Hm(Ω) to Wm+μ,p(Ω), compare the remarks at the end of Subsection 8.4.4. Forconvergence for u ∈Wm,p(Ω) we refer directly to Theorems 8.21–8.23. We start listingthe conditions for linear, quasilinear and fully nonlinear problems.

1. We use in Theorem 8.31 the following spaces, cf. (8.77), often we omit Rq andset p = p′ = 2

Ub = Vb = Wm,p0 (Ω,Rq), V ′

b = W−m,p′

0 (Ω,Rq), for m, q ≥ 1, 1 ≤ p ≤ ∞,

Uhb = Vh

b = Wm,p0,+ (Ωh,Rq), V ′h

b = U ′hb = W−m,p′

+,0 (Ωh,Rq), (8.140)

and for strong equations Uhs = W 2m,p

+ (Ωh),Uhs,b = Uh

s ∩Wm,p0,+ (Ωh),

Vhs = V ′h

s = L2+(Ωh).

2. For μ = 1, 2, and for linear problems we assume, cf. Remark 8.7,

aα,β ∈ C(Ω) ∩Wμ,∞(Ω), u ∈Wm+μ,p(Ω), A : Wm+μ,p(Ω) →W−m+μ,p′.

(8.141)

3. For quasilinear problems we assume, with Nm in (8.17),

Aj , Aα equicontinuous in Ω×N∇m ⊂ Ω× RNm , N∇m bounded, s.t. (8.142)

∀u : ‖u− u0‖V ≤ ρ,V,∇≤mu(x) ∈ N∇m∀x ∈ Ω,∇≤m+ u(x) ∈ N∇m∀x ∈ Ωh

0 ,

with Lipschitz-continuous Aj , Aα with respect to the variables in N∇m . Finallylet for

u ∈Wm+μ,p(Ω), Aα : D(Aα) ∩Wm+μ,p(Ω) →W−m+μ,p′(Ω)) (8.143)


4. For fully nonlinear problems, here only formulated for order 2m = 2, let

G be equicontinuous in Ω×N∇2 ⊂ Ω× RN2 with bounded N∇2 , s.t. (8.144)

∀u : ‖u− u0‖V ≤ ρ,V,∇≤2u(x) ∈ N∇2∀x ∈ Ω,∇≤2+ u(x) ∈ N∇2∀x ∈ Ωh

0 ,

with Lipschitz-continuous G with respect to the variables in N∇2 . Furthermorerequire

for u ∈ H2+μ(Ω), G : D(G) ⊂ H2+μ(Ω) → Hμ(Ω)). (8.145)

We obtain consitency of order 1 and 2, respectively.

Theorem 8.31. Convergence for symmetric and unsymmetric finite difference meth-ods: Let Ub = Vb ⊂ V and Uh

b = Vhb be the spaces and their duals in (8.140),

cf. (8.27)–(8.31), (8.77), and Ph = Ih and Q′h the projectors in Subsection 8.5.2.

For the original linear and nonlinear elliptic problems, Au = f ∈W−m.p′(Ω,Rq) and

G(u) = 0 ∈W−m.p′(Ω,Rq) define the difference methods as listed in Section 8.4, and

impose, for the different cases, the conditions (8.141)–(8.145). Finally, let A orG′(u0) : Wm,p(Ω,Rq) →W−m,p′

(Ω,Rq) be boundedly invertible.

1. Then under the conditions in Chapter 2, a (locally) unique exact solution u0 andfor small enough h, a unique discrete solution uh

0 for these problems exist.2. Stability, and hence convergence, follows for 2 ≤ p <∞ from the Hm

0,+(Ωh)coercivity of the principal part ah

p(.,.) in the ‖ · ‖Hm+

(Ωh) norm.3. If for Uh

0 = Vhb = Wm,p

+ (Ωh) with 2 ≤ p <∞ an inf-sup condition for ahp(.,.) is

valid, the uh0 converge in ‖ · ‖W m,p

+ (Ωh).4. For strong difference equations Theorem 8.21 can be updated. More precisely, we

obtain convergence of order μ, with μ = 1, and 2, for unsymmetric and symmetricdifference methods, cf. (8.105), (8.106), (8.103), for the linear and quasilinearproblems. Symmetric methods, with μ = 2, are worthwhile in cuboidal domains,cf. (8.95).∥∥Phu0 − uh

0

∥∥Uh ≤ Chμ‖u0‖W m+μ,p(Ω) for μ = 1, 2, u0 ∈Wm+μ,p(Ω), (8.146)

with p = 2 and p > 2 for the two previous cases in 2. and 3., respectively. Forfully nonlinear second order problems we find∥∥Phu0 − uh

0

∥∥Uh ≤ Chμ‖u0‖H2+μ(Ω) for u0 ∈ H2+μ(Ω). (8.147)

Theorem 8.32. Difference methods for monotone operators, eigenvalue problems,nonlinear boundary operators, quadrature approximations: Under the conditionsof Theorems 8.29 and 8.31, difference methods applied to the following problems yieldunique converging solutions. For monotone operators the convergence satisfies (4.196)–(4.198). For the other problems (8.146) and (8.147) are valid: variational methods foreigenvalue problems, cf. Section 4.7, for nonlinear boundary operators, cf. Section 5.3,quadrature approximations, cf. Section 5.4.


Solving nonlinear equations in difference methods

As for FEMs and DCGMs we use a discrete Newton’s method based upon the meshindependence principle (MIP). For the MIP we have to determine the linearized form ofthe nonlinear system and its discretization, cf. Subsections 8.4.6, 8.4.7. The derivativeshave to be consistently differentiable and of order μ, hence, additionally to thestandard consistency,

‖(Gh)′(Phu)Phv −Q′h(G′(u)v)‖V′h → 0 for small ‖u− u0‖V and, μ = 1, 2,

‖(Gh)′(Phu)Phv −Q′h(G′(u)v)‖V′h ≤ Chμ(1 + ‖u‖Us

)‖v‖Usfor h→ 0. (8.148)

These ‖v‖Usare usually ‖v‖W m+μ,p(Ω) and the above m,μ, p. The proof of this property

follows essentially the lines of FEMs and DCGMs, so we do not repeat it here, cf.Theorems 5.21, 7.46, 7.47. We transform these results to difference equations:

Theorem 8.33. Discrete Newton method for difference equations Gh: Let u0 ∈Hs(Ω) be an isolated solution of G(u0) = 0, such that G′(u0) : U → V ′ is boundedlyinvertible and let Gh

(uh

0

)= 0 indicate one of the difference equations for the nonlinear

problems in Subsections 8.4.6, 8.4.7. Assume G and G′(u) to be Lipschitz-continuouswith respect to the variables u, for ‖u− u0‖V small enough. Let the conditions inTheorems 8.21–8.23 be satisfied for G and G′. Then the corresponding differenceequation Gh for G is consistently differentiable.

Start the original Newton process with u1 ∈Wm+μ,p(Ω), μ = 1, 2, and the discreteNewton process for i = 1, . . . , with uh

1 := Phu1, for small enough ‖u0 − u1‖U and h.Then in the original Newton method and in

uhi+1 := uh

i −(Gh(uh

i

)′)−1

Gh(uh

i

). (8.149)

the ui+1 and the uhi+1 in (3.84) uniquely exist and converge quadratically to u0 and to

uh0 and uh

i to ui+1 of order μ, such that∥∥uhi+1 − uh

i

∥∥Uh ≤ C

∥∥uhi − uh

i−1


i − Phui

∥∥Uh ≤ Chμ‖ui‖Us

, i = 1, . . . .

8.6 Natural boundary value problems of order 2

We turn to natural boundary, e.g. Neumann conditions, instead of the previousDirichlet case. We only consider second order equations, and their symmetric differencemethods on cuboidal domains, cf. (8.95). Generalizations to order 2m are possible,but require many technicalities, cf. the discussion in Subsection 2.4.2 (2.92) ff. andSubsection 2.4.4, Lions and Magenes [478], p. 120, Oden and Reddy [518], p. 290 ff.Other boundary conditions are discussed, e.g. by Collatz [206], Forsythe and Wasow[319] and Hackbusch [387].

Standard natural boundary conditions as in (8.152) for linear problems (8.151) aregeneralized to quasilinear problems in (8.160). Mixed boundary value problems can beincluded by generalized test functions. As in the previous sections, there is nearly as

8.6. Natural boundary value problems of order 2 611

close a relation between the weak and strong differential equations as in (8.151)–(8.152)for natural boundary conditions, cf. Proposition 8.35. This allows us to prove uniqueexistence of discrete solutions, uh

0 , and their convergence of order 1 very similarly tothe previous sections. For the proof of the full order 2 of convergence of this differencemethod, the technique in Section 5.2 for fully nonlinear elliptic problems with FEMswould have to be employed. We omit that here.

8.6.1 Analysis for natural boundary value problems

As mentioned already, we only study second order elliptic problems in

U := V := H1(Ω),U ′ := H−1(Ω), with Ub := Vb := H10 (Ω),U ′

b := H−10 (Ω). (8.150)

We give a short summary of the analytic results for natural boundary value problemsof second order, cf. Chapter 2. They again allow the transformation from strong toweak forms, cf. (2.24), and Hackbusch [387], Section 7.4.

The As, A and a(·, ·) are maintained and the boundary conditions in (8.43)–(8.45)are modified. The conditions for the coefficients of A are formulated in Summary 8.6,and, for difference methods, in Remark 8.7. We solve Au0 = f ∈ U ′, u0 ∈ U (or foru0 ∈ H2(Ω) : Asu0 = g ∈ L2(Ω), Bau0|∂Ω ∈ L2(∂Ω))

a(u0, v) =n∑

i,j=0

∫Ω

aij∂iu0∂

jvdx = 〈Au0, v〉U ′×U = 〈f, v〉U ′×U (8.151)

= (Asu0, v)L2(Ω) +∫

∂Ω

(Ba u0)vds

with the natural boundary operator

Ba u :=j=1,...,n∑i=0,...,n

νjaij∂iu cf. (2.25). (8.152)

We do not consider the general f ∈ U ′, e.g. in (8.71), but restrict it in (8.151), (8.156),to the form

f(v) = 〈f, v〉U ′×U =∫

Ω

fvdx +∫

∂Ω

ϕvds,∀v ∈ U with (8.153)

f ∈ L2(Ω), ϕ ∈ H−1/2(∂Ω), ‖f‖U ′ =(‖f‖2U ′

b+ ‖ϕ‖2H−1/2(∂Ω)

)1/2

.

For c0 > 0, the extended principal part of a(u, v), in (8.151),

ap,e(u, v) := ap(u, v) + c0(u, v)L2(Ω) : U × U → R, c0 > 0, (8.154)

is U-coercive, cf. Theorems 2.43, 2.89, Corollary 2.44.Let A, and Ap,e : U → U ′, be the bounded linear operators induced by a(u, v), and

ap,e(u, v). Hence Theorem 2.12 implies a unique solution for f ∈ U ′,

∀f ∈ U ′ : ∃1u1 ∈ U : ap,e(u1, v) = 〈f, v〉U ′×U∀v ∈ U , (8.155)


and for

A−1 ∈ L(H−1(Ω),U) ⇒ ∃1u0 ∈ U : a(u0, v) = 〈f, v〉U ′×U∀v ∈ U . (8.156)

Ap,e is coercive and its difference form Ahp,e will turn out to be consistent with

Ap,e. So Theorem 3.29 implies the stability of a boundedly invertible A, as a compactperturbation of Ap,e.

The most important special case, showing some difficulties, is the Laplacian

Asu = −Δu,Au0 = f , a(u, v) =n∑

i=1

∫Ω

∂iu∂iv = (Asu, v)L2(Ω) +∫

∂Ω

v∂u/∂νds

Neumann boundary condition Bau = ∂u/∂ν = ϕ, f ∈ L2(Ω), ϕ ∈ L2(∂Ω). (8.157)

Obviously, for any constant c ∈ R we find −Δc = 0, ∂c/∂ν = 0, so (8.157) isnot uniquely solvable. Hence the kernel, kerF ′(u0) ∼= R of the operator Fu :=(−Δu, ∂u/∂ν) is not empty. By the Fredholm alternative, cf. Theorem 2.21, a uniquesolution exists only for the modified problem

∃1u0 ⊥ kerF ′(u0)⇔ f ⊥ kerF ′(u0) ⇔ 〈f, 1〉U ′×U =∫

Ω

f0dx +∫

∂Ω

ϕds = 0.

This is a special form of conditions relevant in bifurcation. A direct approach isdiscussed in [387]. We will study these problems in [120]. The numerical Liapunov–Schmidt method systematically proves the convergence for these cases. For avoidingthese difficulties, we assume here a boundedly invertible linear operator, A.

Then we solve (8.156), and summarize consequences for the differential equation andthe boundary condition, cf. Remark 2.6. We will partially prove again the results inTheorem 2.43 and Corollary 2.44. The transition in (8.159) from the weak Au0 − f = 0to the strong form is only possible for special f ∈ H−1(Ω), cf. (8.153), so we assumea Lipschitz-continuous ∂Ω, and

Au0 − f = 0 ⇔ Asu0 = f = f0 ∈ L2(Ω), Bau0 = ϕ ∈ H−1/2(∂Ω). (8.158)

Theorem 8.34. Analytical results for natural boundary conditions:

1. The ap,e(·, ·) in (8.154) is V-coercive, the original in (8.156) is V-elliptic, andsatisfies the Fredholm alternative. Regularity results are available in Theorems2.45 and 2.47. The latter yields u0 ∈ H2(Ω) for convex Ω and coefficients satis-fying (2.149), (2.47), and f ∈ L2(Ω), ϕ ∈ H1/2(∂Ω) in (8.158).

2. The solution, u0 ∈ H1(Ω), of (8.156), (8.158), with f in (8.153), satisfies

Au0 = f ∈ L2(Ω), Bau0 = ϕ ∈ H−1/2(∂Ω) or, cf (8.158), for smooth data

u0 ∈ H2(Ω), Asu0 = f ∈ L2(Ω), Bau0 = ϕ ∈ H1/2(∂Ω). (8.159)

3. An analogous result holds for the quasilinear problems in (8.160), (8.161).

Proof. We start testing (8.156) with ∀v ∈ Ub. This eliminates the boundaryterm,

∫∂Ω

ϕvds = 〈ϕ, v〉H−1/2(∂Ω)×H1/2(∂Ω) = 0, by 0 = v|∂Ω ∈ H1/2(∂Ω), and yieldsa(u0, v) = (f , v)L2(Ω)∀v ∈ Ub, hence Au0 − f ∈ L2(Ω), by (8.156), (8.151), and since


L2(Ω) is dense in V. We even get u0 ∈ H2(Ω), for f ∈ L2(Ω), ϕ ∈ H1/2(∂Ω), smoothenough ∂Ω and coefficients, or convex ∂Ω, hence Asu0 = f cf. Theorem 2.45. Next,testing (8.156) with general ∀v ∈ U implies

∫∂Ω

(Ba u− ϕ)vds = 0 ∀v|∂Ω ∈ H1/2(∂Ω).�

For the general quasilinear systems of order 2 we obtain

Gs(�u) :=n∑

k=0

((−1)k>0∂

kAk(x, �u,∇�u) (8.160)

∀�u ∈ V = H2(Ω,Rq), �v ∈ H1(Ω,Rq)

〈Gu, v〉V′×V = a(�u,�v) =∫ n∑

k=0

(Ak(x, �u,∇�u), ∂kv)qdx

−∫

∂Ω

(BA(�u,∇�u), �v

)qds,

where BA(�u,∇�u) :=n∑

k=1

νkAk(·, �u,∇�u),

For these general quasilinear systems of order 2 we obtain, with the precise con-ditions for the Ak discussed in Chapter 2, for difference equations in Remark 8.7,(8.111), and with Ak(·, �u,∇�u)(x) ∈ Rq, BA(�u,∇�u)(x) ∈ Rq:

Gs(�u) :=n∑

k=0

(−1)k>0∂kAk(x, �u,∇�u) ∈ L2(Ω,Rq), ∀�u ∈ H2(Ω,Rq), �v ∈ H1(Ω,Rq)

(Gs(�u), �v)L2(Ω,Rq) +∑

e∈∂Ω

∫e

(BA(�u,∇�u), �v

)qds =

∫Ω

n∑k=0

(Ak(x, �u,∇�u), ∂kv)qdx

= 〈G(�u), �v〉H−1(Ω,Rq)×H1(Ω,Rq),

and �u ∈ H1(Ω,Rq). (8.161)

8.6.2 Difference methods for natural boundary value problems

For these natural boundary value problems we are going to define the correspondingdifference methods. As in Section 8.4 we could formulate the first order unsymmetricforms for domains Ω in (8.98). This would yield similar equations as for Dirichletconditions, however with a modified discrete natural boundary operator Bh

a . Or weassume (8.95) and restrict the presentation to symmetric difference approximations,cf. Hackbusch [387], Subsections 4.7.2, 4.7.3 and Section 7.4. We only present thissecond choice in two different variants. We start by extending the original Ωh by alayer of exterior points. Another possibility, a shifted grid, will only be indicated.

As in Subsection 8.4.3, we have to avoid the ∂jhu

h, and instead use arithmetic means,(∂j− + ∂j

+

)uh/2. This complicates the presentation. It requires modifying the previous


Ωh in (8.96), (8.97). For

Ω :=n∏

i=1

(ai, bi), Ωh := Ω∩Gh, (8.162)

we add to Ωh left and right outer boundary grid points, x ∈ ∂Ωhl and x ∈ ∂Ωh

r both⊂ Gh in the distance h from Ωh, more precisely

∂Ωhl := {x = (x1, . . . , xn) ∈ Gh : ∃j : xj = aj − h,∀i �= j : ai ≤ xi ≤ bi}, (8.163)

∂Ωhr := {x = (x1, . . . , xn) ∈ Gh : ∃j : xj = bj − h,∀i �= j : ai ≤ xi ≤ bi} and

Ωhr := Ωh ∪ ∂Ωh

r , Ωhl := Ωh ∪ ∂Ωh

l ,Ωhe := Ωh

r ∪ ∂Ωhl , ∂Ωh

e := ∂Ωhr ∪ ∂Ωh

l .

The results for unsymmetric difference equations are obtained essentially by choos-ing one of the unsymmetric versions used in the following formulas for the averaging.

Straightforward extensions to systems with �u(x) ∈ Rq are omitted here.In this section the outer normal vectors ν in boundary grid points, x ∈ ∂Ωh, play

a crucial role. Sometimes even the number, #ν(x) ≥ 1, of these unit normals in x, ortheir parerallel shifts to ∂Ωh

e , are relevant:

for x ∈ ∂Ωh, x ∈ interior of a face of ∂Ω : #ν(x) = 1, otherwise #ν(x) > 1. (8.164)

In particular, we find for n = 2 two normals in the vertices, and for n = 3 two normalsin the edges, and three in the vertices. For an interior point x of a face of ∂Ω, hence#ν(x) = 1, this ν = ±ej for one j. For an x in an edge or a vertex x, we need #ν(x)unit vectors ej .

For every j = 1, . . . , n, there are two boundary points x ∈ ∂Ωje ⊂ ∂Ωh, cf. (8.176),

with the outer normal ν = −ej in ∂Ωjl and ν = +ej in ∂Ωj

r, νj = ∓1, cf. (8.163).Consequently, we use in the following formulas the notation ±

(a�,j∂

�±u

h)(· ∓ hej) =

νj(a�,j∂

�±u

h)(· ∓ hν).

Accordingly, we modify, for our symmetric forms, the discrete functions, uh, theSobolev spaces, the inner product, seminorms, norms. This yields, with the

∫Ωh ,∫Ωh

l,∫

Ωhr, cf. (8.25), (8.163), and a new boundary integral, e.g.

uh, vh : Ωhe → R,

∫Ωh

uhdxh := hn∑

x∈Ωh

uh(x),∫

∂Ωhe

uhdsh := hn−1∑

x∈∂Ωhe

uh(x),

(uh, vh)H1±(Ωh

e ) :=12

(∫Ωh

l

n∑j

∂j+u

h∂j+v

hdxh +∫

Ωhr

n∑j=0

∂j−u

h0∂

j−v

hdxh), (8.165)

and

Vh = H1±(Ωh

e

):={uh : Ωh

e → R, ‖uh‖H1±(Ωh

e ) <∞}. (8.166)

The relation between the weak and the strong linear difference operators is basedupon the natural discrete boundary operator Bh

auh and

∫∂Ωh

evhBh

auhdsh, cf. (8.174).

Accordingly, we modify the previous Ahs , A

h, ah(·, ·), fh(·), into the form for natural


boundary conditions motivated by (8.151)–(8.152), (8.153), and Proposition 8.35. Wemaintain the notation Ah, Ah

s , . . . , instead of Ahws, A

hss, . . . , for the weak and strong

form of symmetric difference equations in Subsection 8.4.3:

Ahs : H2

±(Ωh

e

)→ L2(Ωh), Ah : Vh := H1

±(Ωh

e

)→ V ′h, Bh

a : H1±(∂Ωh

e

)→ L2

(∂Ωh

e

),

fh : V ′h → R, uh, uh0 , v

h ∈ Vh, cf. (8.166):

Ahsu

h(x) :=12

n∑j,�=0

(−1)j>0

(∂j−(a�,j∂

�+u

h)

+ ∂j+

(a�j∂

�−u

h))

(x) (8.167)

ah(uh, vh) := 〈Ahuh, vh〉V′h×Vh (8.168)

:=12

⎛⎝∫Ωh

l

n∑j,�=0

a�,j∂�+u

h0∂

j+v

hdxh +∫

Ωhr

n∑j,�=0

a�j∂�−u

h0∂

j−v

hdxh

⎞⎠,

Bhau

h|∂Ωhl

:= Bha,lu

h :=12

⎛⎝ j=1...,n∑�=0,...,n

νj(a�,j∂

�+u

h +(a�j∂

�−u

h)(·+ hej)

)⎞⎠, in x ∈ ∂Ωhl

Bhau

h|∂Ωhr

:= Bha,ru

h :=12

⎛⎝ j=1...,n∑�=0,...,n

νj((a�,j∂

�+u

h)(· − hej) + a�j∂�−u

h)⎞⎠ in x ∈ ∂Ωh

r ,

∫∂Ωh

e

vhBhau

hdsh :=∫

∂Ωhl

vhBha,lu

hdsh +∫

∂Ωhr

vhBha,ru

hdsh, and (8.169)

〈fh, vh〉h :=∫

Ωh

fhvhdxh +∫

∂Ωhe

vhϕh(· − hν/2)dsh, cf.(8.153). (8.170)

Quasilinear difference systems satisfy, with ∇h± =

(∂1±, . . . , ∂

n±)

and Vh = H1±(Ωh,Rq),

〈Gh�uh, �vh〉V′h×Vh = ah(�uh, �vh) (8.171)

:=12

⎛⎝∫Ωh

l

n∑j=0

Aj

(·, �uh,∇h

+�uh)∂j+v

hdxh +∫

Ωhr

n∑j=0

Aj

(·, �uh,∇h

−�uh)∂j−v

hdxh

⎞⎠=∫

Ωh

12

n∑j=0

(�vh, (−1)j>0

(∂j−Aj

(·, �uh,∇h

+�uh)

+ ∂j+Aj

(·, �uh,∇h

−�uh)))

qdxh (8.172)

+

(∫∂Ωh

l

vhBha,lu

hdsh +∫

∂Ωhr

vhBha,ru

hdsh

)

=:(Gh

s�uh, �vh

)L2(Ωh)

+∫

∂Ωhe

vhBhau

hdsh, with (8.173)


Bhau

h|∂Ωhl

:= Bha,lu

h :=12

⎛⎝ n∑j=1

νj(Aj

(·, �uh,∇h

+�uh)

+ Aj

(·, �uh,∇h

−�uh)(·+ hej)

)⎞⎠,

Bhau

h|∂Ωhr

:= Bha,ru

h :=12

⎛⎝ n∑j=1

νj(Aj

(·, �uh,∇h

+�uh)(· − hej) + Aj

(·, �uh,∇h

−�uh) )⎞⎠

Proposition 8.35. For natural boundary conditions, the linear symmetric weakand strong difference operators, Ahuh, Ah

suh, ah(uh, vh), and the discrete boundary

operators, Bhau

h, in (8.167)–(8.169), are related by

ah(uh, vh) = 〈Ahuh, vh〉V′h×Vh =(Ah

suh, vh

)L2(Ωh)

+∫

∂Ωhe

vhBhau

hdsh. (8.174)

Quasilinear difference systems satisfy, with ∇h± =

(∂1±, . . . , ∂

n±)

and Vh = H1±(Ωh,Rq),

cf. (8.171)–(8.173)

〈Gh�uh, �vh〉V′h×Vh = ah(�uh, �vh) =(Gh

s�uh, �vh

)L2(Ωh)

+∫

∂Ωhe

vhBhau

hdsh. (8.175)

Proof. Again, cf. (8.53), we start with the first order result for j > 0 and the simplifiednotation for grid points and function values in Ωh in the ej direction:

Ωj := {xi := x0 + ihej , i = 0, . . . , k}, (8.176)

Ωjr := {xi := x0 + ihej , i = 0, . . . , k + 1},

Ωjl := {xi := x0 + ihej , i = −1, . . . , k},⊂ Gh cf. (8.163).

Then we obtain for j > 0 :

−∑Ωj

vh∂j−u

h = −(

k∑i=0

viui −k−1∑

i′=−1

vi′+1ui′

)/h

= −(

k∑i=0

(viui − vi+1ui)− v0u−1 + vk+1uk

)/h

=k∑

i=0

ui∂j+vi + (v0u−1 − v−1u−1 + v−1u−1 − vk+1uk)/h

=∑Ωj

l

uh∂j+v

h + (v−1u−1 − vk+1uk)/h

= −∑Ωj

vh∂j−u

h (8.177)


and

−∑Ωj

vh∂j+u

h =k∑

i=0

ui∂j−vi +

(v−1u0 − vkuk+1 + vk+1uk+1 − vk+1uk+1

)/h

=∑Ωj

r

uh∂j−v

h +(v−1u0 − vk+1uk+1

)/h = −

∑Ωj

vh∂j+u

h. (8.178)

With ∂0±u

h = uh, we insert for uh in (8.177) the wh :=(∑n

�=0 a�,j∂�+u

h), and in

(8.178) the wh :=(∑n

�=0 a�j∂�−u

h), multiply with hn, and sum for j ≥ 0:∫

Ωh

vhn∑

j,�=0

(−1)j>0∂j−(a�,j∂

�+u

h)dxh =

∫Ωh

l

n∑j,�=0

a�,j∂�+u

h∂j+v

hdxh (8.179)

+∫

∂Ωhl

vh

⎛⎝ j=1...,n∑�=0,...,n

a�,j∂�+u

h

⎞⎠ dsh

−∫

∂Ωhr

vh

⎛⎝ j=1...,n∑�=0,...,n

a�,j∂�+u

h

⎞⎠ (· − hej)dsh

and∑Ωh

vhn∑

j,�=0

(−1)j>0∂j+

(a�j∂

�−u

h)dxh =

∫Ωh

r

n∑j,�=0

a�j∂�−u

h∂j−v

hdxh (8.180)

+∫

∂Ωhl

vh

⎛⎝ j=1...,n∑�=0,...,n

a�j∂�−u

h

⎞⎠ (·+ hej)dsh

−∫

∂Ωhr

vh

⎛⎝ j=1...,n∑�=0,...,n

a�j∂�−u

h

⎞⎠ dsh.

We take the arithmetic mean of the last two equations and find (8.174).Similarly, applying these results to uh = Aj(·, �uh,∇�uh), yields (8.175). �

For the second variant we consider the shifted grid, shown for n = 2 in Figure 8.5,and we modify (8.163). To the inner grid points Ωh ⊂ Ω of the shifted grid Gh, we addthose grid points in Gh with distance ≤ h to Ωh, cf. (8.95)–(8.97):

For Ωh let Ωh := Ωh ∪ {◦ in Figure 8.5} and ∂Ωh := {× in Figure 8.5}. (8.181)

Compared to the first variant, we mainly have to modify the boundary conditions,but do not give the details. For, e.g. Neumann conditions, choose for each x ∈ ∂Ωh,never a vertex of Ω, the next neighbor xh = x∓ hν/2 ∈ Ωh, and the correspondingouter next neighbor xh

±h = x± hν/2 ∈ {◦ in Figure 8.5} ⊂ Ghn \ Ω. Then we require,


: points in Ωh

: points in ∂Ωh

Figure 8.5 Two-dimensional grid for a square, and Neumann conditions.

cf. (8.174),

ϕ(x) = ∂νh/2u

h(x) ≈ ∂

∂νu(x)∀x ∈ ∂Ωh. (8.182)

We might want to avoid outer points xh±h ∈ {◦ in Figure 8.5} outside Ωh. To this

end we combine, in the five-point star, the above xh = x∓ hν/2 ∈ Ωh with thecorresponding xh

±h �∈ Ωh. We replace the value for the uh(xh±h

)by a linear combination

of ϕ(x), and uh(xh), computed from (8.182). For an x ∈ ∂Ωh, next to an edge for n > 2or a vertex of Ωh, for n ≥ 2, e.g. the point • next to (0, 0) in Figure 8.5, there are atleast two directions with neigboring points xh

±h for x to apply this replacement. ForNeumann boundary conditions we are done.

Now we have to solve first the linear then the quasilinear problems.

Ahsu

h0 (x) = fh(x) ∀x ∈ Ωh, fh ∈ H−1

± (Ωh), (8.183)

uh0 ∈ Vh : ah

(uh

0 , vh)

=⟨Ahuh

0 , vh⟩V′h×Vh = 〈fh, vh〉h,∀vh ∈ Vh, (8.184)

Bhau

h0 (x) = ϕh(x− hν/2)∀x ∈ ∂Ωh

e ,

or ∫∂Ωh

e

vhBhau

h0ds

h :=∫

∂Ωhe

vhϕh(· − hν/2)dsh,

ah(uh

0 , vh)

=(Ah

suh0 , v

h)L2

h(Ωh)+∫

∂Ωhe

vhBhau

h0ds

h = 〈fh, vh〉h∀vh ∈ Vh. (8.185)

Either f , ϕ are continuous or we use appropriate approximations as in Proposition 8.5.The original f and ϕ define the corresponding fh := Q

′hf and ϕh := Q′hϕ in (8.170)

and (8.191). The quasilinear difference systems has the form

〈Gh�uh, �vh〉V′h×Vh = ah(�uh, �vh) =(Gh

s�uh, �vh

)L2(Ωh)

+∫

∂Ωhe

vhBhau

hdsh = 0. (8.186)


As in Subsection 8.6.1 we have the two choices of either the weak or the strong for-mulation. The latter essentially yields the classical difference equations and boundaryconditions.

The weak form allows using our variational techniques. For this we prove stability,unique discrete solutions and their convergence for our difference method. The neces-sary relation between the weak Ah, the strong Ah

s , the boundary condition, Bha , the

coercivity and consistency are collected in Propositions 8.36–8.38. They guarantee theclaim in Theorem 8.39.

So we contrast the weak solution, uh0 , cf. Proposition 8.35, similarly to Theorem

8.34, defined by

ah(uh

0 , vh)

=⟨Ahuh

0 , vh⟩V′h×Vh = 〈fh, vh〉∀v ∈ Vh = H1

±(Ωh

e

). (8.187)

The strong solution solves the difference equation and the discrete natural boundaryconditions:(

Ahsu

h0 − fh

)(x) = 0∀x ∈ Ωh and Bh

auh0 (x)− ϕh = 0∀x ∈ ∂Ωh

e . (8.188)

Proposition 8.36. Every solution, uh0 ∈ H2

±(Ωh

e

), of (8.188) satisfies (8.187) as well,

and vice versa. An analogous result holds for quasilinear problems.

Proof. We modify the proof of Theorem 8.34 according to (8.169)–(8.174). WithVh

0 :={vh ∈ Vh : vh|∂Ωh

e= 0}

we obtain

ah(uh

0 , vh)

=⟨Ahuh

0 , vh⟩V′h×Vh =

(Ah

suh0 , v

h)L2

h(Ωh)= (fh, vh)L2

h(Ωh)∀vhVh0 ,

hence the strong difference equation in (8.188). Next we extend testing (8.169) withvh ∈ Vh. By the previous equation, we are left with: for all vh ∈ Vh :∫

∂Ωhe

(Bh

auh0 − ϕh

(· − hν

2

))vhdsh = 0 ⇒ Bh

a uh0 (x)− ϕh

(x− hν

2

)∀x ∈ ∂Ωh

e ,

(8.189)

so a natural boundary condition.Vice versa, a solution uh

0 ∈ H1±(Ωh

e

), satisfying the strong difference equation, and

natural boundary conditions in (8.188), satisfies the weak problem (8.187) as well. Thesame strategy applies to quasilinear problems. �

By an obvious modification of part (3) of the proof of Theorem 8.25 we get

Proposition 8.37. Under the condition (8.38), and with c0 > 0, the extended prin-cipal part

ahp,e(u

h, vh) := ahp(uh, vh) + c0(uh, vh)L2(Ωh) : Vh × Vh → R (8.190)

is Vh-coercive, hence stable; the original ah(·, ·) in (8.169) is Vh-elliptic.

We finally need the consistency of our approximations. To that end, we use theobvious extensions of the Ph, Q

′h in Subsection 8.5.2 to the Ωhe , ∂Ωh

e in this section.


The only not totally obvious modification concerns the boundary integral terms. Wecome back to the boundary term of the fh in (8.170): for

〈ϕh, vh〉h :=∫

∂Ωhe

vhϕh(· − hν/2)dsh = hn−1∑

x∈∂Ωhe

(vh(x)ϕh(x− hν/2), let (8.191)

Q′hb : L2(∂Ω) → V ′h : Q

′hb ϕ(x) := ϕh(x− hν/2) := Ph(Ecϕ)(x− hν/2)∀x ∈ ∂Ωh

e ,

⇒⟨Q

′hb ϕ, vh

⟩V′h×Vh

:=∫

∂Ωhe

vhQ′hb ϕdsh =

∑x∈∂Ωh

e

hn−1vh(x)(PhEcϕ)(x− hν/2).

Proposition 8.38. With the extended Ph, Q′h, as in (8.99), (8.84), (8.93), Q

′hb in

(8.191), and the conditions for the coefficients of A in Summary 8.6 and Remark 8.7,we obtain for

Au = f ∈ V ′, A ∈ L(V,V ′), As ∈ L(H2(Ω), L2(Ω)), (8.192)

fh = Q′hf0, ϕ

h = Q′hb ϕ, u, v ∈ V = H1(Ω),Vh = H1

±(Ωh

e

), (8.193)

consistent symmetric approximations Ah, Ahs , B

ha cf. (8.60), (8.194), for linear opera-

tors, hence,

limh→0

||AhPhu−Q′hAu||V′h = lim

h→0||Ah

sPhu−Q

′hAu||V′h = 0 ∀u ∈ H1(Ω),

limh→0

||BhaP

hu−Q′hb Bau||V′h

b= 0∀u ∈ H2(Ω). (8.194)

More precisely, we obtain, for the weak form and u ∈ H2(Ω), A ∈ L(H2(Ω), L2(Ω)),Ba ∈ L(H2(Ω), L2(∂Ω)), consistency of order 1 for Ah, and for Bh

a ,

||AhPhu−Q′hAu||V′h ≤ Ch‖u‖H2(Ω), ||Bh

aPhu−Q

′hb Bau||V′h

b≤ Ch‖u‖H2(Ω).

For u ∈ H3(Ω) with A ∈ L(H3(Ω),H1(Ω)), and Ba ∈ L(H3(Ω),H1(∂Ω)), we obtainconsistency of orders 2 and 1 for Ah

s , and for Bha ,

||AhsP

hu−Q′hAu||V′h ≤ Ch2‖u‖H3(Ω), ||Bh

aPhu−Q

′hb Bau||V′h

b≤ Ch‖u‖H3(Ω)

With the technique in Section 5.2 we obtain consistency of order 2 for Bha as well.

For possible choices of fh �= Q′hf , and ϕh �= Q

′hb ϕ, the limh→0 ||fh −Q

′hf ||V′h = 0and limh→0 ||ϕh −Q

′hb ϕ||V′h

b= 0 have to be proved.

Analogous results hold for the quasilinear problems.

Proof. Except for ||BhaP

hu−Q′hb Bau||V′h

bwe can argue for the strong form as in

Theorem 8.21. In∫

∂ΩhevhBh

auhdsh we had to shift from x ∈ ∂Ωh to ∞− hν/2. This

reduces the consistency to order 1. Order 2 would be obtained if we could compare Ba

and Bha exactly along ∂Ωh. The technique in Section 5.2, splitting into differential and

boundary ooperators, allows this exact comparison along ∂Ωh, and thus yields order2. We do not repeat that proof. �


We summarize the necessary conditions for our difference methods for naturalboundary conditions, essentially restricted to convergence of order 2.

1. We use the following spaces, cf. (8.77), often we omit the Rq:

V = H1(Ω,Rq), V ′ = H−1(Ω,Rq), for q ≥ 1,

Vh = H1±(Ωh,Rq), Vh

b = H10,±(∂Ωh,Rq), V ′h = H−1

± (Ωh,Rq). (8.195)

2. For linear problems we assume, cf. Remark 8.7,

aij ∈ C(Ω), and A : H3(Ω) → H1(Ω), Ba : H3(Ω) → H1(∂Ω). (8.196)

3. For quasilinear problems we assume, with N1 in (8.17),

Aj equicontinuous in Ω×N∇1 ⊂ Ω× RN1 , N∇1 bounded, s.t. (8.197)

∀u ∈ V : ‖u− u0‖V ≤ ρ,∇≤1u(x) ∈ N∇1∀x ∈ Ω,∇≤1h u(x) ∈ N∇1∀x ∈ Ωh,

with Lipschitz-continuous Aj with respect to the variables in N∇1 . Finally weassume

Aj : D(Aj) ∩H3(Ω) → H1(Ω)). (8.198)

Theorem 8.39. Difference methods for natural boundary conditions:

1. Let A or G′(u0) : H1(Ω,Rq) → H−1(Ω,Rq) be boundedly invertible, for a (locally)unique exact solutions u0 for the exact equations, cf. the conditions in Chapter2. Under the conditions (8.195)–(8.197) we extend Ph = Ih and Q

′h, fromSubsection 8.5.2, to Ωh

e and define Q′hb as in (8.191). For the exact linear and

nonlinear problems, Au = f ∈ H−1(Ω,Rq) and G(u) = 0 ∈ H−1(Ω,Rq) definethe symmetric difference methods as in (8.167)–(8.175), defined on cuboidaldomains, cf. (8.95).

2. A unique discrete solution uh0 , simultaneously for the weak and strong discrete

linear and nonlinear problems exist near u0 for small enough h. It convergesof order 1. With the f in (8.153), and the corresponding fh in (8.170) withfh = Q

′hf0, ϕh = Q

′hb ϕ, in (8.192) we find

Ahuh0 = fh ⇒

∣∣|Phu0 − uh0

∣∣ |Vh ≤ Ch‖u0‖H2(Ω) for u0 ∈ H2(Ω). (8.199)

3. This uh0 satisfies the classical difference and a modified boundary equation:

Ahuh0 = fh ⇒

(Ah

suh0 − fh

)(x) = 0∀x ∈ Ωh, and (8.200)

Bhau

h0 (x) = ϕh(x− hν/2)∀x ∈ ∂Ωh

e .

4. If we maintain, in (8.200), (Ahsu

h0 − fh)(x) = 0, and replace the modified bound-

ary condition by the classical discrete boundary condition, then this implies


convergence of order 2:(Ah

suh0 − fh

)(x) = 0∀x ∈ Ωh, Bh

auh0 (x) = ϕh(x)∀x ∈ ∂Ωh (8.201)

=⇒∣∣|Phu0 − uh

0

∣∣ |Vh ≤ Ch2‖u0‖H3(Ω) for u0 ∈ H2(Ω).

5. We replace the above (8.167) ff., (8.174) for linear problems by (8.171) ff.,(8.175) for quasilinear problems. Then we obtain the corresponding results.

Proof. We only prove first order convergence, and indicate second order. We haveshown that Ah, Ah

p,e : H1±(Ωh

e

)→ H−1

±(Ωh

e

), induced by ah(uh, vh), ah

p,e(uh, vh), and

A,Ap,e induced by a(u, v), ap,e(u, v), are bounded linear operators. By Proposition8.37, the ah

p,e(uh, vh), and ap,e(u, v) are Vh-, and V-coercive. This A is a compact

perturbation of Ap,e with stable Ahp,e. Furthermore, the Ah

p,e, Ah are consistent with

the Ap,e, A, by Proposition 8.38.Hence for a boundedly invertible A, the H1

±(Ωh

e

)-stability of Ah

p,e, and the consis-tencies in Proposition 8.38 and Theorem 3.29 imply the stability of Ah. With A−1 ∈L(H−1(Ω),H1(Ω)), and the corresponding f ∈ H−1(Ω), fh = Q

′hf ∈ H−1±(Ωh

e

), as in

(8.170), we obtain

∃1uh0 ∈ H1

h(Ωh) : ah(uh

0 , vh)

=⟨Ahuh

0 , vh⟩

= 〈fh, vh〉∀vh ∈ H1h(Ωh), (8.202)

with the convergence. By the equivalence in (8.174), we verify the strong differenceequation, and the modified discrete natural boundary condition. This is shown as inthe previous sections.

For second order convergence, we have to replace the above modified boundarycondition in (8.200) by classical discrete natural boundary conditions in (8.201), andsimilarly for the quasilinear case. Then the full operator is split into differential andboundary operators, (As, Ba). The corresponding classical pair of discrete differenceand boundary operators,

(Ah

s , Bha

), is consistent of order 2. This allows the full proof

by modifying the technique in Section 5.2. We omit that here. �

8.7 Other difference methods on curved boundaries

In Section 8.4 we discussed nonsymmetric first order methods for domains withcurved boundaries for general elliptic problems. Here we modify these methods for thefollowing special cases, allowing, by appropriate treatment of the boundary, methodsof higher order. Points more or less near the boundary will have to be distinguished.We summarize the results for problems with Dirichlet boundary conditions, but donot prove them for several reasons:

For FEMs, an extended theory essentially for all types of elliptic equations andsystems is available, and is preferably used and presented in Chapter 4. For thementioned special cases, the proofs for stability, consistency and convergence usetechniques totally different from those in Sections 8.5, 8.6. They do not seem to allowgeneralizing and unifying them towards the problems in Section 8.4.

8.7. Other difference methods on curved boundaries 623

The results in Subsections 8.7.1 and 8.8.1 are only known for special cases. Hack-busch [387] formulates his results only for the Laplacian, but his proofs apply (ashe mentions) to most of the problems in Subsection 8.7.1. A coercivity proof as inthe last sections is not known for the Shortley-Weller method. We refer, cf. (8.210),to the Vh = H1

h(Ωh)-regularity result in [387], Theorems 9.2.3, and 9.2.8 for theprincipal part. Our techniques imply the stability of (8.205) for a boundedly invertibleoperator. For the polynomial interpolation case, even asymptotic expansions for thediscretization error are available, allowing higher order methods via extrapolationor defect corrections. Proving the convergence in the following subsections wouldprobably be possible for special cases by applying the techniques in Section 5.2,splitting the discretization error into errors for the differential and the boundaryoperator. However this is beyond the scope of this chapter.

Finally, Heinrich [395] discusses with appropriate machinery finite difference meth-ods on irregular networks.

8.7.1 The Shortley–Weller–Collatz method for linear equations

We assume Ω as in (8.98), and have to modify the Ωh,Ωh

and ∂Ωh. The originalgrid Gh, the j-th grid line Gh

j , and different types of grid points, Ωh,Ωhi , Ωh are

distinguished, cf. Figure 8.6. With ei, the i-th unit vector, i = 1, · · · , n, let

Gh := {x = (x1, . . . , xn)T : xi/h ∈ Z for i = 1, · · · , n}, (8.203)

Ghj := {x = (x1, . . . , xn)T ∈ Rn : xj ∈ R, xi/h ∈ Z for j �= i = 1, · · · , n},

Ωh := {x ∈ Ω∩Gh : x± hei ∈ Ω for i = 1, · · · , n}, Ωhi := (Ω∩Gh) \ Ωh,

∂Ωh :={P ∈ ∂Ω∩Gh

j for j = 1, · · · , n}, Ω

h:= Ωh ∪Ωh

i ∪ ∂Ωh,Ωhe := Ωh ∪Ωh

i .

We denote Ωh and Ωhi , and ∂Ωh as the set of regular, Ωh

i as the irregular, ∂Ωh as theboundary, Ω

has all mesh points in Ω, and Ωh

e as the essential mesh points in Ω. InFigure 8.6 we have marked these points by •, by �, and by �

Ω

points in Ωh : far from ∂Ω

points in Ωih : close to ∂Ω

points in ∂Ωh (on ∂Ω):

Ωh : = Ωh ∪ Ωih ∪ ∂Ωh

Ωeh : = Ωh ∪ Ωi

h

Figure 8.6 Grid for curved ∂Ω.


We only consider the following special second order equations, and formulate theirclassical strong difference equation, cf. (8.43)–(8.47). Note that we exclude mixedderivatives, and use symmetric formulas

Asu(x) =n∑

j=1

(− aj(x)(∂j)2u(x) + bj(x)∂ju(x)

)+ c(x)u(x) = f(x), (8.204)

and with ∂jh :=

(∂j− + ∂j

+

)/2, and ∂

2ej

h := ∂j−∂

j+, cf. (8.12), (8.13), define

Ahsu

h(x) =n∑

j=1

(− aj(x)∂2ej

h uh(x) + bj(x)∂jhu

h(x))

+ c(x)uh(x) = f(x) (8.205)

∀x ∈ Ωh, uh : Ωh → R, with boundary conditions uh(x) = 0∀x ∈ ∂Ωh, (8.206)

with bounded and continuous aj , bj , j = 1, . . . , n, cf. Remark 8.7.Then Ah

suh is well defined for ∀x ∈ Ωh, but not for x ∈ Ωh

i . The ellipticity condition(8.38) for this As has the form, see (2.16), (2.20),

n∑j=1

aj(x)(ϑj)2 ≥ ε|ϑ|2 or aj(x) ≥ ε > 0,∀x ∈ Ω. (8.207)

There is a long history of dealing with the missing x ∈ Ωhi in (8.205). We discuss

only two different approaches.We choose an x ∈ Ωh

i with at least one neighbor outside of Ω, hence

for x ∈ Ωhi choose the smallest possible 0 < s+

j , s−j ≤ 1, j = 1, · · · , n, (8.208)

with minj=1,··· ,n

{s+

j , s−j

}< 1 s.t. x + s+

j hej , x− s−j hej ∈ Ωh.

Shortley and Weller [588] use, instead of the symmetric differences, ∂2ej

h uh(x),in (8.205), the unsymmetric differences with respect to x′ = x− s−j hej < x < x′′ =x + s+

j hej , with u(x′) = 0 or u(x′′) = 0 for x′ or x′′ ∈ ∂Ωh, [588] replace, cf. [387],(4.8.5), (4.8.6),

∂2ej

h uh(x) by (∂j)2hu(x) := 2h−2(u(x′′)− u(x)

s+j

− u(x)− u(x′)s−j

)/(s+

j + s−j)

with∣∣u′′(x)− (∂j)2hu(x)

∣∣ ≤ hmax{s+

j , s−j

}‖u′′′‖C[x′,x′′]/3 (8.209)

and ∂jhu

h(x) by (∂j)hu(x) := h−1u(x′′)− u(x)s+

j

with |u′(x)− (∂j)hu(x)| ≤ hs+j ‖u′′‖C[x′,x′′].

The previous discrete Sobolev spaces have to be modified accordingly. The specialdifferences in (8.205), corresponding to (8.209), are indicated by special multi-indices,α, so let αc := jei, j = 0, 1, 2, i = 1, . . . , n. Finite differences, ∂αc

h , are chosen as in(8.205), (8.209). With these divided differences, ∂αc

h uh, the discrete Sobolev inner

8.7. Other difference methods on curved boundaries 625

products, norms, and spaces are defined, for m = 0, 1, 2, as

(uh, vh)Hm

h (Ωh):= (uh, vh)

L2h(Ω

h)+

∑0<|αc|≤m

(∂αc

h uh, ∂αc

h vh)

L2h(Ωh

e ), and (8.210)

|vh|2Hk

h(Ωh):=

∑0<|αc|=k

∥∥∥∂αc

h vh∥∥∥2

L2h(Ω

h)), ‖vh‖

Hmh (Ω

h):=

⎛⎝ ∑0≤k≤m

|vh|Hk

h(Ωh)

⎞⎠1/2

.

Applying (8.209) for (8.208) instead of (8.205), we obtain the Shortley–Weller approx-imation for an x ∈ Ωh

i , with minj=1,··· ,n{s+

j , s−j

}< 1, usually for one j :

Ahsu

h(x) :=n∑

j=1

[aj(x)h−2

(− 2s+

j

(s+

j + s−j)uh

(x + s+

j hej

)+

2s+

j s−j

uh(x) (8.211)

− 2s−j(s+

j + s−j)uh

(x− s−j hej

))+ bj(x)

uh(x + s+

j hej

)− uh(x)

s+j h

]+ c(x)uh(x).

Reinterpreted, this is essentially Collatz’s method as well [205, 206], see below. Forcomputations we multiply Ah

SW = Ahs in (8.205), (8.211) by the diagonal matrix Dh

obtaining, cf. [387], Corollary 9.2.10, with s−j s+j := 1 for x ∈ Ωh,

Dh := (diag d(x))x∈Ωhe, d(x) := min

j=1...,n

{1, 2s−j s

+j

}, and A

′hSW := DhAh

SW . (8.212)

Then instead of AhSWuh

0 = fh we have to solve A′hSWuh

0 = Dhfh. Hackbusch’s [387]Theorems 4.3.16, 4.8.4 and 9.2.8 are formulated for Asu = −Δu. His proofs forconsistency in (4.8.11) or Theorem 9.2.14 including his regularity in Theorems 9.2.3,9.2.8 apply to bj = c = 0 in (8.205), (8.211). The extension to boundedly invertibleoperators A : H1

0 (Ω) → H−1(Ω) follows with the previous consistency and regularityof the principal part by Theorem 3.29.

The preceding Shortly–Weller method can be generalized to special strong quasilin-ear equations of the form, cf. (8.64), yielding a linearized form (8.204),

G : D(G) ⊂ H1(Ω) → H−1(Ω), Gh : D(Gh) ⊂ H1h

(Ωh

e

)→ H−1

h

(Ωh

e

),

Gsu0 =n∑

j=0

(−1)j>0∂jAj(·, u0, ∂

ju0) = 0 ∈ L2(Ω), u0 ∈ H2 ∩H10 (Ω), (8.213)

Ghsu

h0 (x) =

n∑j=0

(−1)j>0

(∂j−Aj

(·, uh

0 , ∂j+u

h0

)+ ∂j

+Aj

(·, uh

0 , ∂j−u

h0

) )(x)/2 = 0

∀x ∈ Ωh, and uh0 ∈ Vh

b . (8.214)

For the proof, we combine, as usual, Theorems 3.23, 8.40, 3.29, 3.21 with the aboveconsistency. We summarize these results in

Theorem 8.40. Convergence of the Shortly–Weller method:


1. Let Ω be bounded, and in C2 or convex, and A : H10 (Ω) → H−1

0 (Ω) in (8.204)be boundedly invertible. Then A

′hSW , the matrix defined by the Shortley–Weller

method (8.205), (8.211), (8.212) as A′hSW : H1

h

(Ωh

e

)→ H−1

h

(Ωh

e

), cf. (8.210), is

stable and consistent. The Shortley–Weller equations have, for small enough h,a unique solution uh

0 , converging to the exact solution u0 of Au0 = f :∥∥uh0 − Phu0

∥∥H1

h(Ωhe )≤ Ch‖u0‖H2(Ω). (8.215)

2. Whenever the conditions for an M matrix are satisfied, cf. [387], Theorems 4.8.4,4.8.6, then

maxx∈Ωh

e

∣∣uh0 (x)− u0(x)

∥∥ ≤ Ch2‖u0‖C3L(Ω).

3. Whenever the conditions for well-posed quasilinear equations of the form (8.213)and for a locally unique solution u0 with boundedly invertible G′(u0) are satisfied,cf. Chapter 2, the Shortly–Weller equations (8.214) have, for a small enough stepsize h, a unique solution uh

0 , converging as in (8.215).

8.8 Asymptotic expansions, extrapolation, and defect corrections

This area of research for ordinary and partial differential equations florished between1960 and 1990. Since the interest in these results is pretty low these days, we will onlylist a choice of relevant results and apply them to extrapolation methods.

The following authors have contributed to this area: textbooks are, e.g. Conte andde Boor, Deuflhard and Hohmann, and Fox, Isaacson and Keller, and Stetter [207,284,324,414,596]. Papers are due to, e.g. Axelsson and Layton, Bohmer, with Hemker,Gross, Schmitt, R. Schwarz, and with Stetter, Chibi and Moore, Doedel with Reddien,Fox, Frank and Ueberhuber, Gragg, Hackbusch, Keller and Pereyra, Stetter, Monnet,Pereyra et al., Reinhardt, and Thoma [44,101,103,106,107,110,125,126,130,172,288–291,321–323,325,326,358,381–384,442,496,526,527,548,549,596,597,625]. Extensionsto asymptotic expansions for FEMs are due to Lin, Liu, Rannacher, Zou, [476,477,541].

In particular, Hackbusch’s regularity results applied to defect corrections [382–384]showed that defect corrections yielded the same orders of acccuracy as the extrapola-tion methods. The approach presented in this chapter, with convergence results withrespect to the discrete Sobolev norms, ‖ · ‖Hm

± (Ωh), is a special regularity result as well,now proved for all the previous problems. So the result [384] is valid for our methodsand problems as well.

Here we combine extrapolation techniques or defect corrections with asymptoticexpansions for the discretization error. They are available for cuboidal and curveddomains, cf (8.95) and (8.98). Similarly to extrapolation for numerical integration, cf.e.g. [414, 602, 603], they allow, by linear combinations of numerical results for stepsizes, say h, h/2, h/4, higher order methods via extrapolation. Defect corrections arestrongly related to discrete Newton methods. Deferred corrections are another choice,but are more difficult to explain. So we omit them here, cf. the above papers.

8.8. Asymptotic expansions, extrapolation, and defect corrections 627

8.8.1 A difference method based on polynomial interpolation forlinear, and semilinear equations

The methods discussed now, and the necessary tools for their proofs, fit even lessthan in Subsection 8.7.1, to the previous approach. We include them here, since theirdiscretization error admits an asymptotic expansion.

For smooth Ω, and small enough h we find for x ∈ Ωhi , cf. (8.203), e.g.

x + hej �∈ Ω, the x + sjhej ∈ ∂Ω, 0 < sj < 1, (8.216)

but x− (ν − 1)hej ∈ Ω, ν = 1, · · · , k ≥ 1.

Again we choose the smallest 0 < sj < 1, such that x + sjhej ∈ ∂Ω and the fullsegment {x− t h ej} ⊂ Ω,−sj ≤ t ≤ k − 1.

All the following methods essentially replace uh(x + hej) in (8.216) by the valueof a polynomial interpolating the boundary function ϕ(x + sjhej) in x + sjhej ∈ ∂Ω,and the uh(x) at the other points in (8.216): Gerschgorin [340] replaced uh(x + hej)by a nontrivial ϕ(x + sjhej). Pereyra et al. [527], referring to unpublished notes ofH.-O. Kreiss, use the following polynomial P k

j for k ≥ 2. It is a univariate k-th orderinterpolating polynomial

P xkj+ := P k

j ∈ Πk interpolates: P kj (x + sjhej) = ϕ(x + sjhej),

and P kj (x− (ν − 1)hej) = uh(x− (ν − 1)hej), ν = 1, · · · , k. (8.217)

Then they replace uh(x + hej) by P kj (x + hej). If in (8.205), bj(x) ≡ 0, the above

Shortley–Weller method is obtained for k = 2.Pereyra et al. [527] conjectured an asymptotic expansion for the discretization

error of order k based upon Bramble–Hubbard techniques [139]. Bohmer [106] provedthis expansion ‖u(x)− uh(x)− h2e2(x)− h4e4(x)‖∞ ≤ O(hk+1) for k ≤ 4. Since largevalues for k impose restrictive conditions for Ω, and h, k ≤ 4 is not a really severerestriction. [106, 139] essentially work with M -matrix arguments, not applicable toour previous technique.

Differently, Collatz [205,206] replaced uh(x), not uh(x + hej), by the value of the lin-ear polynomial interpolating in (x + sjhej , ϕ(x + sjhej)), and (x− hej , u

h(x− hej)).Then again (8.211) results. In fact, rescale (4.8.14a) and (4.8.14b) in [387], by theequibounded 2/(s� + sr), and 2/(su + so) and then proceed to the modified (4.8.16),yielding (4.8.7). So [205,206] is included in the last subsection.

For x + hej �∈ Ω, see (8.216) [106,527], we compute the approximate,

uh(x + hej) := P kj (x + hej) =

(ϕ(x + sjhej)−

k∑ν=1

cνju

h(x− (ν − 1)hej)

)/c0j ,

(8.218)with

cνj :=

k∏ν �=�=0

(1− sj)− �

ν − �,


and with our ϕ(x + sjhej) = 0.The situation x− hej �∈ Ω, and x− s−j hej ∈ ∂Ω, 0 < s−j < 1, in (8.216)–(8.218), is

handled analogously by replacing h by −h.Summarizing, we replace in (8.205), e.g. the ∂j

+uh(x) by

(P k

j (x + hej)−uh(x− hej)

)/h and obtain for an x ∈ Ωh

i , x + hej �∈ Ω

f(x) = Ahsu

h(x)

=n∑

j �=i=1

[−aj(x)∂2ej

h uh(x) + bj(x)∂jhu

h(x)]

+ c(x)uh(x)

− aj(x)h−2

([0−

k∑ν=1

cνju

h(x− (ν − 1)hej)

]/c0j − 2uh(x) + uh(x− hej)

)(8.219)

+ bj(x)h−1

([0−

k∑ν=1

cνju

h(x− (ν − 1)hej)

]/c0j − uh(x− hej)

)/2.

By (8.216), the situation x ∈ Ωhi , x + hej �∈ Ω, and x− hej1 �∈ Ω for j1 �= j or even

x± hej �∈ Ω for k = 1 is not excluded. Then in (8.219) the first sum over j �= i =1, . . . , n has to be modified into i = 1, . . . , n, x± hei ∈ Ω, and the missing i have to becollected in the last two lines of (8.219). We avoid the messy formulation.

For (8.219) we need a connectedness condition for the grid Ωh. We define, for a fixedx ∈ Ωh ∪Ωh

i , the set of neighbors as

N(x) :={x ∈ Ωh ∪Ωh

i occurring in (8.205) or (8.219)}.

Then we reformulate (8.205), and (8.219) with σ(x, x) �= 0 as

x ∈ Ωh ∪Ωhi : Ah

suh(x)− f(x) =:

∑x∈N(x)

σ(x, x)uh(x) + Bh(−aj , bi, f, 0 = ϕ, x).

Then the grid Ωh ∪Ωhi is called connected, if for every proper subset, Sh,

∀ Sh ⊂⊂ Ωh ∪Ωhi :

(∪

x∈ShN(x)

)∩((

Ωh ∪Ωhi

)\ Sh

)�= ∅. (8.220)

Furthermore we assume for small enough h the existence of an Ω∗ ⊃ Ω such that

h ≤ min

⎧⎨⎩minx∈Ω

(2∑n

j=1 aj(x)|c(x)|

)1/2

,n

minj=1

minx∈Ω

aj(x)|bj(x)|

⎫⎬⎭ (8.221)

and that

Ω ⊂ Ω∗ ⊃{x± hej : j = 1, · · · , n, x ∈ Ωh ∪Ωh

i

}. (8.222)

We summarize parts of Propositions 4.1, 4.3, and Theorems 4.4, 4.11 in [106]:

Theorem 8.41. Convergence for a Collatz–Pereyra method: For Ω in (8.98) letΩh ∪Ωh

i be connected, let As : C2(Ω) ∩ C(Ω) → C(Ω) in (8.204) satisfy (8.207), and


be boundedly invertible. Let Ahk be the matrix defined by the preceding method, see

(8.216)–(8.219), and let h satisfy (8.221). Then Ahk : L∞(Ω

h) → L∞(Ω

h) is invertible

with bounded∥∥∥(Ah

k

)−1∥∥∥∞≤ C, hence Ah

k is stable in the sense of (8.58). If additionally

for Ω∗ in (8.222) an extension u∗0 ∈ Cq+α(Ω∗), 2 ≤ q ≤ 6, 0 < α ≤ 1 exists, then the

discrete solution uh0 converges according to∥∥uh

0 − u0

∥∥L∞

h (Ωh)≤ Chmin{2,k+1,q−2+α}. (8.223)

In addition let

−aj , bj , c, f ∈ Ck,α(Ω), ϕ ∈ Ck+2,α(∂Ω), ∂Ω ∈ Ck+2,α, 0 ≤ k ≤ 4.

Then, by Theorems 2.38 and 2.39, the exact solution satisfies u0 ∈ Ck+2,α(Ω), andthere exist f2 ∈ Ck+α(Ω), f4 ∈ Ck−2+α(Ω), independent of h, such that the followingasymptotic expansion is valid∥∥uh

0 − (u0 + h2f2 + h4f4)∥∥

L∞h (Ω

h)≤ Chmin{4+α,k+1}. (8.224)

In Bohmer [104] we have extended results in Theorem 8.41 to quasilinear equations

Gsu(x) =n∑

j=1

(−aj

(x, u(x), ∂ju(x)

)(∂j)2u(x)

)− f(x, u(x),∇u(x)) = 0, (8.225)

Ghsu

h(x) =n∑

j=1

(−aj

(x, u(x), ∂j

hu(x))∂

2ej

h uh(x))− f(x, uh(x),∇hu

h(x)) = 0

∀x ∈ Ωh with boundary conditions u(x) = 0∀x ∈ ∂Ω, and uh(x) = 0∀x ∈ ∂Ωh,

with bounded and continuous f, aj , j = 1, . . . , n.Then Gh

s is well defined for ∀x ∈ Ωh, for x ∈ Ωhi the modification as in (8.219)

has to be employed. For the stability we have to consider the derivative G′(u0) of(8.225), evaluated in the exact solution, u0. This again has the form (8.204). Weimpose condition (8.226) for stability, and (8.227) for consistency, and asymptoticexpansions, basic for Theorems 4.5, and 5.3 in [104]:

Let u0 be the unique solution of (8.225), and (8.226)

0 < a ≤ aj(·, u0(·), ∂ju0(·)) in Ω,∀j = 1, . . . , n

0 < Q∗ ≤

⎛⎝∂f

∂u(·, u0(·),∇u0(·))−

n∑j=1

∂aj

∂u(·, u0(·), ∂ju0(·))(∂j)2u0(·)

⎞⎠

×

⎛⎝ n∑j=1

aj(·, u0(·), ∂ju0(·))

⎞⎠−1

≤ Q∗ in Ω, and as the aj , bj


for G′(u0) in (8.204) we get aj(·) := aj(·, u0(·), ∂ju0(·)),∀j = 1, . . . , n

bj(·) := − ∂aj

∂(∂ju)(·, u0(·), ∂ju0(·))(∂j)2u0(·)−

∂f

∂(∂ju)(·, u0(·),∇u0(·)) in Ω,

and assume ‖bj(·)/aj(·)‖L∞(Ω) ≤ P ∗j hence for our case∣∣∣∣( ∂aj

∂(∂ju)(·, u0(·), ∂ju0(·))(∂j)2u0(·) +

∂f

∂(∂ju)(·, u0(·),∇u0(·))

)×(aj(·, u0(·), ∂ju0(·))

)−1∣∣∣ ≤ P ∗

j , for stability, with

aj ,∂f

∂u,∂aj

∂u,

∂aj

∂(∂ju),

∂f

∂(∂ju)(·, u0(·),∇u0(·)) ∈ C(Ω), and, for asymptotics,

u0 ∈ C(Ω) ∩ C2q+2+α(Ω), aj ∈ C2q+α(Ω×B), f ∈ C2q+α(Ω×Bn), ∂Ω ∈ C2q+2+α,

with closed B ⊃ {(x, u0(x), ∂ju0(x))∀x ∈ Ω, j = 1, . . . , n} (8.227)

Bn ⊃ {(x, u0(x),∇u0(x))∀x ∈ Ω}, and 0 < α < 1.

Theorem 8.42. Asymptotic expansions for quasilinear problems: For Ω in (8.98)let Ωh ∪Ωh

i be connected, and Gs : D(Gs) ⊂ C2(Ω) ∩ C(Ω) → C(Ω) in (8.225) satisfy(8.226). Then Gh(·) in (8.225) is stable in Phu0.

Additionally let (8.216), k − 1 ≤ min{3, 2q + α}, (8.227), and the more or lessgeometric type conditions for these types of problems in Agmon et al. [2], Chapter7, be satisfied, implying u0 ∈ C2q+2+α(Ω). Then we obtain the asymptotic expansion∥∥∥∥∥uh

0 −Rhu0 −q∑

i=1

h2ie2i

∥∥∥∥∥L∞(Ω

h)

≤ Chk−1‖u‖C2q+2+α(Ω), (8.228)

with q := max{m ∈ N|2m < k − 1,m ≤ q}. These e2i are independent of h, and

e2i ∈ C2q+2−2i+α(Ω), and e2i|∂Ω = 0, i = 1, . . . , q.

8.8.2 Asymptotic expansions for other methods

Results based upon the earlier sections in this chapter are summarized. Basic isProposition 8.5, where we have proved the asymptotic expansion for

fh(x) :=1hn

∫ch(x)

f(x)dx, |fh(x)− f(x)| ≤ Lh2 as (8.229)∣∣∣∣∣∣fh(x)−

⎛⎝f(x) +2m′,ev∑|α|=2

(h/2)α

(α +�1)!∂αf(x)

⎞⎠∣∣∣∣∣∣ ≤ Ch2m′+1. (8.230)

In Subsection 8.8.1, we have already listed results on asymptotic expansions of thediscretization errors for those special methods. Here we extend, but do not prove,these results to the methods in Sections 8.4–8.6. We obtain asymptotic expansions inpowers of h and h2 for unsymmetric and symmetric difference methods for the Dirichlet


problem on curved and cuboidal domains, respectively. The asymptotic expansions inpowers of h2 are much more interesting for extrapolation methods.

Theorem 8.43. Asymptotic expansion for the difference methods in Sections 8.4–8.6: For unsymmetric and symmetric difference methods on curved and cuboidaldomains, cf. (8.95), with μ = 1 and μ = 2, respectively, we assume the conditionsof Theorems 8.31, 8.39. Furthermore, assume u0 ∈Wμ(r+1),p(Ω), and Lipschitz-continuous G ∈ C

μ(r+1)L with respect to the variables in N∇m . This essentially implies

the additional technical conditions, e.g. the (μr, μ)-smooth asymptotic expansionof the local discretization error, of the AhPhu−Q

′hAu, cf. (8.105), (8.106), andGhPhu−Q

′hGu, cf. (8.113), (8.114), (8.119), cf. Stetter [596] and Bohmer [103].Then for small enough h, unique discrete solutions uh

0 for the linear and nonlinearproblems in Sections 8.4, 8.5 exist, converge, and admit an asymptotic expansion ofthe discretization error. More precisely, there exist eμk ∈Wμ(r+1−k),p(Ω), 1 ≤ k ≤ r,independent of h, such that∥∥∥∥∥Ph

(u0 −

r∑k=1

hμkeμk

)− uh

0

∥∥∥∥∥Vh

≤ Chμr+1‖u0‖W μ(r+1),p(Ω) for u0 ∈Wμ(r+1),p(Ω).

(8.231)

We combine asymptotic expansions with Richardson extrapolation: a few stepsizes, h = h0 > h1 > h2 > h3, often h = h0, h/2, h/4, h/8, are chosen with Ωh ⊂ Ωh1 ⊂Ωh2 ⊂ Ωh3 . The corresponding Ghi

(uhi

0

)= 0, i = 0, 1, . . . , t, defined on Ωhi are com-

puted. Appropriate linear combinations of these uhi require the restriction of the uhi

to Ωh. So define,

for Ωh ⊂ Ωh1 ⊂ Ωh2 . . . : ui : Ωh → Rq : ui(x) := (Rhuhi)(x) := (uhi)(x)∀x ∈ Ωh;

(8.232)next define the Pi,0 := ui, i = 0, 1, . . . , t, and for i = 0, 1, . . . , t− j, j = 1, . . . , t < q,finally Pi,j := Pi+1,j−1 + Pi+1,j−1−Pi,j−1

(hi/hi+j)μ−1 .

Theorem 8.44. High order difference methods by Richardson extrapolation: Let aproblem G(u0) = 0 and its stable, consistent discretization Gh

(uh

0

)= 0 define solu-

tions u0 and uh0 with an asymptotic expansion (8.231), and let h be small enough.

Compute the ui as in (8.232). Then we obtain for u0 ∈Wμ(r+1),p(Ω) and a constantγ := hi/hi−1 > 1 :∥∥Ph

(u0 − hμ

i . . . hμi+j(−1)jeμ(j+1)

)− Pi,j

∥∥Vh

= O(hμ

i hμi . . . h

μi+j

), t < q, (8.233)

for fixed γ :∥∥∥Ph

(u0 − h

μ(j+1)i+j (−1)jγ−j(j−1)/2eμ(j+1)

)− Pi,j

∥∥∥Vh

= O(h

μ(j+1)i+j

).

Defect and deferred correction methods have strong advantages compared toRichardson extrapolation. As indicated we restrict the discussion to defect correc-tions, and here to elliptic problems. The basic version has to solve the nonlinearsystem Gh

(uh

0

)= 0 only once and only for one Ωh. A combination with mesh

refinement strategies is possible by computations on different Ωhi , cf. Bohmer,et al. [11, 12, 101, 102, 104, 107–109, 112, 125, 126]. The basic idea of this method is


a modification of Newton’s method. This has to be combined with the asymptoticexpansions in (8.231). We start with the original discrete problem

define uh0 by Gh

(uh

0

)= fh := Q

′hf, or Ah(uh

0

)= fh. (8.234)

For our linear difference methods, we can update this uh0 iteratively by the defect

correction process. Note that (8.234) is usually a nonlinear system, whereas (8.235) is asequence of linear problems, always with the same linear

(Gh(uh

0

))′. The nonlinearity

is reduced to computing the nonlinear defect Q′hG(Phu

hi

):(

Gh(uh

0

))′ (uh

i+1 − uhi

)= −Q′h

(G(Phu

hi

)− f), i = 1, . . . . (8.235)

Here, it is important that the asymptotic expansion of the uhi is reproduced by Ph, cf.

(8.231), hence with eμk,i, gμk,i, gμk,i independent of h,∥∥∥∥∥Phu0 − uhi −

r∑k=i+1

hμkeμk,i

∥∥∥∥∥Vh

≤ Chμr+1‖u0‖W μ(r+1),p(Ω) with (8.236)

eμk,i ∈Wμ(r+1−k),p(Ω), gμk,i, gμk,i ∈Wμ(r+1−k)−2m,p(Ω), 1 ≤ k ≤ r,

has to imply∥∥∥∥∥Phuhi −(u0 −

r∑k=i+1

hμkeμk,i

)∥∥∥∥∥Vh

≤ Chμr+1‖u0‖W μ(r+1),p(Ω) =⇒

∥∥∥∥∥Q′hG(Phu

hi

)−(Q

′h(G(u0)−

r∑k=i+1

hμkgμk,i

)∥∥∥∥∥V′h

≤ Chμr+1‖u0‖W μ(r+1),p(Ω)

or ∥∥∥∥∥Ghi+1

(uh

i

)−(Q

′h(G(u0)−

r∑k=i+1

hμkgμk,i

)∥∥∥∥∥V′h

≤ Chμr+1‖u0‖W μ(r+1),p(Ω),

where this last Ghi+1

(uh

i

)indicates a difference approximation for Q

′hG(Phu

hi

)of

order μ(i + 1) again with an asymptotic expansion. The Ph can be defined as above,for Gh

i+1

(uh

i

)see e.g. [101,102,104,107].

Theorem 8.45. Defect correction method for Gh: Under the conditions of Theorem8.44, and for our linear difference methods the defect correction method (8.235) forGh in (8.234) yields for small enough h, high order approximations uh

i of the orderμ(i + 1), hence∥∥∥∥∥Phu

hi −(u0 −

r∑k=i+1

hμkeμk,i

)∥∥∥∥∥Vh

≤ Chμr+1‖u0‖W μ(r+1),p(Ω) for i = 0, . . . , r − 1,

and with eμk,i ∈Wμ(r+1−k),p(Ω), 1 ≤ k ≤ r independent of h. (8.237)

8.9. Numerical experiments, v.Karman equations 633

8.9 Numerical experiments for the von Karman equations,with C.S. Chien

The strong form of the von Karman equations is, cf. (2.403)

Gs(u,w) :=

⎛⎝Δ2u −[u,w]

Δ2w + 12 [u, u]

⎞⎠ =

⎛⎝ f1

f2

⎞⎠ (8.238)

in Ω = [0, 1]2 ⊂ R2, with Δ2 = ΔΔ, boundary conditions

u = Δu = 0, w = Δw = 0 on ∂Ω,

and the bracket operator [u, v] := uxxvyy − 2uxyvxy + uyyvxx = [v, u].For demonstrating the power of extrapolation, we choose

f1(x, y) := π4[4 sinπx sinπy − 5 sinπx sin 2πx sin2 πy + 4 cosπx cos 2πx cos2 πy]

and

f2(x, y) := π4[25 sin 2πx sinπy + sin2 πx sin2 πy − cos2 πx cos2 πy].

Then the exact solution is

u(x, y) = sinπx sinπy, w(x, y) = sin 2πx sinπy.

We employ the symmetric difference method for fourth order systems on the squareΩ = [0, 1]2 with Ωh as in (8.95)–(8.97). With step sizes h = 1/8, 1/16, 1/32, 1/64, and1/128, we obtain coefficient matrices of order 98× 98, 450× 450, 1922× 1922, 7938×7938, and 32258× 32258. The nonlinear difference systems corresponding to (8.238)are solved with Newton’s method. In this academic example we have guaranteed allnecessary smoothness conditions.

In Table 8.9 we list the computed ehu := ‖Phu0 − uh

0‖L∞(Ωh), ehw := ‖Phw0 −

wh0‖L∞(Ωh) and determine the squares of the corresponding experimental orders of

convergence, (EOC)2, from the quotients ehu/e

h/2u , eh

w/eh/2w , corresponding to the

expected h2/(h/2)2 = 4, cf. (8.231)In Table 8.10 we compare the errors in the midpoint of the square, in X := (1/2, 1/2)

ehu(X) :=

∣∣(Phu0 − uh0

)(X)∣∣ eh

w(X) :=∣∣(Phw0 − wh

0

)(X)∣∣ . Again, we determine the

Table 8.9: Computational errors and the corresponding experimental orders ofconvergence (EOC)2 of uh, wh approximations in the discrete L∞ norm.

Grid h∥∥eh

u

∥∥L∞(Ωh)

(EOC)2∥∥eh

w

∥∥L∞(Ωh)

(EOC)2

1 1/8 3.4731045E− 02 – 9.6361989E− 02 –2 1/16 8.156673E− 03 4.2600 2.3262711E− 02 4.14233 1/32 2.0079870E− 03 4.0601 5.7658634E− 03 4.03464 1/64 5.0015205E− 04 4.0148 1.4383812E− 03 4.00865 1/128 1.2568853E− 04 3.9793 3.5913250E− 04 4.0051


Table 8.10: Computational errors and the corresponding experimental ordersof convergence, (EOC)2 of the uh(X), wh(X) approximations at the midpointX = (1/2, 1/2).

Grid h∣∣(eh

u

)(X)∣∣ (EOC)2

∣∣(ehw

)(X)∣∣ (EOC)2

1 1/8 3.0417972E− 02 – 8.2382692E− 02 –2 1/16 7.2228714E− 03 4.2113 1.9354054E− 02 4.25663 1/32 1.7835678E− 03 4.0497 4.7722116E− 04 4.05564 1/64 4.4453207E− 04 4.0122 1.1890687E− 04 4.01345 1/128 1.1192101E− 04 3.9718 2.9399067E− 05 4.04455

Table 8.11: Computational errors and the corresponding experimental orders ofconvergence, (EOC)2 of the uh(X) approximations at the midpoint X.

Grid h uh(X) uh(X) |uh(X)− uh(X)| (EOC)2

1 1/8 1.0304179722 0.999491170 5.0883E− 04 –2 1/16 1.0072228714 0.999970465 2.9535E− 05 17.22803 1/32 1.0017835678 0.999998187 1.8130E− 06 16.29064 1/64 1.0004445321 1.000001051 1.0510E− 06 1.72505 1/128 1.0001119210

squares, (EOC)2, corresponding to the expected h2/(h/2)2 = 4, now for theeX,h

u /eX,h/2u , and eX,h

w /eX,h/2w .

In Table 8.11, we compare the errors in the midpoint X, but now for the first extrap-olated approximation. Instead of the Pi,j in (8.232), (8.233), with γ = 2, we use thenotation uh(X) := (4uh/2(X)− uh(X))/3. We list the uh(X), uh(X), |u(X)− uh(X)|and determine the squares, (EOC)2, corresponding to the expected h4/(h/2)4 = 16from the |u(X)− uh(X)|/|u(X)− uh/2(X)|. Similar results hold for wh(X), wh(X),|w(X)− wh(X)|.

9

Variational methods for wavelets,with S. Dahlke

9.1 Introduction

In the previous variational approaches we have used FEs, DCGEs, and grid functionsin this book, and will use radial and spectral functions in Bohmer [120]. In this chapter,the spaces of ansatz and test functions are defined by wavelets. Despite the unifiedconvergence theory, cf. Chapter 3, Petryshyn [528, 532], Stetter [596], Stummel [607–609] Vainikko [644] and Zeidler [676,677], these different methods require us to carefullydistinguish their different approximation characteristics.

Until now, only conforming wavelet Galerkin methods have been applied. We presentthem here for general elliptic, saddle point, and Navier–Stokes equations. This extendssome recent results for wavelet methods. Again, we employ the basic concepts of invert-ibility, compact perturbation, approximation and general discretization theory. Firstresults, for nonsymmetric, noncoercive, boundedly invertible linear operators havebeen derived by Bohmer and Dahlke [122]. Here nonlinear problems and bifurcationare indicated, however not explicitely formulated. Bohmer and Dahlke [122] constitutethe initial form of this chapter. Our standard result for elliptic problems specified forconforming wavelet Galerkin schemes is applied. Let B be a boundedly invertibleoperator with stable discretization Bh and A = B + C a compact perturbation ofB, such that A is boundedly invertible as well. Then Ah is also stable. In fact,the additional condition of consistent Ah, Bh, Ch is always satisfied for conformingmethods. In Chapter 3, cf. [122], we have additionally shown that, under appropriatetechnical conditions, the stability for the nonlinear problem is guaranteed if it is correctfor the linearized operator, cf. Stetter [596] and Bohmer [116], for variational methods.This applies to wavelet methods, if the exact evaluation of all nonlinear terms in theoperator is possible. For wavelet methods this is still a problem for future research,in particular since in the wavelet setting, exact evalutions of nonlinear operators arehard to achieve, see e.g. [69, 251] for details. Therefore we restrict the discussion tononlinear problems, e.g. in (9.13), where the nonlinear terms can be evaluated forwavelet arguments, see Subsection 9.3.5.

The linearized Navier–Stokes equations can be interpreted as compact perturbationsof the Stokes equations, at least for moderate Reynolds numbers. In this sense, weobtain convergence results for wavelet applied to the Navier–Stokes equations on thesame level as the finite element methods. These problems have been carefully studiedby a different approach in, e.g. Girault and Raviart [348] and Temam [622].

636 9. Variational methods for wavelets

For our conforming wavelet methods, convergence is, for linear coercive problems, animmediate consequence of Cea’s Lemma 4.48. If the solution u0 is well approximated,here infvn∈Sn

‖u0 − vn‖L2(Ω)→ 0, with the stability implied by the coercivity, we obtain“optimal” convergence.

During the last few years, wavelets have shown some potential for the numericaltreatment of partial differential and integral equations, c.f. e.g. Barinka et al. [70],Cohen et al. [197,198], Dahlke et al. [236,237,241] and Dahmen [243].

The advantages of wavelet methods can be described as follows. It turns out that asimple diagonal scaling applied to stiffness matrices relative to wavelet bases suffices toproduce uniformly bounded condition numbers. Moreover, for a wide class of integralor pseudo-differential operators the stiffness matrix relative to wavelet bases can beshown to be sufficiently close to sparse matrices so that efficient sparse solvers canbe applied and yield powerful stable and convergent Galerkin schemes. The mostfar-reaching results were obtained for linear self-adjoint and saddle point problems.For these problems, it has even been possible to derive optimal convergent adaptivewavelet schemes, see Cohen et al. [197] and Dahlke et al. [236,237]. Numerical tests forself-adjoint and saddle point problems, e.g. the Stokes problem, with wavelet methodsare documented in Barinka et al. [70] and Dahlke et al. [237].

Adaptive wavelet techniques require highly technical proofs, but do have a lotof advantages whenever they are needed. They can be applied only to boundedlyinvertible (linearized) operators A. This is the general assumption in the citedpapers: Barinka et al. [70], Cohen et al. [197, 198], Dahlke et al. [236] and Dahmen[243]. For a nonsymmetric, boundedly invertible A, for these techniques the useof the symmetric coercive A∗A is unavoidable, but expensive, cf. e.g. Sections 3and 7 in Cohen et al. [198] for further details. Here A∗ denotes the adjoint ofA. This excludes bifurcation problems. There is another difficulty related to A∗A:It is well-known that cond (A∗A) = (cond(A))2 explodes for increasing cond(A).First results for nonsymmetric, boundedly invertible problems that avoid thesedifficulties and which are based on perturbation arguments have been derived byGantumur [332].

The adaptive treatment of problems where the underlying operator is not boundedlyinvertible, i.e. of ill-posed problems, is a very challenging task that has recently beenintensively studied. We refer to Bonesky et al. [131,132] for further details. First resultsfor nonlinear equations have been developed by Cohen et al. [199].

This chapter is organized as follows. In Section 9.2, we briefly discuss the scopeof problems we shall be concerned with. Especially, we introduce the setting ofelliptic problems. In Section 9.3 we present the basic concepts of wavelet analysis,i.e. the discrete wavelet transform, biorthogonal bases, approximations of functionsby wavelets, construction of wavelets on domains and manifolds, and the evaluationof nonlinear functionals applied to wavelet expansions. Section 9.4 discusses thefirst steps, definitions and basic tools, e.g. stability and preconditioning for waveletmethods. Section 9.5 is devoted to wavelet methods for general linear and nonlinearelliptic problems. We explain how suitable Galerkin schemes based on wavelets canbe constructed and discuss their stability and consistency properties. Then, in Section9.6, we apply these results via its linearization to the Navier–Stokes equations. Finally,

9.2. The scope of problems 637

in Section 9.7, we explain how adaptive numerical methods based upon wavelets canbe designed.

The following problems, only presented for FEMs, can be solved and yield thecorresponding results for wavelet methods as well: general convergence theory formonotone operators, cf. Section 4.5, variational methods for eigenvalue problems, cf.Section 4.7, methods for nonlinear boundary operators which can be evaluated forwavelet functions, cf. Section 5.3, quadrature approximate methods, cf. Section 5.4.

All this is limited to problems where the nonlinearities can be evaluated for wavelets,cf. Theorem 9.13. For all the nonlinear problems, considered here, the correspondingnonlinear equations can be solved with Newton’s method, satisfying the mesh inde-pendence principle. Methods for fully nonlinear elliptic problems are impossible withour technique.

In this chapter we use the notation Rd,Cd, . . . , instead of the previous Rn,Cn, . . . ,and restrict the presentation to a Hilbert space setting.

9.2 The scope of problems

In this section, we shall briefly explain the scope of problems we shall be concernedwith. The goal is to derive a stable and convergent wavelet scheme for problems with alinearization, that can be interpreted as boundedly invertible compact perturbationsof an operator equation with stable discretization. Especially, we are interested inproblems related to well-known elliptic and saddle point problems. The Stokes operatorand, for moderate Reynolds numbers, its compact perturbation, the (linearized)Navier–Stokes operator, fall into this category. Therefore we first recall the generalsetting of operator equations.

Suppose that U is a Hilbert space with norm ‖ · ‖U induced by the inner product〈·, ·〉. Let A : U −→ U ′ denote a linear operator into the normed dual U ′ of U . Incontrast to the other methods, we assume throughout that in this chapter U andthe approximating subspaces, SΛ := {ψλ : λ ∈ Λ} ⊂ U , include the appropriateboundary conditions. We shall mainly discuss the case that A can be written as

A = B + C, (9.1)

with a boundedly invertible operator, B ∈ L(U ,U ′), with stable discretization, and acompact C ∈ C(U ,U ′). We start with a short presentation of general elliptic equations.It is well known that, given a Hilbert space U , its dual U ′ and the dual pairing 〈·, ·〉U ′×U ,a general elliptic operator A : U → U ′ induces a continuous and elliptic bilinear form,see Sections 2.3 and 2.4,

a : U × U → R, a(u, v) := 〈Au, v〉U ′×U ∀ u, v ∈ U and (9.2)

a(v, v) ≥ α‖v‖2U − γ‖v‖2WU , α > 0, for Hilbert spaces U ↪→WU =W ′U ↪→ U ′,

usually a Gelfand triple U ↪→WU =W ′U ↪→ U ′ with dense and continuous embedding

U ↪→WU . Furthermore, in A = B + C we usually assume B, inducing a coercive


bilinear form b(·, ·), i.e.

〈Bv, v〉U ′×U = b(v, v) ≥ α‖v‖2U , α > 0, for all v ∈ U . (9.3)

It is known that, for an invertible A, in particular for a B in (9.3), the equation

a(u0, v) = 〈f, v〉U ′×U for all v ∈ U (9.4)

is uniquely solvable.Consequently, typical examples for A and B are given by general elliptic partial

differential operators and their invertible special cases satisfying (9.3). For example,the operator induced by the Poisson equation49

−'u = f, in Ω ⊂ Rd (9.5)

u = 0, on ∂Ω,

would play the role of B. This B = −' is a bounded and boundedly invertiblemapping of H1

0 (Ω) onto its dual H−10 (Ω), i.e. ‖Bu‖U ′ ∼ ‖u‖U . Here “a ∼ b” means

that both quantities can be uniformly bounded by some constant multiple of eachother. Likewise, “<∼” indicates inequality up to constant factors.

We generalize the Laplacian to elliptic operators of order 2m. The generalization tosystems follows exactly Section 4.3.2, cf. (9.13).

For the reader’s convenience we repeat the relevant notation and equations: for themulti-indices α, we defined, see (2.73), partials, vectors and reals as

∂iu =∂u

∂xi,∇u = (∂1u, . . . , ∂du) and ∂αu = ∂α1

1 . . . ∂αd

d u =∂|α|

∂xα11 . . . ∂xd

αdu,

ϑ = (ϑ1, . . . , ϑd) ∈ Rd, ϑα = (ϑ1)α1 · · · (ϑd)αd , ϑi, ∂iu(x), ∂αu(x) ∈ R.

A linear second order elliptic equation, see (2.160), determines the solution u0, suchthat

u0 ∈ U : a(u0, v) = 〈Au0, v〉U ′×U =∫

Ω

d∑i,j=0

aij∂iu0∂

jvdx = 〈f, v〉U ′×U

∀ v ∈ U = H10 (Ω), with aij ∈ L∞(Ω). (9.6)

The linear form f and our elliptic bilinear form a(u, v) are bounded, so

|〈f, v〉U ′×U | ≤ C ′‖v‖U , |a(u, v)| ≤ C‖u‖U‖v‖U . (9.7)

The principal part, ap(u, v), satisfies for all ϑ ∈ Rd

λ|ϑ|2d ≤ λ(x)|ϑ|2d ≤d∑

i,j=1

aij(x)ϑiϑj ≤ Λ|ϑ|2d ∀ x ∈ Ω with λ > 0. (9.8)

49 Note that we use in this chapter Ω ⊂ Rd instead of Ω ⊂ Rn in the previous chapters.

9.3. Wavelet analysis 639

By (2.22), (2.79), ap(u, v) is bounded, and U = H10 (Ω)-coercive, hence there exists

λ∗ > 0, such that

ap(u, u) =∫

Ω

d∑i,j=1

aij∂iu∂judx ≥ λ∗‖u‖2U ,∀u ∈ U , i.e. |ap(u, u)| ∼ ‖u‖2U . (9.9)

Elliptic equations of order 2m require U = Hm0 (Ω). For linear A, determine

u0 ∈ U = Hm0 (Ω) : a(u0, v) :=

∫Ω

∑|α|,|β|≤m

aαβ∂αu0∂

βvdx = 〈f, v〉U ′×U (9.10)

∀v ∈ U with aαβ ∈ L∞(Ω), and for 1 < m = |α| = |β| : aαβ ∈ C(Ω).

The previous a(u, v) is bounded and has to be elliptic, hence the principal part,ap(u, v), is U-coercive, satisfying with λ, λ∗ > 0 and ∀ϑ ∈ Rd, see (2.79), (4.132),

λ|ϑ|2md ≤ λ(x)|ϑ|2m

d ≤ (−1)m∑

|α|=|β|=m

aαβ(x)ϑαϑβ ≤ Λ|ϑ|2md ∀ x ∈ Ω, (9.11)

∀u ∈ U : ap(u, u) =∫

Ω

∑|α|=|β|=m

aαβ∂αu0∂

βudx ≥ λ∗‖u‖2U , i.e. |ap(u, u)| ∼ ‖u‖2U .

Quasilinear elliptic systems of order 2m extend and modify the above equations andconditions, cf. Subsection 2.6.3 ff. In the introduction we have already mentioned thedifficulties in evaluating nonlinear functionals with wavelet arguments, cf. Subsection9.3.5. Therfore, we impose the following condition: we admit only coefficients suchthat

Aα in (9.13), Aα(x, �u, . . . ,∇m�u), can be evaluated for wavelet arguments. (9.12)

Under this condition we consider for U = Hm0 (Ω,Rq), q ≥ 1, the weak form, and

determine �u0 ∈ D(G) ⊂ U�u0 ∈ D(G) s.t.

a(�u0, �v) := 〈G�u0, �v〉U ′×U :=∫Ω

∑|α|≤m(Aα(x, �u0, . . . ,∇m�u0), ∂α�v)qdx

= 〈�f,�v〉U ′×U =∑

|α|≤m(�fα∂α�v)qdx ∀ �v ∈ U = Hm

0 (Ω,Rq),

�fα ∈ L2(Ω,Rq), q ≥ 1, (9.13)

cf. Proposition 2.34. The appropriate conditions are listed in Sections 2.6.4, 2.6.6. Forthe linearized principle part, (9.11) has to be replaced by (2.394).

Applications to saddle point and Navier–Stokes equations will be discussed inSection 9.6.

9.3 Wavelet analysis

Our goal is conforming Galerkin methods for the approximate solution of Au = f orGu = f for operators A,G as in Section 9.2. However, in contrast to conventional


finite element discretizations we will work with trial spaces that not only exhibitthe usual approximation properties and good localization but in addition lead toexpansions of any element in the underlying Hilbert spaces in terms of multiscaleor wavelet bases with certain stability properties. This fact can be used to derive,e.g. very efficient preconditioning strategies as well as powerful adaptive algorithms,see Section 9.4, Theorem 9.6, and Section 9.7. To understand these relationships, weshall first of all briefly recall the basic setting of wavelet analysis as far as it is neededfor our purposes. In Subsection 9.3.1, we want to collect some facts concerning thediscrete wavelet transform. Then, in Subsection 9.3.2, we discuss the biorthogonalwavelet approach. It is one of the most important properties of wavelets that theyform stable bases for scales of smoothness spaces such as Sobolev and Besov spaces.These aspects will be discussed in Section 9.3.3. Then, in Subsection 9.3.4, we brieflyrecall the construction of wavelet bases on bounded domains. Finally, in Subsection9.3.5, we discuss how nonlinear functionals applied to wavelet expansions can beevaluated.

9.3.1 The discrete wavelet transform

In general, a function ψ is called a wavelet if all its scaled, dilated, and integer-translated versions

ψj,k(x) := 2j/2ψ(2jx− k), j, k ∈ Z, (9.14)

form a (Riesz) basis of L2(R). Usually, they are constructed by means of a multireso-lution analysis introduced by Mallat [482]:

Definition 9.1. A sequence {Vj}j∈Z of closed subspaces of L2(R) is called a multires-olution analysis (MRA) of L2(R) if

· · · ⊂ Vj−1 ⊂ Vj ⊂ Vj+1 ⊂ . . . ; (9.15)

∞⋃j=−∞

Vj = L2(R); (9.16)

∞⋂j=−∞

Vj = {0}; (9.17)

f(·) ∈ Vj ⇐⇒ f(2·) ∈ Vj+1; (9.18)

f(·) ∈ V0 ⇐⇒ f(· − k) ∈ V0 for all k ∈ Z. (9.19)

Moreover, we assume that there exists a function ϕ ∈ V0 such that

V0 := span{ϕ(· − k), k ∈ Z} (9.20)


and that ϕ has stable integer translates, i.e., with

�2(Z) :=

⎧⎨⎩c := {ck}k∈Z : ‖c‖�2 :=

(∑k∈Z

c2k

)1/2

<∞

⎫⎬⎭ ,

C1||c||�2 ≤ ||∑k∈Z

ck ϕ(· − k)||L2 ≤ C2||c||�2 , c := {ck}k∈Z ∈ �2(Z). (9.21)

The function ϕ is called the generator of the multiresolution analysis.

The properties (9.15), (9.18), (9.20), and (9.21) immediately imply that ϕ isrefinable, i.e. it satisfies a two-scale relation

ϕ(x) =∑k∈Z

akϕ(2x− k), (9.22)

with the mask a = {ak}k∈Z ∈ �2(Z). A function satisfying a relation of the form (9.22)is sometimes also called a scaling function. Because the union of the spaces {Vj}j∈Z isdense, and their intersection is {0}, it is easy to see that the construction of a waveletbasis reduces to finding a function whose translates span a complement space W0 ofV0 in V1,

V1 = V0 ⊕W0, W0 = span{ψ(· − k) | k ∈ Z}. (9.23)

Indeed, if we define

Wj := {f(·) ∈ L2(R) | f(2−j ·) ∈ W0}, (9.24)

it follows from (9.16), (9.17) and (9.18) that

L2(R) =∞⊕

j=−∞Wj , (9.25)

so that

ψj,k(·) = 2j/2ψ(2j · −k), j, k ∈ Z (9.26)

forms a wavelet basis of L2(R).Obviously, (9.18) and (9.20) imply that the wavelet ψ can be found by means of a

functional equation of the form

ψ(x) =∑k∈Z

bkϕ(2x− k), (9.27)

where the sequence b := {bk}k∈Z has to be judiciously chosen; see, e.g. [173,254,490]for details.

The construction outlined above is quite general. In many applications, it is conve-nient to impose some more conditions, i.e. to require that functions on different scalesare orthogonal with respect to the usual L2 inner product, i.e.

〈ψ(2j• −k), ψ(2j′• −k′)〉 = 0, if j �= j′. (9.28)


This can be achieved if the translates of ψ not only span an (algebraic) complementbut the orthogonal complement,

V0 ⊥ W0, W0 = span{ψ(• − k) | k ∈ Z}. (9.29)

The resulting functions are sometimes called pre-wavelets. The basic properties ofscaling functions and (pre-) wavelets can be summarized as follows:

� Reproduction of polynomials. If ϕ is contained in Cr0(R) := {g | g ∈

Cr(R) and supp g compact}, then every monomial xα has an expansion ofthe form

xα =∑k∈Z

cαkϕ(x− k), α ≤ r. (9.30)

� Oscillations. If the generator ϕ is contained in Cr0(R), then the associated wavelet

ψ has vanishing moments up to order r, i.e.∫R

xαψ(x)dx = 0 for all 0 ≤ α ≤ r. (9.31)

� Approximation. If ϕ ∈ Cr0(R) and f ∈ Hr(R), then the following Jackson-type

inequality holds, cf. Lemma 9.7:

infg∈Vj

‖f − g‖L2(R) ≤ C2−jr|f |Hr . (9.32)

In practice, it is clearly desirable to work with an orthonormal wavelet basis. This canbe realized as follows. Given an �2-stable generator in the sense of (9.21), one maydefine another generator φ by

φ(ξ) :=ϕ(ξ)

(∑

k∈Z |ϕ(ξ + 2πk)|2)1/2. (9.33)

It can be checked directly that the translates of φ are orthonormal and span the samespace V0. The generator φ is also refinable,

φ(x) =∑k∈Z

akφ(2x− k) (9.34)

and it can be shown that the function

ψ(x) =∑k∈Z

bkφ(2x− k), bk := (−1)ka1−k (9.35)

is an orthonormal wavelet with the same regularity properties as the original generatorϕ. However, this approach has a serious disadvantage. If the generator ϕ is compactlysupported, this property will in general not carry over to the resulting wavelet since itgets lost during the orthonormalization procedure (9.33). Therefore the compact sup-port will only be preserved if we can dispense with the orthonormalization procedure,i.e. if the translates of ϕ are already orthonormal. This observation was the startingpoint for the investigations of I. Daubechies who constructed a family φN , N ∈ N, ofgenerators with the following properties [253,254].


Theorem 9.2. Good wavelet generators: There exists a constant β > 0 and a familyφN of generators satisfying φN ∈ CβN (R), supp φN = [0, 2N − 1], and

〈φN (·), φN (· − k)〉 = δ0,k, φN (x) =N2∑

k=N1

akφN (2x− k). (9.36)

Obviously, (9.36) and (9.35) imply that the associated wavelet ψN is also compactlysupported with the same regularity properties as φN .

Given such an orthonormal wavelet basis, any function v ∈ Vj has two equiv-alent representations, the single-scale representation with respect to the functionsφj,k(x) = 2j/2φ(2jx− k) and the multiscale representation which is based on thefunctions φ0,k, ψl,m, k,m ∈ Z, 0 ≤ l < j, ψl,m := 2j/2ψ(2jx−m). From the coefficientsof v in the single-scale representation, the coefficients in the multiscale representationcan easily be obtained by some kind of filtering, and vice versa. Indeed, given

v =∑k∈Z

λjkφj,k

and using the refinement equation (9.34) and the functional equation (9.35), it turnsout that

v =∑l∈Z

2−1/2

(∑k∈Z

ak−2lλjk

)φj−1,l +

∑m∈Z

2−1/2

(∑k∈Z

bk−2lλjk

)ψj−1,m. (9.37)

From (9.37) we observe that the coefficient sequence λj−1 = {λj−1k }k∈Z which describes

the information corresponding to Vj−1 can be obtained by applying the low-pass filterH induced by a to λj ,

λj−1 = Hλj , λj−1l =

∑k∈Z

2−1/2ak−2lλjk. (9.38)

The wavelet spaceWj−1 describes the detailed information added to Vj−1. From (9.37),we can conclude that this information can be obtained by applying the high-pass filterD induced by b to λj :

cj−1 = Dλj , cj−1l = 2−1/2

∑k∈Z

bk−2lλjk. (9.39)

By iterating this decomposition method, we obtain a pyramid algorithm, the so-calledfast wavelet transform:

λ jH

D

H

D

l j − 2

c j − 2 c j − 3c j − 1

l j − 1 l j − 3H

D


A reconstruction algorithm can be obtained in a similar fashion. Indeed, a straight-forward computation shows∑

l∈Z

λjlφj,l =

1√2

∑l∈Z

(∑k∈Z

al−2kλj−1k +

∑n∈Z

bl−2ncj−1n

)φj,l

so that

λjl = 2−1/2

∑k∈Z

al−2kλj−1k + 2−1/2

∑n∈Z

bl−2ncj−1n . (9.40)

Similar decomposition and reconstruction schemes also exist for the pre-wavelet case.There are several methods to generalize this concept to higher dimensions. The

simplest way is to use tensor products. Given a univariate multiresolution analysiswith generator ϕ, it turns out that

φ(x1, . . . , xd) := ϕ(x1) · · ·ϕ(xd) (9.41)

generates a multiresolution analysis of L2(Rd). Let E denote the vertices of the unitcube in Rd. Defining ψ0 := ϕ and ψ1 := ψ, it can be shown that the set Ψ of 2d − 1functions

ψe(x1, . . . , xd) :=d∏

l=1

ψel(xl) e = (e1, . . . , ed) ∈ E\{0}, (9.42)

generates by shifts and dilates a basis of L2(Rd). There also exist multivariate waveletconstructions with respect to nonseparable refinable functions φ satisfying

φ(x) =∑k∈Zd

akφ(2x− k), {ak}k∈Zd ∈ �2(Zd), (9.43)

see, e.g. [420] for details. Analogously to the tensor product case, a family ψi, i =1, . . . , 2d − 1, of wavelets is needed. Each ψi satisfies a functional equation similar to(9.27):

ψi(x) =∑k∈Zd

bikφ(2x− k). (9.44)

The basic properties of wavelets and scaling functions (approximation, oscillation,etc.) carry over to the multivariate case in the usual way.

9.3.2 Biorthogonal bases

Given an orthonormal wavelet basis, the basic calculations are usually quite simple.For instance, the wavelet expansion of a function v ∈ L2(R) can be computed as

v =∑

j,k∈Z

〈v, ψj,k〉ψj,k, ψj,k(x) := 2j/2ψ(2jx− k). (9.45)

However, requiring smoothness and orthonormality is quite restrictive, and con-sequently, as we have already seen above, the resulting wavelets are usually not


compactly supported. It is one of the advantages of the pre-wavelet setting thatthe compact support property of the generator can be preserved. Moreover, sincewe have to deal with weaker conditions, the pre-wavelet approach provides us withmuch more flexibility. Therefore, given a generator ϕ, many different families ofpre-wavelets adapted to a specific application can be constructed. Nevertheless,since orthonormality is lost, one is still interested in finding suitable alternativeswhich in some sense provide a compromise between both concepts. This can beperformed by using the biorthogonal approach. For a given (univariate) waveletbasis {ψj,k, j, k ∈ Z}, ψj,k(x) := 2j/2ψ(2jx− k), one is interested in finding a secondsystem {ψj,k, j, k ∈ Z} satisfying

〈ψj,k(·), ψj′,k′(·)〉 = δj,j′δk,k′ , j, j′, k, k′ ∈ Z. (9.46)

Then all the computations are as simple as in the orthonormal case, i.e.

v =∑

j,k∈Z

〈v, ψj,k〉ψj,k =∑

j′,k′∈Z

〈v, ψj′,k′〉ψj′,k′ . (9.47)

To construct such a biorthogonal system, one needs two sequences of approximationspaces {Vj}j∈Z and {Vj}j∈Z. As for the orthonormal case, one has to find basesfor certain algebraic complement spaces W0 and W0 satisfying the biorthogonalityconditions

V0 ⊥ W0, V0 ⊥ W0, V0 ⊕W0 = V1, V0 ⊕ W0 = V1. (9.48)

This is quite easy if the two generators φ and φ form a dual pair,

〈φ(·), φ(· − k)〉 = δ0,k. (9.49)

Indeed, then two biorthogonal wavelets ψ and ψ can be constructed as

ψ(x) =∑k∈Z

(−1)kd1−kφ(2x− k), ψ(x) =∑k∈Z

(−1)ka1−kφ(2x− k) (9.50)

where

φ(x) =∑k∈Z

akφ(2x− k), φ(x) =∑k∈Z

dkφ(2x− k). (9.51)

Therefore, given a primal generator φ, one has to find a smooth and compactlysupported dual generator φ satisfying (9.49) which is much less restrictive thanthe orthonormal setting. Elegant constructions can be found, e.g. in [202]. There,especially, the important case where the primal generator is a cardinal B-spline,φ = Nm, m ≥ 1, is discussed in detail. Generalizations to higher dimensions alsoexist [201].

For further information on wavelet analysis, the reader is referred to one of theexcellent textbooks on wavelets which have appeared quite recently [173,254,433,490,669].


9.3.3 Wavelets and function spaces

It is one of the most important properties of wavelets that they give rise to stable basesin scales of function spaces. For wavelet applications, the Besov spaces are the mostimportant smoothness spaces. Therefore in this section we introduce the Besov spacesand recall their characterization by means of wavelet expansions. For simplicity, werestrict ourselves to the biorthogonal setting and the univariate case. Similar resultsalso hold for the multivariate and the pre-wavelet setting. We refer to [329] and [490]for details.

Let us start by recalling the definition of Besov spaces. The modulus of smoothnessωr(v, t)Lp(R) of a function v ∈ Lp(R), 0 < p ≤ ∞, is defined by

ωr(v, t)Lp(R) := sup|h|≤t

‖Δrh(v, ·)‖Lp(R) , t > 0,

with Δrh the r-th difference with step h. For s > 0 and 0 < q, p ≤ ∞, the Besov space

Bsq(Lp(R)) is defined as the space of all functions v for which

|v|Bsq(Lp(R)) :=

{(∫∞0

[t−sωr(v, t)Lp(R)]qdt/t)1/q

, 0 < q <∞,

supt≥0 t−sωr(v, t)Lp(R), q =∞ ,

(9.52)

is finite with r := [s] + 1. Then, (9.52) is a (quasi-)seminorm for Bsq(Lp(R)). If we add

‖v‖Lp(R) to (9.52), we obtain a (quasi-)norm for Bsq(Lp(R)). Let P0 be the biorthogonal

projector which maps L2(R) onto V0, i.e.

P0(v) :=∑k∈Z

〈v, φ(· − k)〉φ.

Then, P0 has an extension as a projector to Lp(R), 1 ≤ p ≤ ∞. For each v ∈ Lp(R),we have

v = P0(v) +∑

j≥0,k∈Z

〈v, ψj,k〉ψj,k. (9.53)

The Besov spaces Bsq(Lp(R)) can be characterized by wavelet coefficients provided

the parameters s, p, q satisfy certain restrictions. For simplicity, we shall only discussthe case q = p. Then, the following characterization holds:

Proposition 9.3. Let φ and φ be in Cr(R). If 0 < p ≤ ∞ and r > s > (1/p− 1),then a function v is in the Besov space Bs

p(Lp(R)), if and only if,

v = P0(v) +∑

j≥0,k∈Z

〈v, ψj,k〉ψj,k (9.54)

with

‖P0(v)‖Lp(R) +

⎛⎝ ∑j≥0,k∈Z

2j(s+( 12− 1

p ))p|〈v, ψj,k〉|p⎞⎠1/p

<∞ (9.55)

and (9.55) provides an equivalent (quasi-)norm for Bsp(Lp(R)).


In the case p ≥ 1, this is a standard result and can be found for example in Meyer[490] (Section 10 of Chapter 6). For the general case of p, this can be deduced fromgeneral results in Littlewood–Paley theory (see e.g. Section 4 of Frazier and Jawerth[329]) or proved directly. We also refer to [636] and [195] for further information. Thecondition that s > (1/p− 1) implies that the Besov space Bs

p(Lp(R)) is embedded inLp(R) for some p > 1 so that the wavelet decomposition of v is defined. Also, withthis restriction on s, the Besov space Bs

p(Lp(R)) is equivalent to the nonhomogeneousBesov spaces Bs

p,p defined via Fourier transforms and Littlewood–Paley theory.Another important scale of function spaces we shall be concerned with are the

Sobolev spaces Hs(R). Usually, these spaces are defined by means of weak derivativesor by Fourier transforms, see, e.g. [1] for details. However, it can be shown that forp = q = 2, Sobolev and Besov spaces coincide:

Hs(R) = Bs2(L2(R)), s > 0.

Therefore Proposition 9.3 immediately provides us with a characterization of Sobolevspaces.

Proposition 9.4. Let φ and ψ be in Cr(R), r > s. Then a function v is in the Sobolevspace Hs(R), if and only if,

‖P0(v)‖Lp(R) +

⎛⎝ ∑j≥0,k∈Z

22js|〈v, ψj,k〉|2⎞⎠1/2

<∞ (9.56)

and (9.56) provides an equivalent (quasi-)norm for Hs(R).

9.3.4 Wavelets on domains

Our aim is to employ wavelet bases for the design of efficient numerical schemes.However, usually the operator under consideration is defined on a bounded domainΩ ⊂ Rd, and therefore we are faced with the problem of designing a wavelet basisadapted to this domain. We cannot simply cut all elements of a wavelet basis on thewhole Euclidean plane, for this would destroy almost all of the important propertiesdescribed above. Nevertheless, it is by now well known how to construct suitablebiorthogonal wavelet bases for many cases of interest, see [159,160,204,250]. The firststep is always to construct nested sequences S = {Sj}j≥j0 , S = {Sj}j≥j0 whose unionsare dense in L2(Ω). Then, one has to find suitable bases

Ψj = {ψj,k : k ∈ ∇j}, Ψj = {ψj,k : k ∈ ∇j},

for some complements Wj of Sj in Sj+1 and Wj of Sj in Sj+1, such that thebiorthogonality condition

〈ψj,k, ψj′,k′〉 = δj,j′δk,k′ (9.57)

holds. Moreover, one has to ensure that all the convenient properties of wavelets,especially the characterization of Sobolev spaces according to Propositon 9.4, can still


be saved. What turns out to matter is that

Ψ =⋃

j≥j0

Ψj

forms a Riesz basis of L2(Ω), i.e. every v ∈ L2(Ω) has a unique expansion

v =∞∑

j=j0

∑k∈Jj

〈v, ψj,k〉ψj,k (9.58)

such that

‖v‖L2(Ω) ∼

⎛⎝ ∞∑j=j0

∑k∈∇j

|〈v, ψj,k〉|2⎞⎠ 1

2

, v ∈ L2(Ω), (9.59)

and that both S and S should have some approximation and regularity propertieswhich can be stated in terms of the following pair of estimates. There exists someρ > 0 such that the inverse estimate

‖vn‖Hs(Ω)<∼2ns‖vn‖L2(Ω), vn ∈ Sn, (9.60)

holds for s < ρ. Moreover, there exists some m ≥ ρ such that the direct estimate

infvn∈Sn

‖v − vn‖L2(Ω)<∼2−sn‖v‖Hs(Ω), v ∈ Hs(Ω), (9.61)

holds for s ≤ m, compare with (9.32). Such estimates are known to hold for everyfinite element or spline space. For instance, for piecewise linear finite elements one hasρ = 3/2,m = 2.

It will be convenient to introduce the following notation. Let

J := {λ = (j, k) : k ∈ ∇j , j ≥ j0} =∞⋃

j=j0

({j} × ∇j),

and define

|λ| := j if λ = (j, k) ∈ J, that is, k ∈ ∇j .

Then the following result holds, [242].

Theorem 9.5. Equivalent norms: Suppose that Ψ = {ψλ : λ ∈ J} and Ψ = {ψλ : λ ∈J} are biorthogonal collections in L2(Ω) satisfying (9.59). If both S and S satisfy(9.60) and (9.61) relative to some ρ, ρ′ > 0, ρ ≤ m, ρ′ ≤ m′, then

‖v‖Hs(Ω) ∼(∑

λ∈J

22|λ|s|〈v, ψλ〉|2) 1

2

, s ∈ (−ρ′, ρ), (9.62)

∼(∑

λ∈J

22|λ|s|〈v, ψλ〉|2) 1

2

, s ∈ (−ρ, ρ′), v ∈ Hs(Ω).


It will also be convenient to introduce the diagonal matrix

(Ds)λ,λ′ := 2s|λ|δλ,λ′ . (9.63)

By using (9.63), the norm equivalence (9.62) can be written as

‖v‖Hs ∼ ‖Ds〈v, Ψ〉T ‖�2(J), s ∈ (−ρ′, ρ). (9.64)

All the construction in [159, 160, 204, 250] induce isomorphisms of the form (9.62) atleast for −1/2 < s < 3/2. For special domains such as rectangular domain even muchsharper results are available.

To explain the basic idea, let us briefly recall the construction from [250]. It consistsof three steps:

(a) Construct dual pairs of generator bases

Φj = {φj,k, : k ∈ 'j}, Φj = {φj,k : k ∈ 'j},i.e.

〈Φj , Φj〉 :=(〈φj,k, φj,l〉

)k,l∈�j

= I, (9.65)

whose elements have local support,

diam(suppφj,k) ∼ 2−j , diam(suppφj,k) ∼ 2−j , (9.66)

and such that their spans

Sj = S(Φj) := span Φj , Sj = S(Φj) := span Φj ,

are nested,

S(Φj) ⊂ S(Φj+1), S(Φj) ⊂ S(Φj+1). (9.67)

S(Φj), S(Φj) are referred to as the primal and dual multiresolution spaces.(b) Find a stable basis Ψj of some complement S(Ψj) of S(Φj) in S(Φj+1), i.e.

S(Φj+1) = S(Ψj)⊕ S(Φj), j = j0, . . . . (9.68)

(c) Given such an initial decomposition project the initial basis Ψj into a basis Ψj

which is perpendicular to S(Φj). The union Ψ := Φj0 ∪ ∪∞j=j0

Ψj will be the finalprimal wavelet basis satisfying our requirements.

Note that (9.67) states that each coarse scaling function can be written as alinear combination of fine scale basis functions. Viewing the bases as vectors whosecomponents are the individual scaling functions, (9.67) is equivalent to saying thatthere must be #'j+1 ×#'j matrices Mj,0, Mj,0 such that

ΦTj = ΦT

j+1Mj,0, ΦTj = ΦT

j+1Mj,0. (9.69)

This program can be carried out for various types of domains. As an example, let usconsider the interval Ω = (0, 1). The common strategy is to start with a biorthogonalmultiresolution analysis on R as explained in Subsection 9.3.2. Specifically one oftenchooses the biorthogonal system from [202] where the primal scaling functions consist


of cardinal B-splines. For j ≥ j0 where j0 is fixed and sufficiently large to disentangleendpoint effects, one builds Φj by keeping those translates 2j/2φ(2j · −k), k ∈ Z, thatare fully supported in [0, 1]. These will be refered to as interior basis functions. ForB-splines of order m one adds at each end of the interval m fixed linear combinations ofthe 2j/2φ(2j · −k) in such a way that the resulting collection Φj spans all polynomialsof order m on (0, 1), compare with (9.30). One proceeds in the same way with thedual scaling functions restoring the original order of polynomial exactness whilekeeping #Φj = #Φj . At this point one can verify (9.67), (9.66) and (9.69), i.e.nestedness, refinability and locality. However, only the interior basis functions inheritthe biorthogonality from the line whereas the boundary modifications have perturbedbiorthogonality. One can show that in this spline family of dual multiresolutionsequences one can always biorthogonalize [247], ending up with pairs of generatorbases Φj , Φj satisfying all the properties mentionend above. In fact, there are additonalnoteworthy features:

(a) Exploiting symmetry properties of the original scaling functions, one canarrange the bases to be invariant under the transformation x→ 1− x.

(b) One can arrange that only a single primal and dual basis function differs fromzero at the endpoints of the interval.

(c) According to the above comments, the bases Φj , Φj always consist of threeparts signified by the index sets 'L

j ,'Ij ,'R

j , identifying the left boundary,interior and right boundary basis functions. Only the size of the interior sets'I

j depends on j. The number of boundary functions always stays the same.Moreover, at each endpoint one has a fixed number of scaling relations whichcan be computed a priori and stored. The interior basis functions satisfy theclassical refinement rule from the real line.

(d) The matrices Mj,0, Mj,0 are invariant under reversing the order of rows andcolumns.

(e) The refinement matrices have fixed upper left and lower right blocks (in theleftmost and rightmost column) with the above symmetry properties. Only theinterior block (column) changes its size with growing level j, see Figure 9.1.

This construction can also be modified in order to incorporate homogeneous Dirich-let boundary conditions. There are two ways that suggest themselves:

(I) In view of property (b) above one can simply remove those basis functions fromthe generator bases Φj , Φj that do not vanish at the endpoints of the interval.Obviously, the resulting collections are still biorthogonal and span appropriatesubspaces of H1

0 (0, 1), see [160,204,250].(II) Following [250], remove the two endpoint boundary functions from Φj that

do not vanish at 0 and 1 but discard two interior basis functions fromthe dual collection Φj . It can be shown that the resulting sets can againbe biorthogonalized retaining all the above properties [250]. Note that nowonly Φj ⊂ H1

0 (0, 1) while S(Φj) still contains all polynomials up to a certainorder.


0 2 4 6 8

0

2

4

6

8

10

12

14

16

nz = 300 10 20 30

0

10

20

30

40

50

60

nz = 159

Figure 9.1 Nonzero elements (nz) of the matrix, Mj,0, for a quadratic and cubic wavelet

basis with three and six vanishing moments, respectively.

While (I) is most simple and immediate, option (II) appears to be conceptuallypreferable for the following reasons. Quite in line with the structure of the dualH−1(0, 1), the functional on H1

0 (0, 1) should not be constrained at the endpoints.This is supported by the following observation. Biorthogonality of the wavelets com-bined with the fact that in (II) the dual system retains full polynomial exactnessimmediately ensures that the wavelets have vanishing moments of a certain order,cf. (9.31).

This approach also works for more complicated domains such as polygonal orpolyhedral domains in Rd. In this case, one can use a domain decomposition technique.The domain Ω of interest is devided into nonoverlapping subdomains Ωi,

Ω =N⋃

i=1

Ωi, Ωi ∩ Ωj = ∅, i �= j. (9.70)

Here each Ωi is a smooth parametric image Ωi = κi(�) of the unit cube. In order toconstruct wavelet bases on Ω we follow the recipe from above, i.e. we have to constructfirst dual pairs of biorthogonal bases on the composite domain Ω. This in turn is fairlyeasy be stitching together parametric liftings of generator bases on the unit cube.This is essentially due to the following facts. Firstly, the boundary properties (b) ofthe univariate ingredients confine the gluing process to very few functions associatedwith the domain. Secondly, the symmetric properties (a) leaves convenient flexibility


concerning invariance under rotations in the parametric mappings κi. On the referencedomain, we use tensor products of the scaling functions constructed for the interval,i.e.

ϕ�j,k := ϕj,k1 ⊕ · · · ⊕ ϕj,kj

, k ∈ I1 × · · · × Id, (9.71)

see also (9.41). Then we define a generator bases on Ω by defining

ϕij,k(x) := ϕ�

j,k

(κ−1

i (x)), x ∈ Ω (9.72)

and gluing together those scaling functions across interfaces of subdomains which donot vanish at the interfaces. The analogous construction is applied to the dual system.The result is a global system ΦΩ

j , ΦΩj which is globally continuous and biorthogonal to

the modified inner product

(v, w) :=N∑

i=1

(v ◦ κi, w ◦ κi)

which is equivalent to the canonical inner product on Ω. So far, we have accomplishedstep (a) from above. Step (b) consists of identifying suitable bases for the complementspaces. There are several possibilities described in [250] which work for generatorbases of any order. The result is the so-called composite wavelet basis. Once again,homogeneous Dirichlet boundary conditions can be incorporated. As before, we havethe options (I) and (II).

9.3.5 Evaluation of nonlinear functionals

If we want to discretize nonlinear operator equations by means of wavelets, then sooneror later the problem of evaluating a nonlinear functional applied to a wavelet expansionoccurs. That is, given an expansion uΛ =

∑λ∈Λ dλψλ, one is interested in determining

the coefficients 〈f(uΛ), ψλ〉. Usually, this is a nontrivial problem and many differentsolutions have been suggested. In this chapter, we shall only discuss the approachdescribed in [251], which, in our opinion, leads to the most far-reaching results.

Let u ∈ Bsq(Lτ ) ∩ L∞, τ > (s/d + 1/p)−1, for some q ≤ p with q ∈ (0,∞] and sup-

pose that ‖u‖L∞ ≤ <∞. Furthermore, let a function f ∈ Cm[− , ], m > s withf(0) = 0 be given. Suppose that we have found an approximation in terms of a waveletbasis

uΛ =∑λ∈Λ

duλψλ with ‖u− uΛ‖ < ε. (9.73)

Then one is interested in computing 〈f(uΛ), ψλ〉 with an accuracy of order ε. Expand-ing f(uΛ) with respect to the dual system Ψ leads to

f(uΛ) =∑λ∈J

df(uΛ)λ ψλ, d

f(uΛ)λ = 〈f(uΛ), ψλ〉,

i.e. the quantities 〈f(uΛ), ψλ〉 are just the expansion coefficients of f(uΛ) in terms ofthe dual wavelet bases. Let gΛ =

∑λ∈Λ dλψλ be some approximation of f(uΛ) in the

9.4. Stable discretizations and preconditioning 653

system Ψ satisfying

‖gΛ − f(uΛ)‖L2<∼ε. (9.74)

Using the norm equivalences as stated in Theorem 9.5 (for s = 0), we obtain

‖dΛ − 〈f(uΛ),ΨΛ〉‖�2 ≤ ‖dΛ − 〈f(uΛ),Ψ〉‖�2<∼‖gΛ − f(uΛ)‖L2<∼ε.

This states that a good approximation of f(uΛ) in terms of the dual wavelet basisautomatically yields a good approximation of the desired coefficients. The algorithmin [251] focuses on constructing such an approximation. Roughly speaking, this isdone by first enlarging the set Λ to Λ in such a way that the corresponding spaceSΛ contains a function satisfying (9.74). Under the above assumptions, there exists aconstant C(f, ‖u‖L∞) such that

‖f(u)‖Bsq(Lp) ≤ C(f, ‖u‖L∞)‖u‖Bs

q(Lp), (9.75)

and it can be shown that this guarantees that only a finite enlargement has to be done.In a second step one now constructs the functions gΛ. This is done by performing alocal transformation to scaling functions φλ for all indices in Λ, then approximating〈f(u), φλ〉 (e.g. by quadrature via point evaluations), and decomposing this in anappropriate way.

9.4 Stable discretizations and preconditioning

Our goal is to develop a suitable Galerkin scheme for approximating the solution ofAu = f for an A as in (9.1) which is based on a wavelet basis as introduced in Section9.3. Therefore we consider subspaces of the form

SΛ := {ψλ : λ ∈ Λ} ⊂ U , Λ ⊂ J. (9.76)

We project our problem onto these spaces, i.e. the Galerkin approximation uΛ ∈ SΛ

is defined by

〈AuΛ, v〉 = 〈f, v〉 for all v ∈ SΛ. (9.77)

With the pair of biorthogonal bases Ψ and Ψ we can associate canonical projectorsQΛ and Q′

Λ by

QΛ(v) :=∑λ∈Λ

〈v, ψλ〉ψλ, Q′Λ(v) =

∑λ∈Λ

〈v, ψλ〉ψλ. (9.78)

By means of these projectors, the Galerkin scheme (9.77) may very conveniently bewritten as

Q′ΛAQΛuΛ = Q′

Λf. (9.79)

Obviously, (9.79) is equivalent to a linear system with the stiffness matrix

AΛ := 〈AΨΛ,ΨΛ〉T . (9.80)


The most far-reaching results were obtained for linear operators A : U → U ′, whichare boundedly invertible,

‖Au‖U ′ ∼ ‖u‖U , u ∈ U . (9.81)

In this case, the equation

Au = f (9.82)

clearly has a unique solution, compare again with Section 9.2. Typical examples aresecond order coercive boundary value problems with Dirichlet boundary conditions onsome open domain Ω ⊂ Rd such as the Poisson equation (9.5). In this case U = H1

0 (Ω)and U ′ = H−1

0 (Ω). Other examples are obtained by turning an exterior boundaryvalue problem into a singular integral operator on the boundary Γ of the domain. Fora formulation in terms of the single layer potential operator one obtains for instanceU = H−1/2(Γ) and U ′ = H1/2(Γ). Thus U typically is a Sobolev space and

U ⊂ L2 ⊂ U ′ or U ′ ⊂ L2 ⊂ U .Unless A is a differential operator, one further property of A will matter. WheneverA is an operator with global Schwartz kernel

(Av)(x) =∫

K(x, y)v(y) dy,

we will assume in addition that when (9.81) holds for U = Ht then∣∣∂αx ∂

βy K(x, y)

∣∣ � dist(x, y)−(d+t+|α|+|β|

). (9.83)

This assumption covers a wide range of cases, including Calderon–Zygmund operators,cf. [248].

In any case, to obtain an applicable numerical algorithm, it is essential that theGalerkin scheme has some basic stability properties. By using the projectors QΛ, Q′

Λ

this requirement can be formulated as

‖Q′ΛAuΛ‖U ′ = ‖Q′

ΛAQΛuΛ‖U ′ ∼ ‖uΛ‖U , uΛ ∈ SΛ. (9.84)

Before we study this issue in detail, let us discuss one of its consequences. It is oneof the main advantages of wavelet Galerkin schemes that they give rise to very simplepreconditioning strategies. Essentially, this is a consequence of the norm equivalencesstated in Theorem 9.5. Indeed, if the Galerkin scheme is stable in the sense of (9.84),then a simple diagonal preconditioning involving the matrices introduced in (9.63)gives rise to uniformly bounded condition numbers of the stiffness matrices (9.80).As we shall see below, (9.84) holds under natural conditions for all the problemsconsidered in this chapter. Therefore the following result is very important:

Theorem 9.6. Simple diagonal preconditioning for all problems in this chapter: Sup-pose that the Galerkin scheme (9.79) for A : Ht → H−t is stable (9.84). Furthermore,suppose that the parameters ρ, ρ′ in (9.62) satisfy

|t| < ρ, ρ′. (9.85)


Let DsΛ be the diagonal matrix defined in (9.63). Then the matrices

BΛ := D−tΛ AΛD−t

Λ (9.86)

have uniformly bounded spectral condition numbers,

‖BΛ‖∥∥B−1

Λ

∥∥ = O(1), Λ ⊂ J. (9.87)

Proof. The first step is to introduce the projectors

Qj(v) :=∑

k∈�j

〈v, φj,k〉φj,k, Q′j(v) :=

∑k∈�j

〈v, φj,k〉φj,k. (9.88)

With the aid of these projections, we can define the operators

Σsv :=∞∑

j=j0

2js(Qj −Qj−1)v, where Qj0−1 := 0, (9.89)

which, due to the norm equivalences (9.62), act as shifts in the Sobolev scale,

‖Σsv‖L2(Ω) ∼ ‖v‖Hs(Ω), s ∈ (−ρ′, ρ). (9.90)

It can be shown that

Σ−1s = Σ−s, Σ′

s =∞∑

j=j0

2js(Q′

j −Q′j−1

). (9.91)

Now consider any v ∈ SΛ and set w := Σtv. Thus, by (9.85), (9.90) and (9.84), weobtain

‖w‖L2 = ‖Σtv‖L2 ∼ ‖v‖Ht ∼ ‖Q′ΛAQΛv‖H−t .

Employing the norm equivalence (9.90), now relative to the dual basis, and bearing(9.91) in mind, yields

‖w‖L2 ∼∥∥Σ′

−tQ′ΛAQΛΣ−tw

∥∥L2 .

This means that the operators

At,Λ := Σ′−tQ

′ΛAQΛΣ−t : SΛ → SΛ

are uniformly boundedly invertible, that is,

‖At,Λ‖∥∥A−1

t,Λ

∥∥ = O(1), #Λ →∞. (9.92)

It is now a matter of straightforward calculation to verify that the matrix representa-tion of At,Λ relative to ΨΛ is

〈At,ΛΨΛ,ΨΛ〉T = D−tΛ AΛD−t

Λ , (9.93)

proving the claim. �


Sending #Λ to infinity, the original equation Au = f can be viewed as an infinitediscrete system

D−tAD−td = D−tf , (9.94)

where D−t and A are the infinite counterparts of D−tΛ ,AΛ, respectively, and f :=

〈f,Ψ〉T is the coefficient sequence of f expanded relative to the dual basis. Thesequence d then consists of the wavelet coefficients of the solution,

u = dT Ψ.

The infinite matrix D−tAD−t is, on account of Theorem 9.6, a boundedly invertiblemapping from �2(J) onto �2(J).

In the remaining part of this chapter, we shall mainly discuss Galerkin schemesbased on sets of the form, cf. (9.88),

Λn := {λ ∈ J : |λ| ≤ n}, i.e. QΛn = Qn, SΛn = Sn. (9.95)

Therefore, in this case the approximation spaces consist of the spaces of the underlyingmultiresolution analysis. We project our problem onto these spaces, i.e. the Galerkinapproximation un := uΛn ∈ SΛn for Au0 = f , U = Ht

0(Ω) is defined by

un = uΛn ∈ Sn = SΛn : 〈Aun, v〉 = 〈f, v〉 for all v ∈ Sn = SΛn . (9.96)

These methods correspond to uniform mesh refinements. Later on, in Subsection 9.7,we shall also discuss adaptive numerical schemes based on wavelets, and then we willclearly dispense with these assumptions.

In terms of the adjoint projectors Qn and Q′n,

Qn(v) =∑

λ∈Λn

〈v, ψλ〉ψλ, Q′n(v) =

∑λ∈Λn

〈v, ψλ〉ψλ, (9.97)

the Galerkin scheme (9.96) now reads as

Q′nAu

n = Q′nAQnu

n = Q′nf. (9.98)

(We sometimes use the redundant notation Q′nAQn to indicate self-adjointness.)

For later use, we summarize a generalized version of Propositions 4.50 for thewavelet setting which implies that the operators Qn and Q′

n are projection and linearapproximation operators.

Lemma 9.7. Qn, Q′n are linear approximation and projection operators: The previous

projectors Qn ∈ L(U , Sn), Q′n ∈ L(U ′, Sn), satisfy

∀v ∈ U , f ∈ U ′ : limn→∞

‖Qnv − v‖U = 0 and limn→∞

‖Q′nf − f‖U ′ = 0. (9.99)

More precisely we obtain for U = Hm(Ω), cf. (9.32),

U = Hm(Ω) : ∀v ∈ Hm+s(Ω) : ‖Qnv − v‖U � C2−ns‖v‖Hm+s(Ω). (9.100)


Proof. By using the norm equivalences (9.62), we obtain

‖Qnv − v‖2Hm <∼∑|λ|>n

22|λ|m|〈v, ψλ〉|2

<∼∑|λ|>n

22|λ|(m+s)2−2ns|〈v, ψλ〉|2

<∼ 2−2ns∑|λ|∈J

22|λ|(m+s)|〈v, ψλ〉|2

<∼ 2−2ns‖v‖2Hm+s . �

Let us now come back to the stability requirement as stated in (9.84). For uniformwavelet schemes, this property can be formulated as

‖Q′nAu

n‖U ′ ∼ ‖un‖U , un ∈ Sn. (9.101)

When A is positive definite and self-adjoint, then the property (9.101) holds for anytrial subspace of U . The same is correct, if A induces a coercive bilinear form asB in (9.3). Moreover, in the framework of pseudo-differential operators, sufficientconditions have been derived by Dahmen et al. [248]. Let us briefly recall the definitionof pseudo-differential operators. To do this consider the Fourier inversion formula,cf. Theorem 2.105, and notice that here i ∈ C with i2 = −1,

u(x) = (2π)−d

∫Rd

eix·ξu(ξ)dξ , u ∈ C∞0 (Rd). (9.102)

Differentiating this expression and employing here the notation ∂j =(

∂∂xj

)/i yields

Dβu(x) = (2π)−d

∫Rd

ξβeix·ξu(ξ)dξ . (9.103)

Let P = p(x,D) =∑

|β|≤k aβ(x)Dβ be a differential operator defined on a domainΩ ⊂ Rd. Then

(Pu)(x) = (2π)−d

∫Rd

p(x, ξ)eix·ξu(ξ)dξ . (9.104)

Instead of the polynomial p(x, ξ) one can take a function σ(x, ξ) belonging to a moregeneral class of functions. Then the operator A defined by

(Au)(x) :=∫

Rd

σ(x, ξ)eix·ξu(ξ)dξ (9.105)

is called a (linear) pseudo-differential operator with symbol σ(x, ξ). (For details we referto the standard literature concerning pseudo-differential operators, e.g. Taylor [617].)It turns out that injectivity of A and a Garding inequality also imply stability. More


precisely, it turns out that the uniform Galerkin scheme based on the projectors Qn, Q′n

will be stable and convergent if the following three conditions are satisfied [248]:� A is a pseudo-differential operator with symbol σ and is in the class Sm

1,0, thesubclass of the Hormander class with the property that∣∣Dβ

xDαξ σ(x, ξ)

∣∣ ≤ cα,β(1 + |ξ|)(m−|α|). (9.106)

The symbols in Sm1,0 can be split as a sum of the principal part σ0 representing an

operator of order m and a second part σ1 which represents an operator of orderm′ < m.

� A satisfies a Garding inequality, i.e.

�σ0(x, ξ) ≥ c|ξ|m, ξ ∈ Rd, (9.107)

holds.� A is injective,

Ker A = {0}. (9.108)

In summary, it is by now possible to construct stable wavelet Galerkin schemes for alarge class of problems. One of the aims of this chapter is to investigate to what extentthese stability properties are preserved under perturbations. The relations are clarifiedin the following theorem. This is, for our Hilbert space setting, a known result, seee.g. Zeidler [677], extended to our situation in Chapter 3, and Bohmer et al. [129,567].Nevertheless, we want to repeat the simple and straightforward proof for Theorem 9.8formulated for wavelets.

Theorem 9.8. Stability inherited to compact perturbations: Let B ⊂ L(U ,U ′)and suppose that the biorthogonal wavelet Galerkin scheme Bn := Q′

nBQn is sta-ble. Let A := B + C with C ∈ C(U ,U ′), the set of compact operators from U → U ′.Then

A−1 ∈ L(U ′,U) =⇒ An := Q′nAQn is stable.

Proof. See Chapter 3, and e.g. Bohmer et al. [129, 567]. We determine for anarbitrary u ∈ U and v′ := Cu the unique exact and discrete solutions, u and un, ofthe equations Bu = v′ and Bnun = Q′

nv′ = Q′

nBu. Since Bn is assumed to be stable,the corresponding Galerkin scheme converges, hence for any u ∈ U we obtain, withthe notation T = B−1, Tn = (Bn)−1Q′

n,

limn→∞

∥∥(Bn)−1Q′nCu−B−1Cu

∥∥U = lim

n→∞‖(Tn − T )Cu‖U = 0.

C is compact, so we get

limn→∞

‖(T − Tn)C‖ = 0. (9.109)

Now let un ∈ Sn. Because A is boundedly invertible, we can estimate

‖un‖U ≤ ‖A−1Aun‖U ≤ ‖A−1‖‖Aun‖U ′ ≤ ‖A−1‖‖B(I + B−1C)un‖U ′

≤ ‖A−1‖‖B‖‖(I + TC)un‖U . (9.110)

9.5. Applications to elliptic equations 659

Hence, we obtain

‖Anun‖U ′ = ‖Q′n(B + C)Qnu

n‖U ′

=∥∥Bn

(I + (Bn)−1Q′

nCQn

)un∥∥U ′

≥ 1/‖Bn‖−1∥∥(I + (Bn)−1Q′

nCQn

)un

∥∥U

= 1/‖Bn‖−1‖(I + TnCQn)un‖U≥ 1/‖Bn‖−1(‖(I + TC)Qnu

n‖U − ‖(T − Tn)CQnun‖U ).

By using the fact that A = B + C implies I + B−1C = B−1A, this reduces to

‖Anun‖U ′ ≥ 1/‖Bn‖−1(1/‖A−1B‖ − ‖(T − Tn)C‖

)‖un‖U .

Because of (9.109) and the stability of Bn there exists a positive constant K, inde-pendent of n, such that for all n ≥ n0 the following holds:

‖Anun‖U ′ ≥ K‖un‖U for all un ∈ Sn,

hence An is stable. �

9.5 Applications to elliptic equations

Based upon the previous results, in this section we will employ our general discretiza-tion approach for general convergence results for general linear and semilinear ellipticproblems. Quasilinear problems are still excluded, since they cannot yet be evaluatedfor wavelet arguments. Combining the general stability theory with the conformingwavelets in Theorem 9.11, we obtain our goal, the convergence of (uniform) waveletmethods. The convergence analysis of nonuniform methods, i.e. of adaptive waveletschemes, will be carried out in Section 9.7.

The situation here is very similar to conforming FEMs in Subsection 4.3.3. In fact,only conforming wavelets are employed for our Galerkin methods. For the reader’sconvenience, we update the general approach in Subsection 4.3.3 to the waveletsituation.

For the following it is important that any elliptic operator A induces an ellipticbilinear form a(·, ·). This can be split into the sum of a coercive bilinear form b(·, ·)and its complement c(·, ·) such that the induced operators B,C satisfy A = B + Cwith a compact perturbation, C. For our U = Hm

0 (Ω),m ≥ 1, we use for b(·, ·) theprincipal part ap(·, ·).Wavelet methods for linear elliptic equation of order 2m require U = Hm

0 (Ω). For linearA, determine un

0 ∈ Sn ⊂ U = Hm0 (Ω) such that

a (un0 , v

n) = 〈Aun0 , v

n〉U ′×U =∫

Ω

∑|α|,|β|≤m

aαβ∂αun

0∂βvndx = 〈f, vn〉U ′×U (9.111)

∀ vn ∈ Sn with aαβ ∈ L∞(Ω), and for 1 < m = |α| = |β| : aαβ ∈ C(Ω).


U A, G−→

−→

tested by←→ U

Qn Φn

SnAn, Gn

Sntested by←→ Sn

U ′

Q′n

Figure 9.2 Conforming wavelet methods: Spaces and operators.

(Of course, for preconditioning reasons, we shall always employ the multiscale waveletbases instead of the single-scale bases, compare with Theorem 9.6.) Equation (9.111)defines a bounded, elliptic a(un, vn), with U-coercive principle part, cf. (9.11).

Linear pseudo-differential operators: As already stated in Subsection 9.4, a uniformwavelet Galerkin scheme for operator equations induced by a linear pseudo-differentialoperator, B, in the Hormander class is stable if the fundamental conditions (9.107)and (9.108) are satisfied.

Wavelet methods for quasilinear elliptic equation and systems of order 2m: We mayuse again the spaces Sn of our multiresolution analysis as test and approximationspaces, i.e. we require Sn ⊂ U = Hm

0 (Ω,Rq), q ≥ 1. In fact, we use here and in thefollowing Summary the

same symbols for u0 ∈ U = Hm0 (Ω,Rq), un

0 ∈ Sn ⊂ Hm0 (Ω,Rq), q ≥ 1,

a (un0 , v

n) = 〈Aun0 , v

n〉U ′×U = 〈Anun0 , v

n〉U ′×U = 〈f, vn〉U ′×U (9.112)

∀ vn ∈ Sn,hence, An := Q′nA|Sn = Q′

nAQn|Sn, similarly Gn := Q′

nG|Sn .

We recall and impose the condition (9.12) for the Aα in (9.113): the Aα(x, �u, . . . ,∇m�u)can be evaluated for wavelet arguments. Then we determine

un0 ∈ Sn s.t. a (un

0 , vn) = 〈Gun

0 , vn〉U ′×U :=

∫Ω

∑|α|≤m

(Aα (·, un0 , . . . ,∇mun

0 ) , ∂αvn)q dx

= 〈f, vn〉U ′×U =∑

|α|≤m

(fα, ∂αvn)qdx

∀ vn ∈ Sn ⊂ U = Hm0 (Ω,Rq), q ≥ 1. (9.113)

All these methods are special cases of the general discretization methods inChapter 3. For the proof of the convergence we have to verify the conditions inSubsections 3.3–3.4 for the different spaces and operators constituting these methods.Their relations are sketched in the diagram in Figure 9.2. We summarize the necessaryconditions and definitions. As mentioned, the wavelet methods in this chapter areconforming variational methods.

We discuss wavelet methods for quasilinear problems only under the condition (9.12),but do not discuss wavelet methods for fully nonlinear problems at all. There are stilltoo many problems open in this context.


Summary 9.9. Wavelet methods are special discretization methods:

1. The wavelet method has to be applicable to the problem Au0 = f or Gu0 = 0, cf.Definition 3.12. This requires, throughout a sequence of approximating spacesSn for U , projectors Qn, Q

′n, cf. Lemma 9.7, and the discrete problems An, Gn,

cf. (9.112),

{Sn, Qn, Q′n, G

n}n∈N , with dimSn <∞. (9.114)

This is provided by the spaces of the underlying multiresolution analysis togetherwith the associated biorthogonal projectors. The wavelet spaces Sn are approxi-mating spaces, compare with (9.100):

∀v ∈ U : limn→∞

dist (v, Sn) = 0, e.g. limn→∞

||v −Qnv||U = 0. (9.115)

2. For semi- and quasilinear problems, we require that the Aα(x, u, . . . ,∇mu) in(9.113) can be evaluated for wavelet arguments.

3. Qn ∈ L(U , Sn), Q′n ∈ L(U ′, Sn), are linear approximation operators, such that

∀v ∈ U , f ∈ U ′ : limn→∞

‖Qnv‖U = ‖v‖U , limn→∞

‖Q′nf‖U ′ = ‖f‖U ′ . (9.116)

4. Wavelet methods, cf. (9.112), define a mapping Φn : D(Φn) ⊂ (D(G) ⊂ U →U ′)→ (Sn → Sn), transforming A, f,G into An, fn, Gn as

An := ΦnA := Q′nA|Sn

, fn := Φnf := Q′nf |Sn

, Gn := ΦnG := Q′nG|Sn . (9.117)

5. According to Definition 3.14, we define un, for a chosen u, as Anun = Q′nAu and

Gnun = Q′nGu. Then

Q′nAu−Anun = Q′

nA(u− un) and Q′nGu−Gnun = Q′

n(Gu−Gun) (9.118)

are the so-called (variational) consistency errors, cf. Theorem 9.11.6. The (classical) consistency error or local discretization error in u is

AQnu−Q′nAu or GQnu−Q′

nGu. (9.119)

7. The Gn are called stable in un, and An stable, if for fixed n0, r, S ∈ R+,

uni ∈ Br(un), i = 1, 2,⇒ ‖un

1 − un2‖U ≤ S ‖Gn (un

1 )−Gn (un2 )‖Sn

∀n > n0

(9.120)

For An or An · −fn this reduces to (for any r and un, but for n > n0)

(An)−1 ∈ L(Sn, Sn) exists and ||(An)−1||Sn←Sn≤ S.

The stability of Gn is essentially implied by the stability of its derivative(G′(u0))n, cf. Theorems 9.8, 3.23.

8. For possible quadrature approximations, the Q′n, A

n, Gn have to be replaced byappropriate Q′

n, An, Gn.

9. All our wavelet methods are so-called linear methods in the sense

Φn(B + C) = ΦnB + ΦnC, and Φn(G + f) = ΦnG + Φnf. (9.121)


Theorem 9.10. Existence and convergence for wavelet methods, cf. Theorem 3.21:Let the original problem Gu = 0 have the exact solution u0 ∈ U . Let its discretizationGn = Q′

nG |Sn = Q′nGQn|Sn : Sn → (Sn)′ in (9.117) satisfy

1. Gn : Sn → (Sn)′ is defined and continuous in Br(Qnu0), r > 0, n-independent;2. Gn is consistent with G in Qnu0, hence ‖Q′

nGn(Qnu0)‖U ′ → 0 for n→∞;

3. Gn is stable for Qnu0.

Then the discrete problem Gn(un) = 0 possesses a unique solution un0 ∈ Sn near u0

for all sufficiently large n and un0 converge to u0 as

‖un0 − u0‖U <∼‖Q

′nG

n(Qnu0)‖U ′ → 0. (9.122)

Hence, good convergence, essentially determined by the consistency, is usually onlysatisfied if u0 is smoother than u0 ∈ U .

For our claim of a uniquely determined discrete solution, un0 , we have to verify the

previous conditions 1., 2., 3. in Theorem 9.10. The continuity of G implies that of Gn.So consistency and stability have to be proved.

Theorem 9.11. Variational and classical consistency errors vanish and go to 0,respectively:

1. Wavelet methods applied to linear problems, A ∈ L(U ,U ′), yield vanishing vari-ational consistency errors Q′

nAu0 −Anun0 ≡ 0.

2. For A ∈ L(U ,U ′), the classical consistency or local discretization error is

‖An(Qnu)−Q′nAu‖U ′ = ‖Q′

nA(Qnu− u)‖U ′ <∼‖Qnu− u‖U . (9.123)

This remains correct for G if it is Lipschitz-continuous and (9.12) is satisfied.3. By Lemmas 9.7, the classical consistency error tends to 0, according to‖Q′

nA(Qnu− u)‖U ′ → 0 and ‖Q′n(GQnu−Gu)‖U ′ → 0 for u ∈ U or for

U = Hm(Ω),m ≥ 1 : u ∈ Hm+s(Ω) : ‖Qnu− u‖U ≤ C2−ns‖u‖Hm+s(Ω).

4. The above (9.116) is a consequence of (9.115).

Proof. The variational consistency errors vanish by (9.117) for the exact solutionu0. This is a simple consequence of Sn ⊂ U , e.g. (9.112) implies for A, similarly for G,

〈Au0 −Anun0 , v

n〉U ′×U = a (u0 − un0 , v

n) = 〈f, vn〉 − 〈f, vn〉 = 0 ∀ vn ∈ Sn. (9.124)

For the classical consistency error we use Sn ⊂ U , and (ΦnA)un = Anun = Q′nAu

n.Then we find ‖AnQnu−Q′

nAu‖U ′ ≤ C‖A‖U ′←↩U‖Qnu− u‖U → 0, by Lemma 9.7. �

Finally, the stability of the nonlinear Gn is an immediate consequence of the stabilityof its linearized (Gn)′ (un

0 ) or the invertibility of G′(u0), cf. Theorem 3.23.

Theorem 9.12. Convergence for the wavelet method applied to the above cases:Let Au = f and let G(u) = 0 have (locally) unique solutions u0. Hence A : U → U ′

or, for the nonlinear operator G : D(G) ⊂ U → U ′, the linearization G′(u0) : U → U ′

be boundedly invertible. For nonlinar G in (9.13), (9.113), we impose the condition(9.12), so the Aα(x, �u, . . . ,∇m�u) can be evaluated for wavelet arguments. For the


problems (9.111), (9.13) choose the wavelet methods according to (9.111)–(9.113).They are stable and consistent. Hence for n > n0 the wavelet equations have a(locally) unique solution un

0 as well near the exact u0. If this exact solution satisfiesu0 ∈ Hm+s(Ω), the un

0 ∈ Sn converge according to

‖un0 − u0‖U <∼2−sn‖u0‖Hm+s(Ω). (9.125)

As already indicated in Summary 9.9 7., these results remain correct, if the exactevaluation of the integrals in (9.111), (9.113) is replaced by applying quadratureformulas. The following results are obtained by a straightforward modification of theresults in Section 5.4.

〈Anun, vn〉|S′n×Sn := an(un, vn) := quadr. formula applied to a(un, vn)|Sn×Sn ,

〈fn, vn〉|S′n×Sn := quadr. formula applied to := 〈f, vn〉|S′n×Sn , (9.126)

〈Gnun, vn〉|S′n×Sn := an(un, vn) := quadr. formula applied to a(un, vn)|Sn×Sn .

We require good enough quadrature formulas, such that for smooth enough functionsun, vn ∈ Hm+s(Ω) and coefficients, aij ∈ Hm+s(Ω), terms Aα(un) ∈ H−(m+s)(Ω) forun ∈ Hm+s(Ω), and f ∈ H−(m+s)(Ω) the following quadrature errors hold:

|〈f, vn〉 − 〈fn, vn〉| <∼ 2−sn‖f‖H−(m+s)(Ω) · ‖vn‖Hm+s(Ω), (9.127)

|an(un, vn)− an(un, vn)| <∼ 2−sn · ‖un‖Hm+s(Ω)‖vn‖Hm+s(Ω).

We formulate these quadrature approximate results with other generalizations:

Theorem 9.13. Wavelet methods for monotone operators, eigenvalue problems,nonlinear boundary operators, quadrature approximations: Under the conditions ofTheorem 9.12, wavelet methods applied to the following problems yield unique converg-ing solutions. For monotone operators the convergence has to be modified as in Section4.5. For the remaining problems (9.125) is valid: variational methods for eigenvalueproblems, cf. Section 4.7, and methods for nonlinear boundary operators, cf. Section5.3. Additionally, let the quadrature approximate methods in (9.126) satisfy (9.127), cf.Section 5.4. All these general convergence properties are only valid for problems wherethe nonlinearities can be evaluated for wavelets. Finally, all the nonlinear equations forall problems, considered in this chapter, can be solved with Newton’s method, satisfyingthe mesh independence principle.

We aim for a better convergence with respect to the L2 norm for second orderequations, cf. Theorem 4.60. The solution, φg, of the dual problem,

φg ∈ U s.t. a(w, φg) = (g, w)∀w ∈ U

can be combined with the Aubin–Nitsche Lemma 5.73 for modifying Theorem 5.74:

‖u0 − un0‖U <∼‖u0 − un

0‖Sn sup0�=g∈L2(Ω)

∥∥φg − φng

∥∥Sn /‖g‖L2(Ω) (9.128)

to obtain


Theorem 9.14. Aubin–Nitsche result: Under the conditions of Theorem 9.10 and foran exact solution u0 ∈ H1+s(Ω), the solution of the wavelet method equation satisfies

‖u0 − un0‖L2(Ω)

<∼C2−(s+1)n‖u0‖H1+s(Ω). (9.129)

9.6 Saddle point and (Navier–)Stokes equations

We start with saddle point equations as a generalization of Stokes equations, a specialcase of the Navier–Stokes equations. For the reader’s convenience, we partially repeatSection 2.8.

9.6.1 Saddle point equations

Saddle point problems cf. Subsection 2.8.2 and, for the name, Theorem 2.129, requiretwo pairs of Hilbert spaces U ⊆ WU , M⊆WM, with continuous bilinear forms

a : U × U → R, b : U ×M→ R with U ↪→WU ↪→ U ′,M ↪→WM ↪→M′.(9.130)

Usually, U ↪→WU ↪→ U ′ and M ↪→WM ↪→M′ are Gelfand triples with continuousand dense embeddings U ↪→WU , M ↪→WM.

For given (f, g) ∈ U ′ ×M′ one has to determine a pair (u0, p0) ∈ U ×M such that

a(u0, v) + b(v, p0) = 〈f, v〉U ′×U for all v ∈ U ,b(u0, q) = 〈g, q〉M′×M for all q ∈M.

(9.131)

In general, we assume the bilinear form a(·, ·) to be coercive on the subspace

U0 := {v ∈ U : b(v, q) = 0 ∀ q ∈M}, hence, a(v, v) ≥ α‖v‖2U ∀v ∈ U0, α > 0,(9.132)

compare with (9.3). To ensure that the problem (9.131) is uniquely solvable, we alsohave to assume that U and M fulfill the inf–sup condition:

inf0�=q∈M

sup0�=v∈U

b(v, q)‖v‖U ‖q‖M

≥ β (9.133)

for some constant β > 0. For details, we refer to Subsection 2.8.2, and e.g. to Hackbusch[387]. The following equivalent formulation will be very useful in the sequel. Definingthe operators

A : U → U ′, 〈Au, v〉U ′×U := a(u, v), v ∈ U ,B : M→ U ′, 〈Bp, v〉U ′×U := b(v, p), v ∈ U ,B′ : U →M′, 〈B′u, q〉M′×M := b(u, q), q ∈M,

(9.134)

the problem (9.131) is equivalent to: find (u0, p0) ∈ U ×M such that

Au0 + Bp0 = f in U ′,B′u0 = g in M′.

(9.135)

9.6. Saddle point and (Navier–)Stokes equations 665

If (9.131) is well posed, the operator

B :=(

A BB′ 0

)(9.136)

is bounded and boundedly invertible with respect to the usual graph norm, see againSubsection 2.8.2, and Hackbusch [387] for details,

‖(u, p)‖U×M ∼ ‖B(u, p)‖U ′×M′ where ‖(u, p)‖2U×M := ‖u‖2U + ‖p‖2M. (9.137)

As saddle point problems are defined on product spaces of the form U ×M, weneed two pairs of biorthogonal wavelet bases Ψ = {ψλ : λ ∈ JU}, Ψ = {ψλ : λ ∈ JU}and Θ = {ϑμ : μ ∈ JM}, Θ = {ϑμ : μ ∈ JM} forming Riesz bases for WU and WM,respectively. The second pair of biorthogonal bases Θ and Θ also induces a pair ofprojectors in the sense of (9.79), (9.88):

Pn(q) :=n∑

j=0

∑k∈JM

j

〈q, ϑj,k〉ϑj,k, P ′n(q) :=

n∑j=0

∑k∈JM

j

〈q, ϑj,k〉ϑj,k. (9.138)

(We use the notation Pn here since in our setting the projectiors Qn are related withthe space U .)

In our applications, U ,MU and M,WM are mainly Hilbertian Sobolev spaces onsuitable domains or manifolds Ω1 ⊂ Rd, Ω2 ⊂ Rd′

, i.e.

U = Ht(Ω1), WU = L2(Ω1), M = Hs(Ω2), WU = L20(Ω2), (9.139)

where

L20(Ω) :=

{q ∈ L2(Ω) :

∫Ω

q(x)dx = 0}. (9.140)

Then, we assume that the norm equivalences of the form (9.62) hold for both spaces,cf. Theorem 9.5,

‖v‖Hτ (Ω1) ∼(∑

λ∈JU

22|λ|τ |〈v, ψλ〉|2)1/2

∼(∑

λ∈JU

22|λ|τ |〈v, ψλ〉|2)1/2

, τ ∈ [−t, t],

‖q‖Hζ(Ω2) ∼(∑

λ∈JU

22|μ|ζ |〈q, ϑμ〉|2)1/2

∼(∑

λ∈JU

22|λ|τ |〈v, ϑλ〉|2)1/2

, ζ ∈ [−s, s].

So we choose for (9.135), the trial spaces (UΛ,MΛ) ⊂ (U ,M) with the previous pairof index sets, (JU , JM),

Λ := (ΛU ,ΛM) ⊂ (JU , JM). (9.141)

It is well known [144] that stability of the discretization of (9.135) is ensured if theLadyshenskaja–Babuska–Brezzi (LBB) condition

infqλ∈MΛ

supvλ∈UΛ

b(vλ, qλ)‖vλ‖U ‖qλ‖M

≥ β (9.142)


is satisfied. Quite recently, explicit conditions to check (9.142) in the wavelet contexthave been derived by Dahlke et al. [241], cf. also Dahmen et al. [246], and the conditionsfor FEMs in Subsection 4.6.2. For any subset U ⊆ U we will use the notation

U⊥b := {q ∈M : b(v, q) = 0 for all v ∈ U}, (9.143)

and similarly for M ⊆M

M⊥b := {v ∈ U : b(v, q) = 0 for all q ∈ M}. (9.144)

In terms of these sets, the fundamental result from Dahlke et al. [241] reads asfollows.

Theorem 9.15. Ladyshenskaja–Babuska–Brezzi (LBB) condition: The multiscalespaces UΛ, MΛ defined above fulfill the LBB condition (9.142) provided that one ofthe following equivalent conditions holds (for B c.f. (9.134)):

1. MΛ ⊆ (U , UΛ)⊥b ;2. B(MΛ) ⊆ UΛ = span{ψλ, λ ∈ Λ};3. B′(U , UΛ) ⊆M′ , MΛ and MΛ is defined analogously to UΛ.

For simplicity, we treat (9.135) again by a uniform method, i.e. we consider multi-scale spaces of the form Vn := VJV

n, JV

n = {λ ∈ JV , |λ| ≤ n}, Mn′ :=MJMn′, JM

n′ ={μ ∈ JM, |μ| ≤ n′}. Let Qn and Pn′ denote the associated biorthogonal projectors asdefined in (9.79), (9.88), and (9.138), respectively. Then the resulting Galerkin schemefor (9.135) is given by: find a pair (un

0 , pn′0 )

un0 ∈ Vn := VJV

n, pn′

0 ∈Mn′:=MJM

n′: νQ′

nAun0 + Q′

nBpn′

0 = Q′nf,

P ′n′B′un

0 = P ′n′g. (9.145)

Consequently we obtain for saddle point equations:

Theorem 9.16. Stability and convergence for the saddle point wavelet methods:

1. Assume (9.133), and choose the multiscale spaces Vn and Mn′in such a way

that one of the conditions in Theorem 9.15 is satisfied.2. Then (9.145), the saddle point wavelet method, is stable and consistent for

(9.131) with unique and converging solutions, cf. (9.158).

Adaptive numerical schemes for saddle point problems have been derived in [237],see also Section 9.7.

9.6.2 Navier–Stokes equations

For the Stokes problem let Ω be a bounded, simply connected domain in Rd. Then,given a vector field f ∈ H−1(Ω,Rd) and a function g ∈ L2

0(Ω) one has to determinethe velocity u0 ∈ H1

0 (Ω,Rd) and the pressure p0 ∈ L20(Ω) such that, in the strong

9.6. Saddle point and (Navier–)Stokes equations 667

formulation,

−'u0 +∇p0 = f in Ω, (9.146)

−∇ · u0 = g in Ω.

In the mixed formulation this problem reads as follows: find a solution (u0, p0) ∈(U := H1

0 (Ω,Rd)×(M := L2

0(Ω))

for (9.131),with

a(u, v) := 〈∇u,∇v〉 =d∑

i,j=1

∫Ω

∂ui

∂xj(x)

∂vi

∂xj(x)dx for all u, v ∈ H1

0 (Ω,Rd),

b(v, q) := −〈∇ · v, q〉 = −d∑

i=1

∫Ω

q(x)∂

∂xivi(x)dx for all q ∈ L2

0(Ω).

For further information concerning the theory and the numerical treatment of theStokes equations, the reader is referred to Section 4.6, and, e.g. to Hackbusch [387]and Temam [622]. Our goals is to present a stable and convergent wavelet scheme forthe stationary (nonlinear) Navier–Stokes equation. It has the form

G(u, p) : =

⎛⎝−νΔu +d∑

i=1

ui∂iu +∇ p

div u

⎞⎠ , G(u0, p0) =(f0

)in Ω,

u0 = 0 on ∂Ω,∫

Ω

p0dx = 0. (9.147)

After multiplying with test functions in the usual way, this problem fits into our settingas follows: find a pair (u0, p0) ∈ H1

0 (Ω,Rd)× L20(Ω) such that

νa(u0, v) + d(u0, u0, v) + b(v, p0) = 〈f, v〉 for all v ∈ H10 (Ω,Rd),

b(u0, q) = 〈g, q〉 for all q ∈ L20(Ω), (9.148)

where

d(u, v, w) :=d∑

i,j=1

∫Ω

ui(∂ivj)wjdx. (9.149)

For bounded Ω, and d ≤ 4, see Temam [622], Lemma 1.2, Ch. II, U 1,

d(u, v, w) is a bounded trilinear form on H10 (Ω,Rd)×H1

0 (Ω,Rd)×H10 (Ω,Rd).

To discuss wavelet methods, in particular the stability, for (9.148), we consider itslinearized form. We obtain the derivative G′(u, p) evaluated in (u, p) and applied to(w, r) as

G′(u, p)(w, r) =

⎛⎝−νΔw +d∑

i=1

(wi∂iu + ui∂

iw) +∇ r

div w

⎞⎠ (9.150)


and, with the a(·, ·), b(·, ·), d(·, ·) in (9.148), and the pair of test functions (v, q)

〈G′(u, p)(w, r), (v, q)〉2 :=(νa(w, v) + d(u,w, v) + d(w, u, v) + b(v, r)

b(w, q)

).

(9.151)

In fact, the term∑d

i=1(wi∂iu + ui∂

iw) in (9.150) of lower order derivatives represents,for a moderate Reynolds number, a compact perturbation of the Stokes operator. Thiscan immediately be seen as in Section 4.6.

We shall derive a stable numerical scheme for the treatment of (9.150), (9.151).Some preparations are necessary. We want to solve the problem

〈G′(u, p)(w, r), (v, q)〉2 =(〈f1, v〉〈f2, q〉

)= F (v, q). (9.152)

For fixed u, the continuous bilinear forms d(u, ·, w) and d(·, u, v) define elements inH−1(Ω), hence they induce linear continuous operators, D1, D2,

d1(v) := d(u, ·, v) and d2(v) := d(·, u, v) induce D1, D2 ∈ L(U ,U ′). (9.153)

Therefore we observe that (9.151) can be written as

A(w, r) = B(w, r) + C(w, r) = F (9.154)

where, slightly different from (9.136), cf. (9.134), (9.135),

B =

(νA B

B′ 0

), C =

(D1 + D2 0

0 0

). (9.155)

We treat this problem again by a uniform method, i.e. we consider the same multicalespaces as in Section 9.6.1. Then the resulting Galerkin scheme for the linear (9.154)requires: find a pair (un

0 , pn′

)

un0 ∈ Vn := VJV

n, pn′

0 ∈Mn′:=MJM

n′: (9.156)

νQ′nAu

n0 + Q′

nBpn′

0 + Q′n(D1 + D2)un

0 = Q′nf1,

P ′n′B′un

0 = P ′n′f2.

The stationary (nonlinear) Navier-Stokes equation has the form (9.148), (9.147). Sothe wavelet method requires: find a pair (un

0 , pn′0 ) ∈ Vn ×Mn′

such that

νa (un0 , v

n) + d (un0 , u

n0 , v

n) + b(vn, pn′

0

)= 〈f, vn〉 for all vn ∈ Vn,

b(un

0 , qn′)

= 〈g, qn′〉 for all qn′ ∈Mn′.

(9.157)

Consequently we obtain for the (linearized) Navier–Stokes equations, cf. Subsection4.6.2 and Bohmer et al. [122,129,567].

9.7. Adaptive wavelet methods, by T. Raasch 669

Theorem 9.17. Stability and convergence for the Navier–Stokes wavelet methods:

1. Choose the multiscale spaces Vn and Mn′such that one of the conditions in

Theorem 9.15 is satisfied for the Stokes equations.2. Then the linearized form of the Navier–Stokes operator A in (9.154) represents,

for sufficiently large ν, a compact perturbation of the Stokes operator B in (9.155).For boundedly invertible A, in particular for sufficiently large ν, the Galerkinscheme (9.156) yields stable and consistent Bn and An.

3. The wavelet Navier–Stokes problems (9.156) and (9.157) for the linearized andthe nonlinear equations, possess, for large enough n, unique discrete solutions,un

0 , pn′0 , near the exact solution,u0, p0. For the nonlinear equations the nonlinear

terms have to be evaluatible for wavelet arguments.4. If the exact solution satisfies u0 ∈ H1+s(Ω), p0 ∈ H1+s(Ω), the un

0 , pn′0 converge

according to, cf. [246],

‖un0 − u0‖U + ‖p0 − pn′

0 ‖M<∼2−sn‖u0‖H1+s(Ω) + 2−sn′‖p0‖H1+s(Ω). (9.158)

Proof. Compare Subsection 4.6.2 and Bohmer et al. [122,129,567]. All the operatorsare continuously differentiable. So we show that our wavelet method for both problemssatisfies the conditions of Theorem 3.21. The method is stable by assumption or bythe compact perturbation property of the Navier–Stokes equation, by Theorem 9.8for large enough parameters ν. Finally the conformity of our methods implies theirconsistency. This finishes the proof. �

9.7 Adaptive wavelet methods, by T. Raasch

For the efficient numerical simulation of realistic problems coming from technicalapplications, adaptive approximation methods with a highly nonuniform spatial dis-cretization are of ultimate importance. The core ingredient of any adaptive algorithm isan appropriate coupling of a posteriori error estimators and adaptive space refinement.By following this general strategy, reliable approximations of the unknown solutioncan be provided within a prescribed error tolerance and with a reasonable amount ofcomputational work.

For more than 25 years, adaptive finite element methods have been successfullyused in practical applications. However, a full theoretical comprehension of theirconvergence and complexity properties, even for second order elliptic problems, couldbe acquired only recently. We refer to Morin et al. [499], Stevenson [600] and toChapter 6 for an overview. Since the late 1990s, particular interest has also beendrawn to the analysis of adaptive discretization schemes based on wavelets. Contraryto adaptive finite element methods, the convergence and complexity properties ofadaptive wavelet algorithms could be analyzed already at an early stage of theirdevelopment. This is in particular due to the strong analytic properties of waveletsystems. In the sequel, we will give a brief overview of the current status in adaptivewavelet methods; see also Dahmen [243] and Cohen [195].

Unless otherwise stated, we shall assume in this section that the underlying waveletsystem Ψ = {ψλ}λ∈J is a Riesz basis for the solution space U . By this we mean that


any u ∈ U can be expanded with respect to Ψ into a series u =∑

λ∈J uλψλ with aunique coefficient array u := (uλ)λ∈J , and with the stability condition

cR‖u‖�2(J) ≤ ‖u‖U ≤ CR‖u‖�2(J), (9.159)

where cR, CR > 0 are suitable Riesz constants. Therefore, in contrast to the previoussections, the wavelet system is normalized in U , i.e. we have ‖ψλ‖U � 1 for all λ ∈ J .From the density of span Ψ in U , it follows that the variational problem a(u, v) =〈f, v〉U ′×U for all v ∈ U is equivalent to the infinite system

Au = f , (9.160)

where we define

A :=(a(ψμ, ψλ)

)λ,μ∈J

, f =(〈f, ψλ〉U ′×U

)λ∈J

.

The following observation is of fundamental importance to any adaptive waveletalgorithm:

Theorem 9.18. Bounded bilinear form and A: If the bilinear form a : U × U → R iscontinuous, a(v, w) ≤ β‖v‖U‖w‖U for all v, w ∈ U and some β > 0, then A is boundedon �2(J) with ‖A‖ ≤ βC2

R.Moreover, if a = b + c decomposes into bilinear forms b and c, where b is coercive,

b(v, v) ≥ α‖v‖2U , and c is bounded, c(u, v) ≤ γ‖u‖U‖v‖U with γ < α, then A is bound-edly invertible on �2(J) with ‖A−1‖ ≤ (α− γ)−1c−2

R .

Proof. The claim immediately follows by inserting the respective properties of thebilinear forms and (9.159) into the identity

‖Av‖�2(J) = sup0 �=w∈�2(J)

∣∣a(∑μ vμψμ,∑

λ wλψλ

) ∣∣‖w‖�2(J)

.�

In the mathematical analysis of adaptive wavelet methods, typically the followingthree major issues are addressed.

(i) First of all, one is interested in establishing convergence of the adaptiverefinement strategy. By this we mean that, given a target accuracy ε > 0 asinput argument, the algorithm (u, ε) �→ uε under consideration terminates afterfinitely many steps and outputs a finitely supported coefficient sequence uε,such that

‖u− uε‖�2(J) ≤ ε.

Note that via the Riesz stability (9.159), such convergence estimates areequivalent to controlling the error in the U norm.

(ii) As a second issue, the convergence rate of the adaptive method is usuallyanalyzed in detail, i.e. the ratio between the number of active degrees of freedomand the corresponding discretization error. Results on attainable convergencerates of adaptive approximation schemes and on the conditions to guaranteethem can be used to quantify whether, for a given operator equation, the


additional computational work of adaptive algorithms asymptotically pays offat all, compared to nonadaptive strategies.In order to analyze convergence rates of adaptive approximation schemes indetail, certain tools from nonlinear approximation theory are necessary. Wewill summarize them in Subsection 9.7.1.

(iii) The third issue in the mathematical analysis of an adaptive refinement strategyis related with its computational complexity. In order to scale in an optimalway when applied to a realistic problem, the algorithm should work in linearcomplexity, i.e. the number of floating point and storage operations needed tocompute the output uε should stay proportional to the number of degrees offreedom #suppuε. In that case, a reasonable response time of the adaptivealgorithm can be expected for small target accuracies ε also.

Implementable algorithms which feature one or more of these key properties typicallyhave an iterative structure and make use of a posteriori error estimators. Choosingthe infinite dimensional system (9.160) as a starting point, the following two majorstrategies for the development of adaptive wavelet schemes for stationary problemshave been pursued in the literature.

(i) A first natural strategy is to consider adaptive wavelet–Galerkin methods. Theyselect certain finite subsets Λ ⊂ J and compute the solutions uΛ ∈ �2(Λ) to theassociated Galerkin systems

PΛAIΛuΛ = PΛf ,

where PΛ : �2(J) → �2(Λ) denotes the restriction operator and IΛ = P∗Λ :

�2(Λ) → �2(J) is the trivial inclusion. The particular choice of the active indexsets Λ is steered by the biggest coefficients in the current discrete residual

rΛ = r(IΛuΛ) = f −AIΛuΛ ∈ �2(J)

or in an approximation thereof. This strategy leads to the adaptive wavelet–Galerkin algorithms considered in [70,197,203,236,333], and we note that it issimilar to analogous residual-based refinement strategies in a finite element set-ting [499,600]. In any of these variants, the actual computation of approximateresiduals is made feasible by exploiting certain matrix compression propertiesof wavelet systems, discussed in Subsection 9.7.2. However, we remark that forwavelet–Galerkin methods, it is mostly assumed that the bilinear form a andhence A are symmetric. This is due to the fact that convergence is usuallyproven in the energy norm, see Subsection 9.7.3 for details. Only for a compactantisymmetric part of a, has convergence of adaptive wavelet–Galerkin methodsrecently been analyzed in [332].

(ii) A second approach, initially propagated in [198], is focused on the approximateapplication of well-known iterative methods for finite dimensional systems toinfinite dimensional systems like (9.160). For linear iterative schemes, like thedamped Richardson iteration and the steepest descent method, converge canbe verified also in an infinite-dimensional setting [161, 198] by straightforward


arguments. Moreover, by replacing the evaluations of the infinitely supportedright-hand side f and of biinfinite matrix-vector products Av by inexactcomputable versions thereof, inexact linear iterations were derived in [161,198]and in subsequent related papers. We refer to Subsection 9.7.4 for an overview.For several reasons, adaptive descent iterations are more flexible than adaptiveGalerkin methods. On the one hand, linear descent iterations of the aforemen-tioned type can be readily applied also to more general problems than thosewith a linear symmetric operator A. Generalizations towards nonsymmetricelliptic equations have been discussed in [198]. By using adaptive variants ofthe classical Uzawa iteration, also saddle point problems can be handled [237].Moreover, the analysis of best tree approximation methods in [69, 196, 200]has enabled the development of adaptive wavelet algorithms for semilinearstationary equations, based on an adaptive Newton-type iteration [199]. Werefer to Subsection 9.7.5 for details.In order to improve the linear convergence properties of iterations like theRichardson and the steepest descent method, one may also consider adap-tive variants of more complicated Krylov iterations. As an example, inexactconjugate gradient methods have been analyzed in [252], although their numer-ical performance usually does not outperform the simpler steepest descentscheme.Another important feature is that, when choosing approximate iterativeschemes, one has the possibility of replacing the wavelet basis Ψ by a redundantsystem, e.g. by a wavelet frame. Frames are usually easier to construct thanbases, in particular on nontrivially shaped bounded domains. Moreover, thematrix compression properties of wavelet bases can be preserved when usingsuitable frame systems. It must be noted that in a frame discretization, the dis-crete system (9.160) may become singular and the coefficient representation ofa given function u will no longer be unique. However, these potential drawbackscan be handled in practice, e.g. for Richardson iterations [239,538,598] and forthe steepest descent method [240]. Moreover, domain decomposition methodsin �2(J) can be used to improve the quantitative performance of wavelet framediscretizations considerably, see [601,663] for details.

9.7.1 Nonlinear approximation with wavelet systems

The underlying principle of adaptive wavelet methods consists in approximating theunknown solution u in the norm of U by finite linear combinations from the ansatzsystem Ψ, such that the number of nonzero coefficients is as small as possible. Byusing the Riesz stability (9.159), we may confine the discussion to the sequence spaceregime, approximating the unknown coefficient array u in �2(J) by sparsely populatedsequences.

As a benchmark, the most economical approximations of a given v ∈ �2(J) are thebest N-term approximations vN , defined by discarding in v all but the N largestcoefficients in absolute value. The error of best N -term approximation of v is then


defined as

σN (v) := ‖v − vN‖�2(I).

Obviously, an upper bound for the convergence rate of an adaptive discretizationscheme is given by the largest value s > 0, such that the best N -term approximationerror of the unknown coefficient array u decays like σN (u) � N−s, as N tendsto infinity. All sequences with this property are usually collected in the nonlinearapproximation space

As :={v ∈ �2(J) : ‖v‖As := ‖v‖�2(J) + sup

N∈NNsσN (v) <∞

}.

Note that ‖ · ‖As is in general only a quasinorm.50 It is well known that for ε > 0,0 < p ≤ 2− ε and s = 1

p − 12 > 0, we have the continuous and dense embeddings

�p(J) ↪→ As ↪→ �p+ε(J), (9.161)

see [286, 331] for a proof. Therefore, As is also frequently referred to as the weak �p

space �pw(J), and As may be endowed with several equivalent quasinorms; we referto [286] for details.

In order to ensure that the unknown coefficient array u is contained in As, certainnonclassical regularity assumptions on the solution u play a crucial role. Wavelettheory tells us, see [195], that if Ψ is a wavelet Riesz basis for U = Ht(Ω) which issufficiently smooth and has polynomial order r, then for all 0 < s < (r − t)/d we have

v ∈ �p(J) if and only if v =∑λ∈J

vλψλ ∈ Bsd+tp (Lp(Ω)), (9.162)

where p =(s + 1

2

)−1 and d is the spatial dimension of the domain Ω. Therefore, com-bining (9.162) with the embedding (9.161), we see that whenever the unknown solutionu is contained in the Besov space Bsd+t

p (Lp(Ω)), then the nonlinear Ht-approximationrate of u with respect to the wavelet system Ψ is at least s, ‖u‖As � ‖u‖Bsd+t

p (Lp(Ω)).The maximum nonlinear Ht-convergence rate that can be expected by only assumingappropriate smoothness conditions on the unknown solution u is (r − t)/d.

Remark 9.19.

1. In order to obtain an Ht-approximation rate s > 0 for linear approximationmethods with uniform grid refinement, it would be necessary that u belongs tothe smaller space Hsd+t(Ω). However, recent results from regularity theory ofPDEs show that there exist several relevant classes of operator equations whosesolutions are much smoother in the scale Bsd+t

p (Lp(Ω)) than in the Sobolev scale,see [233, 235, 238] and the references therein. For example, corner singularitiesof polygonal domains in R2 have infinitely high regularity in the Besov scaleB2s+t

p (Lp(Ω)), whereas their Sobolev regularity is strongly limited [234]. Forproblems with those kinds of solutions, nonlinear approximation methods will

50 Here the triangle inequality is replaced by ∃C > 0. ‖x + y‖ ≤ C(‖x‖ + ‖y‖)∀x, y ∈ As


therefore asymptotically outperform linear approximation strategies with uniformrefinement.

2. Note that we will not address the approximation of u in weaker norms than thatof U . This is due to the fact that, to the best of our knowledge, a counterpart ofNitsche duality arguments is not known in the case of nonlinear approximationmethods.

In practical computations, where the target quantity v is only implicitly given,we will not be able to compute a best N -term approximation vN exactly. Instead,it is easier to aim at the computation of an approximate or near best N -termapproximation vN , by which we mean that #suppvN ≤ N and

‖v − vN‖�2 ≤ CσN (v),

with C ≥ 1 being a uniform constant which only depends on the particular algorithm.Due to the fact that the best N -term approximation of an element v ∈ As up toaccuracy ε needs at most ε−1/s‖v‖1/s

As entries, a convergent algorithm (u, ε) �→ uε iscalled asymptotically optimal if, for any u from the unit ball of As, the output uε

fulfills

#suppuε � ε−1/s,

as ε tends to 0. The algorithm has asymptotically optimal complexity if, moreover,the number of floating point and storage operations to compute uε are bounded by aconstant multiple of #suppuε. The algorithm is called s∗-optimal and of s∗-optimalcomplexity, respectively, if these implications hold true for all 0 < s < s∗.

In many adaptive schemes, computational optimality is ensured by inserting certainthresholding operations into the algorithm. The rationale is to thin out the currentiterate in such a way that the perturbed algorithm still converges, but it additionallyfeatures an optimal work/accuracy balance. It is therefore of crucial importance toperform thresholding operations onto a given finitely supported vector v in a compu-tationally optimal way. The naive approach of sorting all entries of v in modulus andthen collecting the most important ones has the disadvantage of log-linear complexity.For this reason, sorting operations were not taken into account in the initial complexityanalysis of adaptive wavelet methods [197]. However, it has been shown in [69, 598]that quasisorting is sufficient to compute an approximate best N -term approximationvN in linear complexity. The following binning variant of hard thresholding is thereforeused in current adaptive wavelet schemes.

Algorithm 9.1.COARSE[v, ε] → vε

q := �log2((#suppv)1/2‖v‖2/ε)�sort nonzero entries of v into bins:for all λ ∈ suppv do

if |vλ| ≤ 2−q‖v‖2 thenput (λ, vλ) into the bin Vq

elseput (λ, vλ) into that bin Vi with 2−(i+1)‖v‖2 < |vλ| ≤ 2−i‖v‖2


end ifend doaggregate large entries until tolerance is met:vε := 0; err := ‖v‖2

2

for i = 0 to q dofor all (λ, vλ) ∈ Vi do

if err ≤ ε2 then return vε

(vε)λ := vλ; err := err − |vλ|2end for

end for

The computational optimality of the subroutine COARSE has been analyzed in[69,598].

Theorem 9.20. vε has few nonzero entries: Let v ∈ �2(J) be finitely supported andε > 0. Then for the output vε := COARSE[v, ε], it holds that ‖v − vε‖�2 ≤ ε and vε

has at most #suppvε � inf{N : σN (v) ≤ ε} nonzero entries. Moreover, the numberof arithmetic operations and storage locations needed to compute vε is bounded by aconstant multiple of #suppv + max{log(ε−1‖v‖�2), 1} � ε−1/s‖v‖1/s

As .

Moreover, finitely supported approximations to a given vector v ∈ As can be thinnedout efficiently by an application of the subroutine COARSE, at the price of a slightlyenlarged error, see [197,598].

Theorem 9.21. Good approximation by COARSE with few nonzero entries: Let θ <1/3 be fixed and s > 0. Then, for any ε > 0, v ∈ As, and a finitely supported w ∈ �2(J)with ‖v −w‖�2 ≤ θε, the output w := COARSE[w, (1− θ)ε] fulfills ‖v − w‖�2 ≤ ε andthe number of nonzero entries in w is bounded by

#suppw � ε−1/s‖v‖1/sAs .

Consequently, there is a constant C > 0 which only depends on s > 0, such that

‖w‖As ≤ C‖v‖As .

However, we note that thresholding routines like COARSE are not a necessaryingredient of any adaptive scheme. Recently, it has been proved that certain variantsof adaptive wavelet–Galerkin methods attain optimal convergence rates also withoutintermediate coarsening of the iterands, see [333].

9.7.2 Wavelet matrix compression

In almost all variants of adaptive wavelet methods, it is exploited that the biinfinitesystem matrix A is not arbitrarily structured but it often features certain compress-ibility properties. By this we mean that A can be approximated well by sparse matriceswith a finite number of entries per row and column.

Such compressibility properties can indeed be observed and verified, see [91, 197,245, 571, 599], whenever the underlying operator is local or at least pseudo-local, i.e.for differential equations and certain classes of integral equations. In both cases, using


the locality and the vanishing moment properties of wavelet systems, off-diagonaldecay estimates of the following type can be shown for individual matrix entries,see [243,571,599]:∣∣a(ψλ′ , ψλ)

∣∣ � 2−||λ|−|λ′||σ(1 + δ(λ, λ′))−β

, for all λ, λ′ ∈ J. (9.163)

Here σ > d/2 and β > d depend on the particular wavelet system Ψ and on theoperator, and δ is given by

δ(λ, λ′) := 2min{|λ|,|λ′|}dist(suppψλ, suppψλ′), for all λ, λ′ ∈ J.

In the special case of a differential operator, the decay parameter β > 0 in (9.163) maybe chosen arbitrarily large, i.e. the off-diagonal decay is essentially exponential in thelevel distance ||λ| − |λ′||,∣∣a(ψλ′ , ψλ)

∣∣ � 2−||λ|−|λ′||σ, for all λ, λ′ ∈ J. (9.164)

Moreover, it is important here that the wavelets ψλ are normalized in the solutionspace U = Ht(Ω). Otherwise, in the case of an L2 normalization, the left-hand sides of(9.163) and of (9.164) would entail also the normalization factors ‖ψλ‖−1

Ht(Ω) � 2−|λ|t.For an appropriate formalization of compressibility from the viewpoint of the full

system matrix, we recall the following definition from [197] and related works. Abounded operator A : �2(J) → �2(J) is called s∗-compressible, when for each j ∈ Nthere exists an infinite matrix Aj with at most αj2j nonzero entries per row andcolumn and

∑∞j=0 αj <∞, such that

∑∞j=0 2js‖A−Aj‖ <∞ holds for any 0 < s <

s∗. Equivalently, for any 0 < s < s∗ and any N ∈ N, there exists an infinite matrixwith at most N nonzero entries per row and column and with distance to A of orderN−s.

Remark 9.22.

1. We do not assume that the nonzero entries of Aj coincide with those of A, i.e.quadrature errors are allowed.

2. Any s∗-compressible A is bounded from As to As, 0 < s < s∗ [331].

In general, the particular compression rules mostly work by retaining in Aj onlythose entries Aλ,λ′ (or approximations thereof), for which the level difference is belowthe threshold ||λ| − |λ′|| ≤ j/d. For differential operators this is already sufficient, atleast for a certain range of the compressibility exponent s∗. In the case of singularintegral operators, additional thresholding techniques have to be applied per column;we refer to [243,571,599] for details.

In order to estimate the associated computational work, an s∗-compressible matrixA is called s∗-computable, if the computation of any column of Aj takes at most O(2j)floating point operations. An interpretation is that the average computational cost ofeach individual entry of Aj should be constant. This notion of computability wasintroduced in [334, 335] in order to verify s∗-optimal complexity of adaptive waveletschemes for differential and singular integral operators, also taking the quadraturesubproblems into account.


Within an adaptive numerical scheme, the matrices Aj will be used to realizethe multiplication of A with a given finitely supported vector v, up to a prescribederror tolerance ε. In particular, the estimate ‖A−Aj‖ ≤ Cj2−js for 0 < s < s∗ canbe used to construct a numerical routine APPLY for approximate matrix-vectormultiplication. The initial definition of APPLY goes back to [197]. However, analogousto the COARSE Algorithm 9.1 from Subsection 9.7.1, also the APPLY subroutine hasundergone a certain evolution in the literature. In Algorithm 9.2 below we present themost advanced binning variant from [598], for which also optimal complexity estimatescould be established.

Algorithm 9.2.APPLY[A,v, ε] → wε

q := �log2((#suppv)1/2‖v‖2‖A‖2/ε)�sort nonzero entries of v into bins:for all λ ∈ suppv do

if |vλ| ≤ 2−q‖v‖2 thenput (λ, vλ) into the bin Vq

elseput (λ, vλ) into that bin Vi with 2−(i+1)‖v‖2 < |vλ| ≤ 2−i‖v‖2

end ifend doregroup large nonzero entries of v into segments v[k]:

k := 0; v[0] := 0; err := ‖v‖22 ; i := 0

while err > ε2/(4‖A‖2) doif Vi = ∅ then i := i + 1if #suppv[k] > 2k − �2k−1� then k := k + 1; v[k] := 0select (λ, vλ) ∈ Vi and remove it from bin Vi

(v[k])λ := vλ; err := err − |vλ|2end dol := kcompute approximation to Av:

compute smallest j ≥ l with∑l

k=0 Cj−k2−(j−k)s‖v[k]‖2 ≤ ε/2

wε :=∑l

k=0 Aj−kv[k]

Theorem 9.23. Bounded operations, storage, nonzero entries by APPLY: Let A bes∗–compressible and v be finitely supported. Then, for any ε > 0, the output wε :=APPLY[A,v, ε] fulfills ‖Av −wε‖�2 ≤ ε and wε has at most #suppwε � ε−1/s‖v‖1/s

As

nonzero entries. Moreover, the number of arithmetic operations and storage locationsneeded to compute wε is bounded by a constant multiple of ε−1/s‖v‖1/s

As + #suppv.

It remains to specify the concrete value of s∗ for which compressibility and com-putability of A can be expected, being of crucial importance to the convergenceproperties of an adaptive wavelet scheme. In order not to spoil the theoretically optimalvalue (r − t)/d, it is desirable that the system matrix A is s∗-computable for valuesof s∗ strictly greater than (r − t)/d.

The first results on the compressibility of operators in wavelet coordinates canbe found in [197, 243], assuming only generic properties of the wavelet system, like


smoothness and vanishing moments, and certain properties of the operator. However,the results from [197, 243], on the actual range of s∗ were still suboptimal. It turnedout shortly afterwards that the restriction to spline wavelet systems can substantiallyimprove the compressibility properties of the system matrix. This is mostly due tothe fact that spline wavelets are locally smooth, away from their lower-dimensionalsingular support. Therefore, in the case of spline wavelets more entries of A can beexpected to have small modulus, compared to the case of generic wavelet systems.Considering the Laplace operator, compressibility of order s∗ = (γ − t)/d has beenproved in [70,237], where

γ = sup{ν : ψλ ∈ Hν(Ω) for all λ ∈ J}

is the critical Sobolev smoothness of Ψ. The most far-reaching compressibilityresults, however, have been obtained in [599]. For spline wavelet discretizations ofdifferential operators and of integral operators with Schwartz kernel, the resultsfrom [599] show that A is indeed s∗-compressible for s∗ > (r − t)/d. In particular,there exist compression rules A �→ Aj , such that ‖A−Aj‖ � 2−js for s largerthan any value for which u ∈ As can be expected. Therefore, the theoreticallyoptimal convergence rate will not be limited by the compressibility of the systemmatrix.

Concerning computability, in the early results on the convergence rates of adaptivewavelet schemes [197], it was assumed that the entries of A are available exactlyat uniformly bounded computational cost. However, using spline wavelets, this isonly possible for differential operators with constant coefficients. The computationalcomplexity of the adaptive matrix-vector multiplication APPLY in the case of generaldifferential operators and of integral operators with Schwartz kernel was discussedrecently in [335] and [334], respectively. Computing the nonzero entries of Aj bysuitable quadrature rules with an error proportional to the compression error of merelydropping entries from A, the computational optimality of APPLY could be verifiedfor a large range of compressible operators as well.

9.7.3 Adaptive wavelet–Galerkin methods

In the history of adaptive wavelet methods, convergence was first established forcertain Galerkin strategies [197, 236] Due to the fact that the respective proofs areusing the energy norm, we will assume in this subsection that the bilinear forma : U × U → R is coercive and symmetric. Moreover, we denote with |||v|||:= 〈Av,v〉1/2

the discrete energy norm of a given coefficient sequence v ∈ �2(J). The following normequivalence is well known:

‖A−1‖−1/2‖v‖�2(J) ≤ |||v||| ≤ ‖A‖1/2‖v‖�2(J), for all v ∈ �2(J).

Therefore, convergence of the coefficient arrays in the discrete energy norm is equiva-lent to convergence in �2(J) and, by the Riesz basis property (9.159), to U-convergenceof the expansions.

Adaptive Galerkin methods typically have an iterative structure. Starting with aninitial Galerkin set Λ0, they compute a sequence of nested index sets Λj , j ∈ N0.


The iteration stops as soon as the associated Galerkin solution uΛj∈ U is considered

sufficiently good by an embedded a posteriori error estimator. During the iteration,successive Galerkin approximations uΛj

are chosen to realize a certain saturationproperty, i.e. the new iterate uΛj+1 and the corresponding coefficient sequence uΛj+1

are strictly better than the previous ones:

|||u− uΛj+1 ||| ≤ ρ|||u− uΛj|||, ρ ∈ (0, 1). (9.165)

In order to predict a new Galerkin set Λj+1 with a prescribed linear error reduction(9.165), the current discrete residual rΛj

= f −AIΛjuΛj

plays a key role. The followinglemma points out that we have to determine the positions of the biggest entries of rΛj

.The result is implicitly used in [197,236], and the proof can be found in [331,333]:

Lemma 9.24. Let 0 < μ ≤ 1 and w ∈ �2(J) with finite support suppw ⊂ Λ ⊂ J besuch that ∥∥PΛr(w)

∥∥�2(J)

≥ μ∥∥r(w)

∥∥�2(J)

. (9.166)

Then the Galerkin solution uΛ associated with Λ fulfills

|||u− IΛuΛ||| ≤(1− μ2

κ(A)

)1/2

|||u−w|||,

with κ(A) = ‖A‖‖A−1‖.

In fact, if w = IΛjuΛj

is the current iterate, an application of Lemma 9.24 showsthat the new Galerkin set Λj+1 ⊃ Λj should fulfill (9.166) with

μ =(κ(A)(1− ρ2)

)1/2.

Note that due to the constraint μ ≤ 1 in Lemma 9.24, the attainable error reductionper iteration is bounded from below by

ρ ≥(1− 1

κ(A)

)1/2

.

However, it is still a nontrivial task to detect the biggest entries in the discrete residualsince rΛj

will almost always be an �2 sequence with infinite support. Moreover, inorder to end up with an optimal computational complexity, the new index set Λj+1

should ideally be the minimal superset of Λj with the prescribed saturation property.Both tasks can in fact be accomplished in practice, by the development of a suitablenumerical subroutine GROW for the extension process. In [236], approximations of thediscrete residual rΛj

were constructed by the aforementioned adaptive matrix-vectormultiplication and by inexact evaluations of the right-hand side f , ending up with aconvergent Galerkin scheme. In [197], the problem of computational optimality wasaddressed. By interleaving the extension steps with certain thresholding operations, itwas shown that the resulting sequence of Galerkin approximations shows an optimalconvergence rate. Moreover, the corresponding algorithm was numerically tested andcompared with finite element schemes in [70]. The experiments revealed that in manycases and for judicious parameter choice, the intermediate thresholding steps could


even be avoided. This observation was analyzed and justified for a specific variant ofGROW in [333]; we cite the corresponding main result.

Theorem 9.25. Bounds for operations and storage by GROW: There exists animplementable algorithm GROW[w, ν, ε] → (Λ, ν) with the following properties:

1. For finitely supported w ∈ �2(J), ν ≥ ‖r(w)‖�2(J) and ε > 0, GROW terminateswith ν ≥ ‖r(w)‖�2(J) and ν � min{ν, ε}. If u ∈ As for some s < s∗, then thenumber of floating point and storage operations for the call is bounded by aconstant multiple of min{ν, ν}−1/s(‖w‖As + ‖u‖As + ν1/2(#suppw + 1)).

2. If GROW terminates with ν > ε, then∥∥PΛr(w)∥∥

�2(J)≥ α− ω

1 + ων

and

#(Λ \ suppw) � ν−1/s‖u‖1/sAs .

Here 0 < ω < α are internal parameters of GROW with (α + ω)/(1− ω) <κ(A)−1/2.

Therefore, unless the GROW subroutine outputs a new Galerkin set with residualerror below the target accuracy ε, the new iterate uΛj+1 fulfills (9.166) for μ = α−ω

1+ω .In consequence, the error reduction rate of the global iteration is at least ρ =

(1−

(α− ω)2/((1 + ω)2κ(A)))1/2.

9.7.4 Adaptive descent iterations

A conceptually different approach to the discretization of stationary elliptic equationshas been proposed in [198]. Instead of cutting out finite portions of the equivalent�2 problem (9.160) and then solving them with standard algorithms from numericallinear algebra, the strategy advocated in [198] proceeds the other way around. Theidea is to directly apply well-known iterative schemes in the infinite dimensionalsetting, preferably with fixed error reduction per iteration step. Only after that, are allinfinite dimensional quantities replaced by finitely supported and computable ones. Inparticular, one has to work with an inexact right-hand side f ≈ f , approximate matrix-vector operations, and appropriately matched tolerances for subroutines like APPLYor COARSE.

Among the relevant iterative schemes are, e.g. Richardson iteration

u(n+1) = u(n) + αr(n), r(n) = f −Au(n), n = 0, 1, . . . (9.167)

with fixed relaxation parameter α > 0, and the closely related steepest descent method

u(n+1) = u(n) + αnr(n), αn =〈r(n), r(n)〉〈Ar(n), r(n)〉 , n = 0, 1, . . . (9.168)

with a posteriori choice of the descent parameter αn. Both iterative schemes canbe shown to converge also in an �2 setting. For instance, convergence of the exact


Richardson iteration (9.167) follows from the contractivity

ρ = ρ(α) = ‖I− αA‖ < 1,

which can be achieved whenever 0 < α < 2/‖A‖. In that case, we have linear errorreduction per iteration,

‖u− u(n+1)‖�2(J) ≤ ρ‖u− u(n)‖�2(J),

and the exact solution u has the series representation

u = A−1f = α∞∑

n=0

(I− αA)nf .

If A is symmetric, then the optimal relaxation parameter α = arg minα ρ(α) can becomputed as

α =2

‖A‖+ ‖A−1‖−1,

with

ρ(α) =κ(A)− 1κ(A) + 1

.

In practical applications, however, we will almost never be able to determine α exactly.Implementations of the Richardson iteration (9.167) will therefore often suffer fromsuboptimal performance. In contrast, the steepest descent iteration (9.168) automati-cally adapts the descent parameter to the current iterate, and it usually outperformsthe Richardson iteration exactly for this reason. We refer to [240, 252] for illustrativenumerical experiments.

The convergence and complexity analysis of the resulting inexact iterations followsgeneral principles. Usually, it is exploited that the exact iteration

u(n+1) = G(u(n)), n = 0, 1, . . . (9.169)

has at least linear error reduction

‖u− u(n+1)‖ ≤ ρ‖u− u(n)‖, n = 0, 1, . . . (9.170)

for some 0 < ρ < 1, and ‖ · ‖ is either the �2 norm or the discrete energy norm. Manyimportant schemes have this property: we mention Richardson iteration (9.167), thesteepest descent method (9.168) and other Krylov methods, like Chebyshev iteration[331] or the conjugate gradient method [252]. In practice, the exact iteration mappingG is replaced by an inexact algorithm ITER[w, ε] → vε whose output vε fulfills∥∥G(w)− vε

∥∥ ≤ ε, for all ε > 0.

Linear convergence of the perturbed but implementable iteration

v(n+1) = ITER[v(n), εn], n = 0, 1, . . . (9.171)

with a slightly larger error reduction constant can easily be shown.


Lemma 9.26. Assume that the exact iteration (9.169) has linear error reduction(9.170) with ρ < ρ < 1. Then for all tolerances εn ≤ ‖u− v(0)‖(ρ− ρ)ρn, the iteratesfrom (9.171) fulfill the error estimate

‖u− v(n)‖ ≤ ‖u− v(0)‖ρn, n = 0, 1, . . .

Proof. The claim immediately follows from the induction

‖u− v(n+1)‖ ≤∥∥u−G(v(n))

∥∥+∥∥G(v(n))− ITER[v(n), εn]

∥∥≤ ρ‖u− v(n)‖+ εn

≤ ρ‖u− v(0)‖ρn + (ρ− ρ)‖u− v(0)‖ρn

= ‖u− v(0)‖ρn+1. �

However, with the simple iteration (9.171) it is in general impossible to verifyeither convergence rates or linear computational complexity of the overall algorithm.In order to end up with an optimal convergence rate, one may consider intermediatethresholding operations with the COARSE subroutine. Note that Theorems 9.20 and9.21 have analogs when the �2 norm is replaced by the discrete energy norm. Due tothe fact that coarsening of a given iterate slightly enlarges the approximation error,the tolerances have to be adjusted carefully to preserve convergence. We thereforeconsider the following generic SOLVE Algorithm 9.3, see also [198,331].

Algorithm 9.3.SOLVE[v(0), ε] → vε

let θ < 1/3 be fixed and K ∈ N such that ρK + (ρ − ρ)∑K−1

i=0 ρi ≤ θρn := 0w(0,0) := v(0)

ε0 := ‖A−1‖‖f − Av(0)‖while εn > ε do

for i = 1 to K dow(n,i) := ITER

[w(n,i−1), (ρ − ρ)εn

]end dov(n+1) := COARSE

[w(n,K), (1 − θ)ρεn

]εn+1 := ρεn

n := n + 1end dovε := v(n)

Theorem 9.27. Estimates for errors and support in SOLVE: Assume that the exactiteration (9.169) fulfills (9.170) with ρ < ρ < 1. Then SOLVE[v(0), ε] → vε terminateswith ‖u− vε‖ ≤ ε. If, moreover, u ∈ As for some s > 0, then #suppvε � ε−1/s‖u‖1/s

As .

Proof. Concerning convergence, it is sufficient to verify that

‖u− v(n)‖ ≤ εn = ε0ρn = ‖A−1‖‖f −Av(0)‖ρn. (9.172)


For n = 0, this is trivial. Otherwise, after the K iterations in the while loop, then withan analogous argument as in Lemma 9.26,

‖u−w(n,K)‖ ≤ ρ‖u−w(n,K−1)‖+ (ρ− ρ)εn

≤ ρK‖u− v(n)‖+ (ρ− ρ)εn∑K−1

i=0 ρi

≤(ρK + (ρ− ρ)

∑K−1i=0 ρi

)εn ≤ θρεn.

By the properties of COARSE, see Theorem 9.20, we can deduce (9.172),

‖u− v(n+1)‖ ≤ θρεn + (1− θ)ρεn = εn+1.

As an immediate consequence of (9.172) and Theorem 9.21, we finally estimate

#suppvε � (ρεn)−1/s‖u‖1/sAs � ε−1/s‖u‖1/s

As . �

In order to obtain an algorithm with linear computational complexity, furtherassumptions on the subroutine ITER are necessary. The following theorem is adaptedfrom [331].

Theorem 9.28. Estimates for operations, storage and support in SOLVE: In thesituation of Theorem 9.27, assume that w = ITER[v, ε] fulfills

#suppw � #suppv + ε−1/s‖u‖1/sAs , ‖w‖As � (#suppv)1/sε + ‖u‖As ,

where the number of floating point and storage operations needed to compute wis bounded by a constant multiple of ε−1/s‖u‖1/s

As + #suppv + 1. Then the outputvε = SOLVE[v(0), ε] can be computed with at most an absolute multiple of ε−1/s‖u‖1/s

As

floating point and storage operations, as ε tends to zero.

Remark 9.29. The generic algorithm SOLVE requires knowledge of the error reduc-tion constant ρ < 1 or at least an approximation thereof. For a particular iterativescheme like Richardson iteration or the steepest descent method, such estimates areavailable whenever the spectral bounds ‖A‖ and ‖A−1‖ can be approximated.

9.7.5 Nonlinear stationary problems

The paradigm which has been outlined in the previous subsection, namely thetreatment of the original operator equation in the sequence space regime by infinite-dimensional iterations and by a subsequent discretization of the single iteration step,may be readily applied to problems with nonlinearities also. Starting out from [199],the adaptive discretization of certain nonlinear operator equations has been discussedrecently in the literature.

Assume that we are looking for a solution u ∈ U to

F (u) = 0, (9.173)


where F : U → U ′ has a continuous Frechet derivative F ′ : U → U ′ which is invertiblein a neighborhood of u,

‖F ′(v)w‖U ′ � ‖w‖U , for all v ≈ u,w ∈ U .Using a wavelet Riesz basis Ψ = {ψλ}λ∈J for U , (9.173) has an equivalent reformula-tion as the infinite nonlinear system

F(u) = 0,

where

F(v) =(〈F (v), ψλ〉U ′×U

)λ∈J

, v =∑λ∈J

vλψλ.

Such systems of equations can be handled by iterative schemes of the form

u(n+1) = u(n) −RnF(u(n)), n = 0, 1, . . . (9.174)

Here the Rn : U → U are suitable bounded linear operators. As typical examples,we mention Richardson iteration with stationary choice Rn = αI, or the Newtonmethod with nonstationary Rn = F′(u(n))−1. Note that the discrete mapping F hasthe Frechet derivative

F′(v) =(〈F ′(v)ψλ, ψμ〉U ′×U

)μ,λ∈J

.

The convergence properties of the exact method (9.174) have been analyzed in [199],for several classes of nonlinearities and iterative strategies. In case we are dealing withsemilinear operator equations

Au + g(u) = f, (9.175)

where A : U → U ′ is boundedly invertible and g : U → U ′ is monotone,

〈v − w, g(v)− g(w)〉U×U ′ ≥ 0, for all v, w ∈ U ,global convergence of the exact discrete Richardson iteration

u(n+1) = u(n) − α(Au(n) + g(u(n))− f

), n = 0, 1, . . .

can be verified for sufficiently small values of α > 0. For general nonlinear problems like(9.173), the simplified Newton iteration Rn = F′(u(0))−1 has been discussed in [199].As a first-order scheme, it at least shows linear error reduction (9.170) with rate ρ < 1in the vicinity of the exact solution u.

When it comes to concrete numerical implementations, the question arises of how toefficiently realize the application of the nonlinearity in (9.174). As already mentioned inSubsection 9.3.5, tree approximation techniques are an appropriate tool for this kind ofproblem. By imposing a tree structure on their output arguments, numerical routinesfor thresholding and for the adaptive application of nonlinear mappings to finitelysupported sequences have been developed in [199], based on the findings from [251].We cite the corresponding main results in the following theorems.

Thresholding of finitely supported vectors can be performed with a subroutineTCOARSE, i.e. tree coarsening.


Theorem 9.30. Quasi-optimal tree in TCOARSE: There exists an implementablealgorithm TCOARSE[v, ε] → vε with the following properties:

1. For finitely supported v ∈ �2(J) and ε > 0, TCOARSE computes a finitely sup-ported vε ∈ �2(J), such that ‖v − vε‖�2(J) ≤ ε and suppvε is a quasioptimal tree,i.e. suppvε is a tree with

#suppvε ≤ C inf{#T : T ⊂ J is a tree and ‖v − IT PT v‖�2(J) ≤ ε

}.

2. The number of floating point and storage operations to compute the output vε

stays proportional to

inf{#T : T is a tree with suppv ⊂ T ⊂ J

}.

Due to its similarity with the COARSE subroutine from Subsection 9.7.1, results likeTheorems 9.20 and 9.21 carry over to the case of tree coarsening with TCOARSE.

The inexact evaluation of the nonlinearity is done with a subroutine EVAL. Asoutlined in Subsection 9.3.5, the algorithmic flow of EVAL decomposes into two steps.The first step is the recovery step, where an estimate for the support of the output iscomputed. The concrete entries indexed by the recovery set are computed in a secondstep. The basic properties of EVAL read as follows, see [199].

Theorem 9.31. Approximation of nonlinear F(v) by wε: There exists an imple-mentable algorithm EVAL[v, ε] → wε with the following properties:

1. For finitely supported v ∈ �2(J) and ε > 0, EVAL computes a finitely supportedwε ∈ �2(J), such that ‖F(v)−wε‖�2(J) ≤ ε and suppwε is a tree with

#suppwε ≤ C inf{#suppw : suppw is a tree and ‖F(v)−w‖�2(J) ≤ ε

}.

2. The number of floating point and storage operations to compute the output wε

stays proportional to #suppwε + #supp v + 1

By combining the subroutines TCOARSE, EVAL and APPLY, an implementaleadaptive iteration has been designed in [199]. At least in the case of semilinearproblems (9.175), the overall algorithm converges at an optimal rate and with linearcomputational complexity. In [69, 72], the recovery algorithm is explained in moredetail, with an explicit analysis of all computational ingredients and with extensivenumerical experiments for nonlinearities depending on function values only. As aninteresting variant, we mention that the recovery algorithm can also be applied toadaptive quadrature schemes for the stiffness matrix A of linear problems; see [71] fordetails.

Moreover, in [435], it has been shown that also in the case of redundant ansatzsystems, tree approximation techniques can be used to obtain adaptive recoveryalgorithms with optimal computational complexity.

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Index

(−1)j>0 := 1 for j = 0 else := −1, 40α = (α1, . . . , αn) ∈ Nn

0 , 54

∇0u = u, 55∇u = ∇1u, 54ϑ = (ϑ1, . . . , ϑn) ∈ Rn, 54(Ai, Bi), 68(·, ·)n, 37(·, ·)n Euclidean scalar product, 456(u, v)H1(Ω), 36

(u, v)H2(Ω), 36

(u, v)Hm(Ω,Rq), 119

‖ u ‖Ck(Ω), 58

‖ u ‖Ck,γ(Ω), 58

‖ u ‖Hi(Ω), 36

‖u‖W m,p(Ω,Rq), 119

‖ v ‖Hi(Ω), i = 0, 1, . . ., 36

‖ v ‖W m,p(Ω,Rq), 119A, 37A∗, 47Ad, 47Ac, 37Ap, 40As, 37, 40B, 41Ba, 41C2m+s,γ(Ω), 58

D≤k, 55Dk, 55h, 293H1(Ω), 36H2(Ω), 36Hm(Ω, Rq), 119

H−1k (T h

c ), 351

H−m(Ω) =??H−m0 (Ω)???, 39

P h, Q′h, 591

W 2m+s,p(Ω), 58W m,p(Ω, Rq), 119W s,p(∂Ω) Sobolev space, 25% Euclidean product, 43C: complex numbers, 36Δu, 36, 119N: natural numbers {1, 2, . . .}, 36N0: {0} ∪ N = {0, 1, 2, . . .}, 36Ω, 36, 119Ω0 ∈ C0,1, 75Ω0 ⊂⊂ Ω1 for open Ω0, Ω1 means Ω0 ⊂ Ω1,

75Q+: positive rational numbers, 36Q: rational numbers, 36

R+,0 = R+ ∪ {0}, 140R: real numbers, 36RNk , 119Rnk , 119R+: positive real numbers, 36Z: integer numbers, 36α, 55Ω, 19L(U ,V ′), 46N (A), 46NL

(Ub,V ′

b

), 184

O(xα), 5O(hp), 229R(A), 46UT h = W k,p(T h), 231V = Hm

0 (Ω, Rq)−coercivity and ellipticity, 121Q set of all vertices T , 235δij = 1fori = j, else= 0, 213�, 233vε with few nonzero entries, 675div �w, 114

index(A), 60∇k, Θk ∈ Rnk , 54∇≤k, Θ≤k ∈ RNk , 55∂u, 54∂Ω, 36, 119

curved, 395polygonal, 395

∂αu, 54∂i∂ju, 36, 119∂iu, 36, 119ϑ = (ϑ1, . . . , ϑn)T ∈ Rn, 40ϑα, 54ϑi, 54| · |n, 37|ϑ|n = |ϑ|, 56|α|, 54| · |n Euclidean norm, 456a<∼b, 638a ∼ b, 638a(·, ·), 37a := b: a is defined by b, 36a =: b: b is defined by a, 36a ∼ b, 423, 638ad(u, v), 47ap(·, ·), 40as(·, ·), 37, 40ei, i-th unit vector, 30

f ∈ W−m,p′0 (Ω), 63

f ∈ H−1k (Ω), 351

f ∈ W−m,p′0 (Ω), 63

734 Index

�f ∈ W−1,p′0 (Ω, Rq), 119

〈�f,�v〉V′×V , 119

o(x), 5o(hp), 229vh �∈ C(Ω) ⇒ vh �∈ H1(Ω), 372vl, vr, [v], {v}, 462X ↪→ Y, 21h.o.t., 5T h non–degenerate, 229

a posteriori error estimator, 669a priori error bound, 424a.e. =almost everywhere, 20, 40adaptive approximation method, 669

convergence, 670convergence rate, 670

adaptive Finite Element Method, 420adaptive wavelet-technique, 636advantage

wavelet methods, 636algorithm

APPLY, 677COARSE, 674SOLVE, 682

almost everywhere (a.e.), 20, 40analysis

natural boundary conditions, 611, 612anti-crime transformation, 190, 527approximating spaces

admissible Galerkin, 272bi-dual pair, 181

approximationcubature, 346edge singularities,boundary layers, 234external, 180five star, 566good by COARSE, few nonzero entries, 675internal, 180nonlinear F(v) by wε, 685quadrature, 305, 346saturated data approximation, 439

approximation projector, P h, Q′h, 184

approximation schemeexternal, 592

asymptotic expansionextrapolation, defect correction, 626

asymptotic optimality, 674atlas, 23, 25Aubin–Nitsche result, 664

Banach spaceapproximate, 334reflexive, V = V ′′, 15, 52

basisbiorthogonal, 644interpolation, 223, 367nodal, 213stable and local, 241stable local, 249

basis functionsinterior, 650

Besov space, 646, 673best N -term approximation, 672bifurcation

multiple, 11bilinear form, 157Vh

b -coercive, 601

Vhb -elliptic, 601

approximate, 346bounded, 45coercive, 45, 50, 70, 121, 321, 637coercivity implies invertibility, 49coinciding strong and weak, 338continuous, 45difference weak, 576dual, 45elliptic, 50, 70, 121, 637induced linear operator, 45strong, 39, 66, 159, 160, 320, 337strong and weak form, 36symmetric, 46weak, 40, 66, 160, 320, 321, 337, 576

biorthogonality of wavelets + full polynomialexactness imply vanishing moments, 651

blue refinement, 436bound

upper a priori, 430lower a priori, 431a preori, 423

boundarycurved, 250

isoparametric, 253polynomial interpolation, 252

parametrization, 24boundary derivative

complementary tangential, 75boundary condition, 5, 8, 588

(2.177) - (2.179), 83Dirichlet, 42, 56, 70, 101, 132, 145, 388, 397,

403, 414, 531, 546, 564, 594Dirichlet and/or Neumann, 531excluded tangential derivative, 42general, 56, 70, 75homogeneous Dirichlet, 258included, 637incorporated, 47modified, 75natural, 8, 41, 42, 71, 74, 388, 403, 614, 617natural, Neumann, 610Neumann, 8, 56, 70, 612results for 2m = 2 and Bi in (2.185), (2.186),

83transformation into homogeneous, 57vanishing Dirichlet, 569

boundary layers, 423boundary operator

Dirichlet BD, 56natural, 41

Index 735

natural discrete, 614principal part, 75

boundary value problem, 5, 8, 459Dirichlet, 42, 56, 66, 86existence, uniqueness, regularity, 67first, 66generalized Neumann, 66generalized Robin, 66linear, 472natural, 44, 66normal, 56, 59, 61second, 67semilinear, 474third, 66

bounded operations, storage, nonzero entries byAPPLY, 677

bounded bilinear form and A, 670boundedness, 498

for the Laplacian, 495SIPG, NIPG, IIPG, GIPG, 501

bounds for operations, storage by GROW, 680bubble

edge, 432element, 432

carr(σ), 25carr u = supp u, 591case

μ, 400σ, 400

Cea’s Lemma, 424center of gravity, 450Chladny sound figures, 10, 290chunkiness parameter, 230, 428co-ordinate system, 25coercivity, 49, 94, 99, 387

H10 (Ω), 41

Hm(Ω), 71Hm

0 (Ω), 71Hm

0 (Ω) ⇒ W m,p0 (Ω), p < 2, 155, 191

Hm0 (Ω) �⇒ W m,p

0 (Ω), 2 < p, 155, 191W m,p

0 (Ω), 155, 191V, 121Vh quasi-linear, 502Vh for the Laplacian, 495Vh

b −, 603discrete, 416discrete, relations, 387in Vh general linear, semi-linear case, 499isoparametric, 416modified, 105NIPG, SIPG, IIPGMs, 497, 500nonlinear, 103, 143

coercivity implies invertibility, 49collar

Dirichlet BVP, 137estimate, 137norm, 137

compact, 14pre-, 14

relatively, 14complement

generalized orthogonal, 16complexity of MAKECONFORM, 443complexity of SOLVE, 446compressibility, 675computational complexity, 671condition

uniform Legendre, 160boundary and quadrature, 396boundedness, 94Brezzi-Babuska, 47Caratheodory, 98, 103, 131, 141, 142, 144Caratheodory type, 128Caratheodory, growth, 145complementary, 59, 82controllable growth, 128, 478ellipticity, 94exterior sphere, 110for linearization, 151growth, 98, 103, 110, 141–144

controllable, 129first, 94fourth, 94natural, 129second, 94third, 94

H, 304,307,311,317,341Hm, 335,340inf-sup-, 48Ladyshenskaja–Babuska–Brezzi, 166, 283,

477Ladyshenskaja–Babuska–Brezzi (LBB), 665Legendre, 120Legendre–Hadamard, 121, 128modified strong Legendre, 136, 339modified Brezzi–Babuska, 166modified ellipticity, 157monotone strict, 103, 143monotone strict uniform, 103monotone uniform, 103, 143natural boundary, 66natural growth, 128, 478stability, consistency for DCGMs, 529strong Legendre, 128, 131, 144strong Legendre–Hadamard, 122, 133, 266uniform Legendre–Hadamard, 161violated boundary, 378violated continuity, 378

condition numbersuniformly bounded, 636

conforming finite elements, 428conjecture, smooth splitted splines, 250connected, 628conservative

flux, 466consistency, 174

Jσh , 491

�h, 505�h, 506

736 Index

consistency (cont.)ah for general linear, 511ah for the Laplacian, 507ch and bh, 503classical, 185, 187, 590difference methods

natural boundary condition, 620differentiable

quasilinear DCGMs, 536differentiable, semilinear DCGMs, 537for violated boundary conditions, 392for violated Dirichlet conditions, 393fully nonlinear, 599general linear ah,, 513linear operator, 596linearized quasilinear, 532of order p in u, 187quadrature for quasilinear problems, 363quasilinear ah in [409], 523quasilinear equations, systems, 598quasilinear system ah, 518quasilinear systems, 526semilinear ah, 513variational, 185, 188

consistency errorvariational, 392

consistentflux-, 466

consistent differentiability, 416, 486, 610constant

depending upon parameters, 212Poincare’s, 21, 430shape regularity, 239

continuationembedding method, 206

continuityHolder, 20Lipschitz, 20modified Lipschitz, 514uniform, 387

controllable and natural growth conditions,128

convergence, 190, 268adaptive FEM, 440order p, 190, 268radial basis function, 595symmetric, unsymmetric finite difference

method, 609weak, 15

convergence theorygeneral for monotone operators, 277, 279general for quasilinear operators, 277, 279Navier–Stokes wavelet method, 669saddle point wavelet method, 666solution for monotone operators, 279solution for quasilinear operators, 279

corner point, 421corner singularity, 421corner stone of modern analysis, 37

coveringopen, 14

cubature formulabivariate, 390

cuboid, 214

data approximation error, 439DCGM

hp, 539, 540hp−, convergence, 546hp−variants, 538coercive linearized principal part, 494consistency, 503, 507, 511, 531convergence, 527, 531convergence of hp, 542convergence via monotony, 531estimate, inverse and interpolation error, 487general form, 468, 473generalization to systems, 482geometry of the mesh, 486IIPG form, 480incomplete form, 468, 473inverse estimate, 486linearized consistency, 534linearized IIPG form, 533NIPG or SIPG or GIPG, 480non symmetric form, 468, 473nonstandard, 471, 530numerical experiences, 546standard, 471, 477, 529symmetric form, 468, 473

derivative, 32directional, 28Frechet, 28Gateaux, 28inverse function, 31normal, 10partial, 36, 54, 230, 264, 638relation:Frechet/Gateaux-, 29weak, 20

diagram, 267diffeomorphism

C1, 31local C1, 31

differencebackward, 562, 564forward, 562, 564nonsymmetric, 562normal, 564, 565operator, 563, 568partial, 566symmetric, 563, 565

difference approximationsymmetric, 613

difference equation, 574classical form, 575coercivity for elliptic, 600consistency, 593divergence form, 574, 575

Index 737

elliptic 2. order, 574natural form, 575partial summation, 575, 578symmetric, 581, 615symmetric form, 560, 573unsymmetric form, 561

difference methodasymptotic expansion, 631convergence, 588, 609

symmetric, unsymmetric method, 609strong (classical) form, 609

curved boundaries, 622elliptic

nonlinear order 2m, 584elliptic linear order 2

Dirichlet, 583fully nonlinear order 2, 587natural boundary conditions, 611, 613, 621nonlinear order 2m, 586polynomial interpolation, 626Shortley–Weller–Collatz, 623simple examples, 562solving nonlinear equation, 610strong form, 609symmetric, 581unsymmetric/symmetric, 567variational, 589

differentiabilitypartial, 30

differentiableFrechet, 28Gateaux, 28consistent, 207, 344, 532partially, 30

differential operatorlinearlinear of order 2m, 56, 62linear system

order 2m, 133order 2m, 70

dilemma the G, Gh, 196Dirac measure, 449Dirichlet system, 75discontinuous Galerkin methods (DCGMs),

455Discrete Brezzi–Babuska condition:, 272discrete solution

quasilinear, unique, convergence, 275unique existence,convergence, 269

discrete spaceLebesgue, 569

discretizationDCGM, 460, 469quasilinear system, 478

discretization and preconditioning, 653discretization errors

difference, 189discretization method

linear, 269, 661wavelets, 661

discretization schemesinner and outer, 185

distanceA, B, 254

domaincurved, 461Lipschitz, 23partition of, 461polygonal, 251polyhedral, 459, 461reference, 213star-shaped, 229, 487

dual pair, 645dual problem, 452dualgenerator, 645duality argument, 449

edge jump, 430edge residual, 430eigen

function, 290space, 290value problem linear operator, 289

eigenvalue, 11, 51, 60, 64, 73, 81, 83, 89, 169, 290variational method, xxiii

eigenvalues, 109element

Cr, 223reference, 213

element residual, 430elliptic, 94, 96

Hm(Ω), 71Hm

0 (Ω), 71V, 121Vh

b , 601linear system order 2, 117strongly, 56strongly 2m−, 136uniformly, 56, 89

elliptic equationquasilinear, nonlinear coercive, 106

elliptic difference equationstability,convergence, 604

elliptic differential operatororder 2, 35

elliptic equationexistence quasilinear, 91fully nonlinear order 2, 108linear, order 2m, 639linear, order 2, 638nonlinear, 77nonuniformly, 91quasilinear

divergent, 88, 100non divergent, 88order 2, 88

special quasilinear, 149special semilinear, 149uniformly, 91weak, order 2m, 69

738 Index

elliptic operatorcharacteristic polynomial, 56definition nonlinear, 79fully nonlinear, 81linear

Fredholm alternative, 54order 2, 64order 2m, 58regular solution, 54

nonlinear, linearization, 147order 2m, m, 80quasilinear, 80semilinear, 78, 80symbol, 56

elliptic problemgeneral linear DCGM, 472general, Dirichlet condition, 572semilinear and quasilinear, 474

elliptic system, 114characteristic polynomial, 115divergent quasilinear order 2m, 137fully nonlinear order 2m, 146linear, nonlinear, 112local solvability, nonlinear, 147nonlinear order 2m, 586order, 115quasilinear, 477quasilinear of order 2m, 639quasilinear, variational method, 125symbol, 115

ellipticityVh

b −, 602fully nonlinear, 112

embedding, 21compact, 21, 50continuous, 21, 50dense, 50

equationBratu, 78, 87differential, 5elliptic

divergent quasilinear, 273fully nonlinear form, 306general quasilinear, 306

fully nonlinear, xxiii, 151, 156Karman, 137Lame, 117linear order 2m, 83minimal surface, 12, 103Monge-Ampere, 13, 109, 307

regular solution, 111monotone convergence, xxNavier–Stokes, 13, 163, 211Navier–Stokes, steady compressible, 554non coercive, 148non divergent, xxnon divergent quasilinear, 156nonuniformly elliptic, 35, 57, 90order 2m, 336quasilinear

divergent, 151

reaction-diffusion, 12saddle point, 664Stokes, 114surface with Gauss curvature, 13, 110, 307uniformly elliptic, 35, 89von Karman, 116von Karman, numerical experiments, 632

equilibrium, 4error

(classical) consistency, 187, 268, 270, 590(classical) consistency, nonconf. FEMs, 270(global) discretization, 190(variational) consistency, 268boundary, 468classical consistency, 179, 189, 296, 373, 389,

401, 485, 662classical consistency=local discretization, 275classical discretization, 412conforming variational, 188, 189consistency, 187, 207, 417FE interpolation

global estimates, 232local estimates, 232tensor FEs, 232

interpolation, 189, 229, 241, 424local discretization, 187, 268, 270, 661,

662local discretization, nonconf. FEMs, 270nonconforming variational, 189nonconforming, variational,classical

consistency, 270quantities of interest, 449variational consistency, 179, 186, 188, 269,

296, 389, 393, 401, 661, 662variational discretization, 412variational,classical consistency,conforming

FEM, 269error bound

a priori, 423a preori, 423

error estimatea posteriori, 433other types, 447quantities of interest, 449

error representation formula, 431errors

wavelet consistency, vanish, 662estimate

hpinverse and error estimate, 540computable a posteriori, 438consistency, 414errors and support in SOLVE, 682interior, 65, 136inverse, 229, 232, 237, 240, 257inverse, nonquasiuniform triangulation, 233lower a posteriori error, 432operations, storage, support in SOLVE, 683point error, 449quantities of interest, 449upper a posteriori error, 430

estimator

Index 739

other type, 447eststimation

quantities of interest, 449Euclidean product

in Rn, 43Euclidean scalar product and norm in Rn,

37Euler system

strong form, 127weak form, 127

examples nonlinear PDEs, 10excluded tangential derivative, 43, 75existence, 145

interior, 65expansion, 640extension

bilinear form, 373energy, 37Friedrichs, 36from Ω to Ωh

0 , 311extremal problem, 49

FE, 213Pn

d−1(T h), 223

affine equivalent, 221approximating spaces, 222approximation theory, 212Argyris, 218Bell, 218bilinear, biquadratic, 219condition

interpolation points on edges, 400conforming, 218, 258cubic Hermite, 227discontinuous Crouzeix–Raviart, 177equivalent, 221

affine, 221example, 221, 258Hermite, 215interpolation, 221interpolation error, 256isoparametric, 251, 254isoparametric polynomial, 253Langrange, 215linear Lagrange, 226Mortar, 234non–degenerate, 229nonconforming, 189, 218, 368nonconforming, conditions, 390of degree, 223order 2m, 271polynomial, 214

at least degree, 223degree or order, 223

quadrilateral, hexahedral, 235rectangular K, 214simple computation, 219smooth

curved domain, 238polyhedral domain, 238

space, 221spaces, updated, 237standard case, 390

curved boundary, 396discontinuous FEs, 400

subdivision, 213tensor product, 232triangular, 215triangular K, 214uniformly star-shaped, 229violated boundary condition, 371, 392, 397violated continuity, 399violating continuity condition, 372violating continuity, consistency, 403

FEMhp, 448adaptive, 434bubble function, 284conforming, 209, 210, 257, 274, 275, 282conforming generalized Petrov–Galerkin,

199conforming,basic idea, 258consistency, 336convergence, 336‖ u0 − uh

0 ‖L2(Ω), 272

linear, quasilinear problem, 271convergence of adaptive, 439convergence, adaptive, 438convergence, quadrature with crimes, 411crimes

linear problems, 380quasilinear problems, 380

crimes, quasilinear, quadrature, 413discrete Newton method, MIP, 276equation

order 2m, 332general convergence theory, 266general linear equations, 264general linear systems, 264impacts of variational crimes, 373isoparametric, 414, 417

quasilinear, 418linear, 276linear + quadratic, 284linear elliptic system

order 2, 266order 2m, 266

linear elliptic equationorder 2, 264order 2m, 265

linear, sparse equation, 342main idea, 258mixed, 211, 282Mixed for Navier–Stokes operator, 286mixed, nonlinear Navier–Stokes,convergence,

288nonconforming, 210, 283, 297, 371nonlinear boundary conditions, 345

equation, 345system, 345

740 Index

FEM (cont.)notation, requirement, 428optimal mesh, 451quadrature

second order linear, 350second order nonlinear, 357systems order 2m, 361

quadrature for nonconforming, 412reducing costs, 262second order, violating boundary condition

and/or continuity, 411semi-conforming, 308semi-conforming, fully nonlinear, 309solution of nonlinear equations, 413sparse linear equations, 342sparse nonlinear system, 342stability, 336stability nonconforming, 406stable

linear, quasilinear problem, 271stable refinement, triangulations in R2, 436summary isoparametric second order

problems, 415system

fully nonlinear, 297general quasilinear, 297order 2m, 332

variational crimes, 297, 368consistency, 368convergence, 368example, 370stability, 368

violating boundary conditions, 371Fichera corner, 423finite element, xvflux

inviscid (Euler), 556numerical, 465viscous, 556

flux term, 459form

bilinear, 41, 45bounded bilinear, 41bounded linear, bilinear, 14continuous, 63divergence, 63, 66, 86linear

special representation, 63strong, 36strong quasilinear, 80strong quasilinear difference, 625weak, 37, 76, 86, 101, 132, 583weak difference, 584, 585

formulafundamental Green’s, 37, 101generalized Green’s, 41Green’s, 37, 77, 459

Frechet derivative, 27Fredholm alternative, 51, 52, 59, 66, 67, 73, 134,

151, 155, 157, 290

for order 2m, 137divergence form, 64

Fredholm-index, 18fully nonlinear problem

existence, uniqueness, 110order 2 and 2m, 108

functioncut-off, 230Dirac delta, 213mesh-size, 425shape, 213step, 591

functionalregular integral, 126

Galerkin approximation, 424Galerkin method, 659

adaptive, 671Gelfand triple, 50, 637general approach, 451general discretization method

DCGM, 482geometric problem

quasilinear, fully nonlinear, 12geometrically stable, 436geometry of the mesh, 487GIPG, 468global Schwartz kernel, 654Gram determinant, 235green refinement, 436Green’s function

approximate, 449Garding inequality, 658

hanging node, 435Hilbert space setting, 458hyper plane

non–degenerate, 215

IIPG, 457, 468, 529inclusion, 21index, 52index A, 82inequality

Cauchy–Schwarz, 19discrete Chebyshev, 18discrete Holder, 19Garding, 50, 70, 411, 608Holder, 19, 231, 349, 392modified Poincare–Friedrich,

397multivariate Markov, 240Sobolev, 571

inf–sup condition (BB), 664integral

discrete, 569, 601integral mean value, 571, 591interior node property, 438interior regularity, 124interpolant

Index 741

FE, simple inductive computation, 219interpolation

error, 232interpolation operator, 424, 429invariance property, 563inverse estimate, 432invertibility, equibounded, xviisomorphism, 16iteration

adaptive descent, 680iterative method, 671

Jacobian, 235jumping coefficients, 423

Ladyshenskaja–Babuska–Brezzi (LBB)condition, 666

Lagrange function, 451Laplacian, 10LDC, 477Lebesgue-/Sobolev space

discrete, 570lemma

Aubin–Nitsche, 272Bramble-Hilbert, 230, 349Cea, 259generalized Aubin–Nitsche, 404nonconforming Aubin–Nitsche, 405Strang, 378

Liapunov-Schmidt methodnumerical, 612

linear approximation method, 673linear complexity, 674linear conforming finite elements, 429linear form

approximate, 346special representation, 63, 260

linear operatorVh

b -regular, 601

Vhb -stable, 601

linearizationbounded, 155, 191

fully nonlinear systems, 161linearized problem, 32local error indicator, 433local solvability and regularity, 112locality, 239longest node bisection, 437lower bound

a posteriori, 431

mappingisoparametric, 254

mapping properties, 235mask, 641matrix

M, 179mass, 257, 262real symmetric, 109

stiffness, 257, 262symmetric, 81weakly diagonal dominant, 179

matrix compression, 676mean value

symmetric, 563measure

positive, 71mesh

anisotropically graded, 236optimal, 425

mesh function, 237mesh independence principle, xviii, 205, 561mesh parameter, 234mesh refinement

necessary nonuniform, 425optimal, 425uniform, 656

mesh sequenceshape-regular, 234

method(linear) discretization, 183adaptive wavelet, 678optimal FEM, 442, 451hp FEM, 448all discretization

quasilinear equations+systems, monotoneoperators, 279

applicable discretization, 184, 316applicable Galerkin, 272approximation

conforming, 180Galerkin, 180generalized Petrov–Galerkin, 180nonconforming, 180Petrov–Galerkin, 180

conforming wavelet Galerkin, 635DCGMs, 538defect correction, 631, 632deferred correction, 631difference, xxivdifference equations, 610difference for eigenvalue problems, 609difference for monotone operators, 609difference for nonlinear boundary operators,

609difference for quadrature approximations, 609discontinuous Galerkin (DCGM), xvii, xxivdiscontinuous Galerkin (DCGMs), 212, 297,

455discrete Newton, 207eigenvalue problems, 531, 663existence,convergence, 662finite difference, xviifinite element, xvii

approximation theory, 212general interior penalty Galerkin (GIPG),

468generalized gradient, 280

742 Index

method (cont.)incomplete interior penalty Galerkin (IIPG),

468linear discretization, 184local discontinuous Galerkin, 477mesh free or radial basis, xviimonotone operators, 531, 663Newton–Kantorovich, 207non symmetric interior penalty Galerkin

(NIPG), 468, 529nonconforming approximation Galerkin, 311nonlinear boundary operators, 531, 663other, nonlinear boundary conditions, 345Petrov–Galerkin

generalized, 211quadrature approximations, 531, 663quasilinear problems, 660Runge–Kutta, 185space discretization, xxi, xxvspectral, xviisymmetric, 587symmetric interior penalty Galerkin (SIPG),

468wavelet, xvii, xxiv

minimal surface equation, 153minimum

potential, 4MIP, 205mixed formulation, 667model

nonlinear, 3model problem for DCGMs, 459model problem, 421modulus of smoothness, 646Monge -Ampere operator

G(uh) defined and continuous, 197monotone, 99

strictly, 99uniform, 103, 143, 274

multi-indices, 54, 230, 264, 638multiresolution analysis, 640multiresolution spaces

dual, 649primal, 649

multiscale, 640

Navier–Stokes operatorlinearization, 286

near best N -term approximation, 674Nemyckii operator, 586newest node bisection, 436NIPG, 468, 529

coercivity linear, semilinear, 499, 500node hanging, 436nondegenerate triangulations, 328nonuniqueness

interior, 65norm, 69

broken Sobolev, 462

discrete dual, 569discrete Sobolev, 569discrete Sobolev,‖uh‖H1

±(Ωhe ), 614

dual, 15dual Au, 48energy, 41, 46eqivalent to Sobolev, 21equivalent, 41, 46Lebesgue discrete, 569penalty, 468, 469, 484, 493scaled Sobolev, 367semi, 230Sobolev, 36, 55, 119, 230Sobolev–Slobodeckij, 23

normsequivalent, 648

notationoften used, 259

number of unknowns, N (h), 425

operator

P h, Q′h, 591

Qn, Q′n linear approximation, 656

ν−Stokes, 167, 169adjoint, 17, 47approximate, 346bijective, 16bounded, 97, 278bounded linear, 13Calderon–Zygmund, 654classical, strong and weak form, 36coercive, 97, 278compact, 13, 14continuous, 97, 278continuous linear, 13difference weak, 576dual, 16, 47elliptic, 40, 56, 81, 388elliptic nonlinear, 80estimates for nonlinear, 99, 141existence for the quasilinear, 91extension, 16, 22extension on Ω, 246Fredholm, 18, 51Fredholm-index, 18global interpolation, 222hemi continuous, 97, 278induced, 46induced boundary, 42, 66induced linear, 45injective, 16interpolation, 424isoparametric interpolation, 254, 256kernel, 14linear, 41linear, continuous, bounded, 14, 41linearization, quasilinear, order 2m, coercive,

158linearized Navier–Stokes, 167, 286

Index 743

linearized Navier-Stokes, 114, 167Lipschitz-continuous, 97, 278local interpolation, 213, 222monotone, 34, 97, 278natural boundary, 42, 66Navier-Stokes, 114, 167Nemyckii, 96, 98, 141Nemyckii for systems, 140nonlinear, 96nonlinear system, 140nonlinearcoercive, 97, 278norm, 14null space, 14properties for nonlinear, 98, 140properties of Nemyckii, 100properties of the trace, 26properties,compact, 15pseudo–differential, 657quasilinear, 81, 150

Hm0 − coercive linearization, 158

quasilinear order 2, 85range, 14saddle point problems, 163self-adjoint, 17semilinear, 81, 105semilinear in nonlinear spaces, 86, 150semilinear non autonomous, 150semilinear Order 2m, 149special linear, 14stable, 97, 278stationary Navier–Stokes, 281Stokes, 116, 163strictly monotone, 97, 278strongly continuous, 97, 278strongly elliptic, 40, 56, 81strongly monotone, 97, 278surjective, 16symmetric, 17trace or restriction, 26uniformly elliptic, 56, 81, 109, 307,

320uniformly monotone, 97, 278weak, 160

optimal complexity, 443optimal mesh

for 2D corner singularity, 427optimality, 442order, 115, 133orthogonality

Galerkin, 440outflow boundary point, 423

paralellotope, 214parameter

chunkiness, 230parametrization of ∂Ω, 25partition of unity, 25PDE

divergence form, 34

divergent, 34penalty

interior, 457penalty function

interior discontinuities, 468penalty norm, 491plate

buckling, 11pollution error, 434polynomialPd = P1

d , 215Pn

d , 215

Pd = P2d , 215

characteristic, 40average Taylor, 230, 243bilinear,biquadratic, 219degree d, 214

positive, 99asymptotic, 99

pre–wavelets, 642primal generator, 645principal part, 40, 64, 67, 73, 76, 80, 81, 106,

122, 134, 151, 158, 160Vh

b −coercive, 602discrete, 574elliptic operator, 56, 70, 133symbol, 115system, 115, 120, 133, 266

principal symbolsystem, 120

principlemesh independence, 486

problemand Stokes, 282convection-diffusion, 458dual, 272, 404, 663existence and uniqueness for semilinear,

106extremal, 49linearization, 151nonlinear stationary, 683reaction-diffusion, 458saddle point, 163, 165, 166, 282, 664

unique solution, 166scope of, 637unique discrete solution++, 283variational, 49, 64, 67, 126

producttensor, 219

projector

Q′h = Qh′

, 260, 270, 385, 401

approximation, P h, Q′h = Qh′

,184

orthogonal, 17property

k-linear algebraic, 342Galerkin orthogonality, 472

propositionuseful, 366

744 Index

quadratic eigenvalue problem, 423quadrature formula

Gauss, 390Gauss–Lobatto, 390Gauss–Radau, 390Gauss–type, 390high order, 390

quasi uniform, 229, 233quasi-interpolant, 241quasi-norm, 673quasi-optimal tree in TCOARSE, 684quasilinear

problem violated boundary conditions,399

quasilinear equation, 629special DCGM, 481

quasilinear operatorspecial, 150

quasilinear systemorder 2, 2m, 157

quasiuniformlocally, 234

radial basis functions, 595red refinement, 436redundant system, 672refinable, 641refinement, red-green, 436

blue, 436newest node bisection, 436longest node bisection, 437

regular discretisation, 428regularity, 146

in Ω, 65bootstrap arguments, 84in Morrey spaces, 125inherited, 84inherited to perturbations, 85results of Ladyzenskaja/Uralceva,

93solutions of FEMs, 327

regularity for order 2m, 137relation

two–scale, 641residual error

estimator, Poisson problem, 429residuals, 433Reynolds number, 164, 281Richardson extrapolation

high order difference, 631Richardson iteration, 680Riesz basis, 648, 669rod

axial load, 7axial loaded, 15bending, 3clamped end, 3loaded, 27perp load, 3

simply supported end, 7stationary position, minimum, 8

saturation property, 679scalar product, 69

discrete Sobolev, 569, 614Lebesgue discrete, 569Sobolev, 36, 55, 119

scale representationmulti, 643single, 643

scaling function, 641Schwartz kernel, 678semi norm, 55, 231

broken Sobolev, 462discrete, 569, 614Sobolev, 36, 119

sequenceapproximating spaces, 267, 661

setadmissible, 126dense, compact, 14

set of marked triangles Ah, 435shape factor, 428Shortley–Weller approximation, 625Shortly–Weller Method

convergence, 625singular solutions

general structure, 422singularities

corner, 421edge, 423

SIPG, 468, 529smooth manifold, 25Sobolev are Banach spaces, 20Sobolev space, 20, 422, 647

broken, 231discrete, 566, 568discretization in discrete, 589fractional order, 23trace, fractional order, 27

solutionΩ convex, 74hp-DCG approximate, 540classical, 38, 459, 475discrete approximate, 471, 474existence, 106, 110existence and uniqueness, 86, 87existence and uniqueness semilinear, 84existence/regularity, 96for convex domain, 74, 124for fully nonlinear, 111for quasilinear, 110generalized, 104, 121, 134, 143

weak Euler system, 129interior regularity, 75linear and semilinear, 88regular, 60, 65, 73, 74, 95, 112, 123, 130, 139,

146

Index 745

singular, 421smooth data imply smooth solution, 170spurious, 87strong, 38, 42strong and weak, 133unique, 93uniqueness, 106, 110viscosity, 296, 299, 587weak, 38, 42, 121, 134, 460, 475weak discrete, 374weak=strong, 42

spaceadmissible approximating, 180approximating, 484

bi–dual, 180conforming, 180Galerkin, 180generalized Petrov–Galerkin, 180nonconforming, 180Petrov–Galerkin, 180

Banach, 14Banach, L(X ,Y), 18Banach, dense, 14broken Sobolev, 462discrete Sobolev, 569, 624discrete Sobolev,H1

±(Ωhe ), 614

dual, 14Holder, 58Holder/Sobolev, 58Morrey, 125nonconforming approximating, 311reflexive Banach, 47, 49Sobolev, 36, 69, 119surplus, 441

space refinementadaptive, 669uniform, 673

spectrumbilinear form, 291operator, 51, 52

splinesuper spline subspace, 247

splittingstable, 250

stability, 174, 190, 191, 239, 268, 485, 661bound, 191compactly perturbed discretization inherits,

408discrete W m,p(Ω) norm, 155elliptic operators, 406in uh, 191, 268, 485, 590proofs, 194threshold, 191

stability and consistency yield convergence, 189stability implies existing A−1, 204stability inherited to compact perturbations,

658star of a vertex, 240star-shaped, 230

state vector, 556steepest descent method, 680step size, 562

maximal, 229Stokes problem

unique discretesolutions,convergence, 285

unique, solution, regularity, 166strategy

fixed energy fraction, 435maximum, 435

subdivision, 229, 231cf. triangulation, 214isoparametric, 254

supp(σ), 25symbol, 657

elliptic operator, 70symmetric formula, 563system

bounded nonlinearity, 145Dirichlet, 56, 62, 132, 145elliptic, 115Euler, 34, 126fully nonlinear, 158quasilinear divergent, 140linear elliptic

order 2m, 132nonlinear

order m, 112, 147order 2m, 115

nonlinear elliptic, 115normal, 56order 2m, 336quasilinear, 144

divergent, 158quasilinear elliptic, 157strongly elliptic, 120uniform Legendre condition, 382uniformly elliptic, 120, 266, 382weak Euler, 129

tensor products, 644term

penalty, 464tetrahedra/bricks, 233The Stokes problem, 666theorem

(local) inverse function, 31Aubin–Nitsche, 272Banach, open mapping, 16closed range, 18extension of linear forms, operators, 16Gauss integral, 27Garding, 71Hahn–Banach Extension, 16implicit function, 31mean value, 30modified Sobolev–Stein extension, 246nonconforming Aubin–Nitsche, 405

746 Index

theorem (cont.)Riesz–representation, 17Riesz–Schauder, 51Sobolev embedding, 22, 350Sobolev–Stein extension, 22trace, 26transformation, 367

theoryRiesz–Schauder, 51

trace from Ω to ∂Ω, 75transform

discrete Fourier, 603Fourier, 134inverse discrete Fourier, 602, 603

transformationanti crime, xvi, xxiii, 293, 408, 605

tree approximation, 684triangles/quadrilaterals

curvilinear, 233triangulation, 214, 251, 254, 428,

461admissible, 214locally quasi-uniform, 540non–degenerate, 229, 231quasi-uniform, 229, 239shape-regular, 540

Tychonov regularization, 57

unique existence, 169uniqueness

in the small, 95interior, 65

uniqueness results, 91unisolvent, 213unit ball, 23upper bound

a posteriori, 430

variablesnodal, 213

variational crimes, 175variational formulation, 462

discretization, 475variational problem

regular, 126vertex

interior, 247vertices, 214

wavelet, 640analysis, 639bases, 647biorthogonal, 647composite, 652dual, 652evaluation of nonlinear functionals, 652generator, 643good, 643nonlinear approximation, 672orthonormal, 642sdiagonal preconditioning for all problems,

654wavelet method

conforming, 660convergence, 662linear elliptic equation

order 2m, 659not for fully nonlinear problems, 660

wavelet transformdiscrete, 640fast, 643

wavelets, 669conforming, 659elliptic equations, 659function spaces, 645on domains, 647

weak �pw(J) space, 673

weak form, 423welldefined G, Gh for the Monge-Ampere

operator, 196

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