SOME BOUNDARY ELEMENT METHODS FOR HEAT CONDUCTION PROBLEMS

MARTTIHAMINA

Mathematics Division

OULU 2000

OULUN YLIOPISTO, OULU 2000

SOME BOUNDARY ELEMENT METHODS FOR HEAT CONDUCTION PROBLEMS

MARTTI HAMINA

Academic Dissertation to be presented with the assent of the Faculty of Science, University of Oulu, for public discussion in Raahensali (Auditorium L 10), Linnanmaa, on June 21st, 2000, at 12 noon.

Copyright © 2000Oulu University Library, 2000

OULU UNIVERSITY LIBRARYOULU 2000

ALSO AVAILABLE IN PRINTED FORMAT

Manuscript received 28 March 2000Accepted 12 April 2000

Communicated by Professor Arvet Pedas Professor Rolf Stenberg

ISBN 951-42-5614-X

ISBN 951-42-5613-1ISSN 0355-3191 (URL: http://herkules.oulu.fi/issn03553191/)

Hamina Martti, Some boundary element methods for heat conductionproblemsMathematics Division, University of Oulu, P.O. Box 4500, FIN 90014 Oulu, Fin-land2000Oulu, Finland(Manuscript received 28 March 2000)

Abstract

This thesis summarizes certain boundary element methods applied to some initial andboundary value problems. Our model problem is the two-dimensional homogeneous heatconduction problem with vanishing initial data. We use the heat potential representationof the solution. The given boundary conditions, as well as the choice of the representationformula, yield various boundary integral equations. For the sake of simplicity, we use thedirect boundary integral approach, where the unknown boundary density appearing inthe boundary integral equation is a quantity of physical meaning.

We consider two different sets of boundary conditions, the Dirichlet problem, where theboundary temperature is given and the Neumann problem, where the heat flux across theboundary is given. Even a nonlinear Neumann condition satisfying certain monotonicityand growth conditions is possible. The approach yields a nonlinear boundary integralequation of the second kind.

In the stationary case, the model problem reduces to a potential problem with anonlinear Neumann condition. We use the spaces of smoothest splines as trial functions.The nonlinearity is approximated by using the L2-orthogonal projection. The resultingcollocation scheme retains the optimal L2-convergence. Numerical experiments are inagreement with this result. This approach generalizes to the time dependent case. Thetrial functions are tensor products of piecewise linear and piecewise constant splines. Theproposed projection method uses interpolation with respect to the space variable andthe orthogonal projection with respect to the time variable. Compared to the Galerkinmethod, this approach simplifies the realization of the discrete matrix equations. Inaddition, the rate of the convergence is of optimal order.

On the other hand, the Dirichlet problem, where the boundary temperature is given,leads to a single layer heat operator equation of the first kind. In the first approach,we use tensor products of piecewise linear splines as trial functions with collocation atthe nodal points. Stability and suboptimal L2-convergence of the method were provedin the case of a circular domain. Numerical experiments indicate the expected quadraticL2-convergence.

Later, a Petrov-Galerkin approach was proposed, where the trial functions were tensorproducts of piecewise linear and piecewise constant splines. The resulting approximativescheme is stable and convergent. The analysis has been carried out in the cases ofthe single layer heat operator and the hypersingular heat operator. The rate of theconvergence with respect to the L2-norm is also here of suboptimal order.

Keywords: collocation, boundary integrals, heat conduction

Acknowledgements

The basis of this research was laid during the years 1989-1991 when I was workingas a researcher in the project: ”Analysis of the Boundary Element Method”. Theproject was funded by the Academy of Finland. Professor Jukka Saranen was thehead of our team. I wish to express my gratitude to the Academy of Finland forfinancial support.

This research was carried out under the supervision of Professor Jukka Saranen.His encouragement has been of great importance during the evolution of my work.I am deeply indebted to my instructor for his guidance and advice. I also wish toexpress my thanks to Dr Keijo Ruotsalainen for his collaboration.

I am also very grateful to Professor Rolf Stenberg and Professor Arvet Pedasfor refereeing the manuscript.

The linguistic form of the English language was revised by Mr Gordon Roberts,to whom I express my thanks for his contribution.

The Mathematics Division in the Faculty of Technology at the University ofOulu has been a stimulating working environment. For this I am grateful to thestaff of the Division. In particular, I wish to express my thanks to Professor JuhaniNieminen and Professor Seppo Seikkala for their encouragement during this work.

Finally, I wish to express my warmest thanks to my parents for their constantsupport.

Oulu, March 2000 Martti Hamina

List of the original articles

This thesis is a summary of the work published in the following five articles:

I Hamina M & Saranen J (1994) On the spline collocation method for thesingle-layer heat operator equation. Math Comp 62: 41-64.

II Hamina M (1997) A collocation type projection method for the single layerheat operator equation. Preprint, University of Oulu, December 1997.

III Hamina M (2000) An approximation method for the hypersingular heat op-erator equation. J Comput Appl Math 115: 229-243.

IV Hamina M, Ruotsalainen K & Saranen J (1992) The numerical approxi-mation of the solution of a nonlinear boundary integral equation with thecollocation method. J Integral Equations Appl 4: 95-115.

V Hamina M (1997) On the numerical solution of a non-linear heat conductionproblem. Integral Methods in Science and Engineering, Vol 2, (ConstandaC, Saranen J, Seikkala S eds), Addison Wesley Longman Inc: 93-98.

These articles are referred to in the text by their Roman numerals.

Contents

AbstractAcknowledgementsList of the original articlesContents1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.1 Recent history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2 Boundary integral formulations . . . . . . . . . . . . . . . . . . . . 141.3 Reduction to an integral equation . . . . . . . . . . . . . . . . . . . 171.4 Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.5 Mapping properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.6 Spline spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.6.1 Inverse properties . . . . . . . . . . . . . . . . . . . . . . . . 241.6.2 Approximation properties . . . . . . . . . . . . . . . . . . . 25

1.7 Some Petrov-Galerkin approximations . . . . . . . . . . . . . . . . 271.7.1 Reformulation of the collocation problem . . . . . . . . . . 291.7.2 Reformulation of the Petrov-Galerkin method . . . . . . . . 30

1.8 Description of the nonlinearity . . . . . . . . . . . . . . . . . . . . 332 Abstract theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.1 Linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.2 Nonlinear equations . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3 Summary of the original articles . . . . . . . . . . . . . . . . . . . . . . 423.1 Article I: On the spline collocation method for the single layer heat

operator equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.2 Article II: A collocation type projection method for the single layer

heat operator equation . . . . . . . . . . . . . . . . . . . . . . . . . 453.3 Article III: An approximation method for the hypersingular heat

operator equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.4 Article IV: The numerical approximation of the solution of a non-

linear boundary integral equation with the collocation method . . 503.5 Article V: On the numerical solution of a non-linear heat conduction

problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

1 Introduction

Initial and boundary value problems are seldom analytically solvable. Thereforenumerical methods are of great importance. The Finite Element Method has beenamong the most popular computational techniques. However, in some instancesthe Boundary Element Method has evolved as a powerful alternative for finiteelements. Compared to the finite element method, the most important feature ofthe boundary element method is that it only requires discretization of the boundaryrather than that of the whole volume. Thus, the boundary element method isadvantageous when the domain extends to infinity or the shape of the boundaryis complex. The most severe restriction of the boundary element approach is thatit is applicable only to such problems where the fundamental solution is available.

The boundary and initial value problem of the heat equation is the most usu-al model problem of the so-called parabolic boundary and initial value problems.This work is devoted to the boundary element solution of the homogeneous heatequation. An exellent classical treatment of partial differential equations of theparabolic type is given by Friedman [22]. Pogorzelski’s book [45] serves as a clas-sical introduction to integral equation methods. Some textbooks concerning themodern treatment of integral equation methods and boundary element methodsare Atkinson [8], and Chen & Zhou [13].

1.1 Recent history

The mathematical theory of the boundary element method has developed vastlyduring the last twenty years. The most familiar discretization methods are thespline Galerkin and spline collocation methods. Other well known approaches arethe Nystrom method, trigonometric collocation and quadrature methods. During1976-1983 a rather complete theory of the Galerkin boundary element methodon smooth boundaries was developed. The theory was based on the concept ofstrongly elliptic pseudodifferential operators. Thus, the analysis covered very largeclasses of boundary element methods. The practical implementation, as well as

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the effect of the numerical integration, has been discussed in references [26], [27]and [35], for example. In the presence of singularities (domains with corners,mixed boundary conditions) theoretical progress has been based on the works ofCostabel, Stephan, Chandler and several other authors.

In the Galerkin approach, the boundary integral equation is satisfied in theaverage sense. From the practical point of view, this means double integrationover the boundary of the domain. Another approach, easier to implement, is thecollocation method where the boundary integral equation is fulfilled on certain,so-called, collocation points. However, the theoretical analysis of the collocationmethod is much more difficult. There are two main lines of research. One is basedon the equivalence between the collocation method and a certain Galerkin method.The other relies on direct Fourier analysis.

The equivalent Galerkin method is available only when the trial functions aresplines of an odd degree. The analysis allows quasi-uniform meshes and yield-s asymptotic error estimates with respect to some nonstandard Sobolev norms,which depend on the order of the operator and on the degree of the trial functions.The method was developed by Arnold & Wendland [5] for the one-dimensionalcase and extended by Arnold & Saranen [4] for biperiodic problems. These ideasgeneralize also to the case of certain parabolic boundary integral equations, as wasshown by Hamina & Saranen in [24], [25] and paper I.

The direct Fourier analysis is applicable if the meshes are uniform. This ap-proach has been applied to the one-dimensional spline collocation in papers [6],[56], [54]. We emphasize that the convergence of the spline collocation in higherdimensions is not generally known for quasi-uniform meshes, even in the case ofelliptic problems. There are results in some special cases where the equation isgiven on a torus [4], [15], [47], square or cube [17], [46]. Recently Costabel & Sara-nen [18] have obtained optimal order error estimates for convolutional parabolicboundary integral equations by using Fourier techniques.

For time dependent problems, the papers by Arnold & Noon [3] and Noon [37]were of crucial importance. They discovered that the single layer heat operatoris coercive in certain anisotropic Sobolev spaces. Their analysis applied to threedimensional heat conduction problems. Corresponding results in two space di-mensions were given by Hsiao & Saranen [28] and Costabel [14]. The coercivityproperty was at once a rigorous basis for the analysis of Galerkin boundary ele-ment methods. The fundamental mapping properties of the associated boundaryintegral operators as well as the solvability theory was developed by Costabel [14]and Hsiao & Saranen [29].

The first theoretical result concerning the spline collocation approximation ofthe single layer heat operator equation was in a conference paper by Hamina &Saranen, [24]. The given results cover the case where the spatial domain is a disk.Later, Hamina & Saranen in paper I analysed a method where the trial func-tions were tensor products of piecewise linear splines defined on a quasi-uniformspace-time mesh. Stability and suboptimal convergence of the method as well asnumerical results were reported. The extension of the approach to more generalboundary curves remained to be done. In their conference article, Hamalainen& Saranen [32] reported a compact perturbation technique which generalizes the

13

analysis to more general boundary curves, provided that the parametrization iswith respect to the arc length parameter. A complete discussion of this topic ispresented by Hamalainen [31].

In paper II Hamina suggested a Petrov-Galerkin approach, where interpola-tion was used with respect to the space variable, and orthogonal projection wasused with respect to the time variable. He applied the method to the single layerheat operator equation as well as to the hypersingular heat operator equation. Inboth cases, quasi-uniform meshes are allowed. For the single layer heat operatorequation (paper II), the trial functions were tensor products of piecewise linear(space) and piecewise constant splines. The hypersingular heat operator decreasessmoothness. Therefore, numerical approximation of the solution of the hypersin-gular heat operator equation (paper III) requires a smoother boundary density.The lowest order trial functions are tensor products of piecewise cubic (space) andpiecewise linear splines. In articles II and III, stability and suboptimal convergenceof the resulting Petrov-Galerkin method was proved. In the case of the hypersin-gular heat operator equation, a stability condition between the mesh parametersappears in contrast to the case of the single layer heat operator equation which isunconditionally stable.

It is remarkable here that, in the one-dimensional case, optimal order error es-timates with respect to the L2-norm are available, in contrast to the multidimen-sional and time-dependent cases where the proof of optimal order error estimatesis not known.

Another line of research started when Ruotsalainen & Wendland [53] converteda nonlinear Neumann boundary value problem for the Laplacian to a nonlinearboundary integral equation. They proved that the resulting nonlinear integral e-quation is uniquely solvable, and the spline Galerkin boundary element method isstable and convergent. Their approach was based on the theory of monotone oper-ators. Later, the idea was extended to the collocation method by Ruotsalainen &Saranen [52], and to Lp-theory by Eggermont & Saranen [20]. In their conferencepaper Hamina, Ruotsalainen & Saranen [23] proposed an approximate colloca-tion scheme such that the nonlinearity is replaced by its L2-orthogonal projection.An asymptotic error estimate of order O(h

32 ) with respect to the H

12 -norm was

presented for the solution of the resulting approximate collocation equation. Anumerical example was also given. The proofs, as well as several extensions, -optimal order L2-convergence, the effect of the numerical integration - were pub-lished in paper IV. The proof of the L2-convergence is based on the results bySaranen [55].

Ruotsalainen [48] considered a nonlinear mixed boundary value problem for theLaplacian in the plane. He converted the problem into a system of boundary in-tegral equations and proved quasi-optimality estimates for the Galerkin boundaryelement solutions for this nonlinear operator equation. In article [49] Ruotsalainenadapted the analysis to cases where the nonlinearity satisfies a polynomial growthcondition. He considered the indirect formulation which transforms the boundaryvalue problem for the Laplacian into a nonlinear boundary integral equation. Theresults on stability and optimal order error estimates for the Galerkin method inLp spaces are given. The corresponding convergence results for the collocation

14

method are presented in Ruotsalainen [50].Analogously to the elliptic case, the Neumann problem for the homogeneous

heat equation with vanishing initial data can be reduced to a nonlinear boundaryintegral equation. Therefore, it was natural to extend the linear theory to certainnonlinear problems by applying the theory of monotone operators to boundaryintegral operators acting on anisotropic Sobolev spaces. Hsiao & Saranen [30]obtained stability and optimal order convergence for the Galerkin approximation.Ruotsalainen & Saranen [52] presented convergence results for the Galerkin ap-proximation when the nonlinearity had a polynomial growth condition. In paperV Hamina proposed a method where interpolation is used with respect to the s-pace variable and orthogonal projection is used with respect to the time variable.The trial functions are tensor products of piecewise linear (space) and piecewiseconstant splines. Hamina obtained stability and optimal order convergence forthe resulting Petrov-Galerkin approximation. In contrast to the above mentionedPetrov-Galerkin method, stability and convergence of the collocation method isstill an open problem.

1.2 Boundary integral formulations

Considering the heat conduction problem, we first introduce a boundary integralapproach for solution of the homogeneous heat equation with the given nonlinearNeumann type boundary condition and vanishing initial data. Let Ω ⊂ R2 be abounded domain with the smooth boundary Γ = ∂Ω. With 0 < T < ∞, we havethe heat conduction problem, (see [30], [52] and paper V) ∂tΦ−∆Φ = 0, in QT = Ω× (0, T ),

∂nΦ(x, t) = −F (x, t,Φ(x, t)) + gNΓ(x, t), on ΣT = Γ× (0, T ),Φ(x, 0) = 0, x ∈ Ω.

(1.1)

In the stationary case, the model problem reduces to the following potential equa-tion with a given nonlinear Neumann type boundary condition (see [53], [51] andpaper IV)

∆Φ = 0, in Ω,∂nΦ(x) = −F (x,Φ(x)) + gNΓ(x), on Γ. (1.2)

Another time dependent model problem is the homogeneous heat equation withthe given Dirichlet type boundary condition and vanishing initial data (see [14],[29], and paper I) ∂tΦ−∆Φ = 0, in QT = Ω× (0, T ),

Φ(x) = gDΓ(x, t), on ΣT = Γ× (0, T ),Φ(x, 0) = 0, x ∈ Ω.

(1.3)

The classical single layer and double layer heat potentials V and W are definedby the expressions

15

(Vσ)(x, t) =

t∫0

∫Γ

σ(y, τ)E(x− y, t− τ)dΓy dτ , ((x, t) ∈ QT ∪QcT ), (1.4)

(Wµ)(x, t) =

t∫0

∫Γ

µ(y, τ)∂nyE(x− y, t− τ)dΓy dτ , ((x, t) ∈ QT ∪QcT ), (1.5)

where QcT = Ωc × (0, T ), Ωc = R2\Ω, ∂ny is the exterior normal derivative and

E(x, t) =

1

4πte− |x|

2

4t , t > 0,0, t ≤ 0,

(1.6)

denotes the fundamental solution of the two-dimensional heat equation. For suffi-ciently smooth boundary densities σ and µ, the heat potential

Φ(x, t) = (Vσ)(x, t)− (Wµ)(x, t), ((x, t) ∈ QT ), (1.7)

combined with the boundary behaviour of the single layer and double layer poten-tial and their normal derivatives ∂nV, ∂nW yields

Φ|ΣT = SΓσ + (12I −DΓ)µ (1.8)

∂nΦ|ΣT = ( 12I +D′Γ)σ +HΓµ. (1.9)

This is, in fact, a compact way to write the well known boundary behaviour

Vσ|ΣT = SΓσ, (1.10)Wµ|ΣT = (DΓ − 1

2I)µ, (interior limit). (1.11)

Here, the boundary integral operators SΓ, DΓ, D′Γ and HΓ with correspondingrepresentations are the single layer heat operator

(SΓσ)(x, t) =

t∫0

∫Γ

σ(y, τ)E(x− y, t− τ)dΓy dτ , (x ∈ Γ) (1.12)

the double layer heat operator

(DΓµ)(x, t) =

t∫0

∫Γ

µ(y, τ)∂nyE(x− y, t− τ)dΓy dτ , (x ∈ Γ) (1.13)

the spatial adjoint of the double layer heat operator

(D′Γσ)(x, t) =

t∫0

∫Γ

σ(y, τ)∂nxE(x− y, t− τ)dΓy dτ , (x ∈ Γ) (1.14)

16

and the hypersingular heat operator

(HΓµ)(x, t) = −∂nx

t∫0

∫Γ

µ(y, τ)∂nyE(x− y, t− τ)dΓy dτ , (1.15)

= −t∫

0

∫Γ

µ(y, τ)∂nx∂nyE(x− y, t− τ)dΓy dτ , (x ∈ Γ).

The direct boundary integral method is, in the case of the Dirichlet problem (1.3),based on the heat potential

Φ = Vσ −WgDΓ (1.16)

combined with the boundary relation (1.8). This approach yields the single layerheat operator equation of the first kind (see papers I and II)

SΓσ = ( 12I +DΓ)gDΓ. (1.17)

For the linear Neumann problem (1.1) with F = 0, the heat potential

Φ = VgNΓ −Wµ (1.18)

together with the boundary relation (1.9) yields the hypersingular heat operatorequation of the first kind (see [29] and paper III)

HΓµ = ( 12I −D

′Γ)gNΓ. (1.19)

In the following, the Nemitsky operator is defined by F(µ)(x, t) = F (x, t, µ(x, t)).The properties of the nonlinearity are given later after fixing the spaces of func-tions. In the direct boundary integral method, the heat potential Φ in (1.1) isgiven by means of the representation

Φ = V(gNΓ −F(µ))−Wµ, (1.20)

which by (1.10) and (1.11) yields the nonlinear boundary integral equation of thesecond kind, (see [30] and paper V)

( 12I +DΓ)µ+ SΓF(µ) = SΓgNΓ. (1.21)

We observe that the linear case F = 0 reduces to the linear boundary integralequation of the second kind

( 12I +DΓ)µ = SΓgNΓ, (1.22)

which has a well established theory [14], [16], [29], [40].In certain anisotropic Sobolev spaces, equations (1.17), (1.19), (1.21) and (1.22)

are uniquely solvable and the solutions admit the interpretations

Φ∣∣ΣT

= µ, (1.23)

∂nΦ∣∣ΣT

= σ. (1.24)

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On the other hand, the indirect single layer potential approach

Φ = V(σ), (1.25)

yields for the boundary density the nonlinear boundary integral equation

( 12I +D′Γ)σ + F(SΓσ) = gNΓ. (1.26)

This approach has been applied to the case of the boundary value problem of theLaplacian with a nonlinear boundary condition [48], [49], [50] as well as to thecase of the boundary and initial value problem of the heat equation [52]. In thesearticles, unique solvability of the resulting nonlinear operator equation has beenestablished, as well as stability and convergence of the chosen numerical methodin appropriate spaces of functions.

1.3 Reduction to an integral equation

For linear problems, we perform the analysis in spaces of functions which areperiodic with respect to the space variable. In the case of nonlinear problems, thisreduction is not used.

We assume that the boundary curve has the smooth, regular, one-periodic para-metric representation x : R→ Γ such that the Jacobian |x′(θ)| is strictly positive.Let R2

T = R× (0, T ). Then, the associated heat operators can be written as

(Su)(θ, t) =

t∫0

1∫0

u(ϕ, τ)E(x(θ)− x(ϕ), t− τ)|x′(ϕ)|dϕ dτ ,

(Du)(θ, t) =

t∫0

1∫0

u(ϕ, τ)nϕ · (x(θ)− x(ϕ))

2(t− τ)E(x(θ)− x(ϕ), t− τ)|x′(ϕ)|dϕ dτ ,

(D′u)(θ, t) =

t∫0

1∫0

u(ϕ, τ)nθ · (x(θ)− x(ϕ))

2(t− τ)E(x(θ)− x(ϕ), t− τ)|x′(ϕ)|dϕ dτ ,

(Hu)(θ, t) = −t∫

0

1∫0

u(ϕ, τ)∂n(θ)∂n(ϕ)E(x(θ)− x(ϕ), t− τ)|x′(ϕ)|dϕ dτ .

We emphasize that the kernel function corresponding to the hypersingular heatoperator has strong singularity, and appropriate regularization methods are neededin actual numerical computations.

Using the notations gD(θ, t) = gDΓ(x(θ), t), gN(θ, t) = gNΓ(x(θ), t), we ob-tain (SΓσ)(x(θ), t) = (Su)(θ, t), with u(θ, t) = σ(x(θ), t) and (HΓµ)(x(θ), t) =(Hu)(θ, t) with u(θ, t) = µ(x(θ), t). Thus, equations (1.17) and (1.19) on the

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boundary transform to ordinary integral equations

(Su)(θ, t) = 12gD(θ, t) + (DgD)(θ, t), (1.27)

(Hu)(θ, t) = 12gN(θ, t)− (D′gN)(θ, t), (1.28)

over the domain (θ, t) ∈ R× [0, T ].We recall that sufficiently smooth functions, which are one-periodic with respect

to variable θ, admit the Fourier representation

u(θ, t) =1

2π

∑n∈Z

∫R

u(n, η)ein2πθ+iηt dη,

where the Fourier coefficient is defined by u(n, η) =1∫0

∫R

u(θ, t)e−in2πθ−iηt dt dθ.

In terms of the Heaviside step function H, the heat kernel corresponding to thecircular boundary of radius ρ, defined by

Eρ(θ − ϕ, t− τ) = H(t− τ)exp

(− ρ2 sin2 π(θ−ϕ)

(t−τ)

)4π(t− τ)

,

is a function of the difference of the coordinates. Therefore, the associated bound-ary integral operators Sρ, Dρ, D′ρ and Hρ are of a convolutional type

(Sρu)(θ, t) = 12π

∑n∈Z

∫R

s(n, η)u(n, η)ein2πθ+iηt dη, (1.29)

(Dρu)(θ, t) = 12π

∑n∈Z

∫R

d(n, η)u(n, η)ein2πθ+iηt dη, (1.30)

(D′ρu)(θ, t) = 12π

∑n∈Z

∫R

d′(n, η)u(n, η)ein2πθ+iηt dη, (1.31)

(Hρu)(θ, t) = 12π

∑n∈Z

∫R

h(n, η)u(n, η)ein2πθ+iηt dη. (1.32)

As an example, we give here the explicit representation formula of the singlelayer heat operator

(Sρu)(θ, t) =

t∫0

1∫0

u(ϕ, τ)exp

(− ρ2 sin2 π(θ−ϕ)

(t−τ)

)4π(t− τ)

ρ dϕdτ

= ρ2π

+∞∫−∞

1∫0

u(ϕ, η)K0(2ρ√

iη sinπ|θ − ϕ|) dϕ dη

= ρ2π

∑n∈Z

∫R

In(ρ√

iη) Kn(ρ√

iη)u(n, η)ein2πθ+iηt dη.

Here In and Kn are the modified Bessel functions, [1]. See also [39]. For thehypersingular heat operator, representation (1.32) follows from the fact that thekernel

−∂n(θ)∂n(ϕ)Eρ(θ−ϕ, t− τ) = −∂n(θ)∂n(ϕ)

(exp(− ρ2 sin2 π(θ−ϕ)

t−τ)

4π(t− τ))

= h(θ−ϕ, t− τ)

is a function of the difference of the coordinates. Similar considerations are truefor Dρ and D′ρ.

19

1.4 Spaces

Let Hs(Γ), s ∈ R (see [2]) be the usual Sobolev space equipped with the norm

‖u‖s = (u , u)12s . In particular, L2(Γ) = H0(Γ) and (u , v)0,Γ =

∫Γu(y)v(y)dΓy.

For s > 0 the norm is given by

‖u‖2s =m∑k=0

‖Dku‖20 +∫Γ

∫Γ

|Dmu(x)−Dmu(y)|2

|x− y|1+2δdΓxdΓy,

where s = m + δ, 0 < δ < 1 and the derivative is taken with respect to the arclength. The negative order Sobolev spaces are defined by duality with respect tothe L2- inner product.

Theoretical analysis of the heat equation leads to the concept of anisotropicSobolev spaces. Here, we concentrate on the anisotropic spaces on the space-timeboundary. These are defined as intersection spaces ([33] p. 87, [34] p. 8).

Hr,s(ΣT ) = H0((0, T );Hr(Γ)) ∩Hs((0, T );H0(Γ)), (r, s ≥ 0).

In particular, we describe the vanishing initial condition in terms of the subspace

Hr,s00 (ΣT ) = u | u = U |ΣT : U ∈ Hr,s(ΣT ), U(·, t) = 0, t < 0, (0 < T <∞).

The negative order spaces H−r,−s00 (ΣT ) for 0 < r < 1, 0 < s < 12 are defined by

duality H−r,−s00 (ΣT ) = (Hr,s00 (ΣT ))′.

Having introduced the parametric representation for the boundary curve, itis enough to consider functions which are one-periodic with respect to the spatialvariable. First, for any r ∈ R, let Hr be the Sobolev space of one-periodic functionson R. The anisotropic spaces are defined by

Hr,s := H0(R;Hr) ∩Hs(R;H0), (r, s ≥ 0).

In terms of the Fourier transform with respect to the time variable

(Ftu)(θ, η) =

∞∫−∞

u(θ, t)e−iηt dt,

the norm of the space Hr,s is given by

‖u‖2Hr,s =

∞∫−∞

(‖(Ftu)(·, η)‖2Hr + |η|2s‖(Ftu)(·, η)‖2H0) dη. (1.33)

The space Hr,s(R2T ), r, s ≥ 0, 0 < T ≤ ∞ is the space of restrictions to R2

T offunctions belonging to Hr,s equipped with the infimum norm

Hr,s(R2T ) = u | u = U |

R2T

: U ∈ Hr,s , (1.34)

‖u‖Hr,s(R2T ) = inf ‖U‖Hr,s | u = U |

R2T. (1.35)

20

Again, we have the Hilbert space

Hr,s(R2T ) = H0((0, T );Hr) ∩Hs((0, T );H0), (0 < T ≤ ∞)

endowed with the norm

‖u‖2Hr,s(R2T ) =

T∫0

‖u(·, t)‖2Hr dt+ ‖u‖2Hs((0,T );H0), (0 < T <∞).

We also use equivalent norms defined by

‖u‖2Hr,s(R2T ) =

T∫0

‖u(·, t)‖2Hr dt+

T∫0

T∫0

‖u(·, t)− u(·, τ)‖2H0

|t− τ |1+2sdt dτ , (0 < s < 1)

and

‖u‖2Hr,s(R2T ) =

T∫0

‖u(·, t)‖2Hr dt+

1∫0

‖u(θ, ·)‖2Hs(0,T ) dθ, (0 ≤ s ≤ 1). (1.36)

Moreover, we introduce the subspace

Hr,s00 (R2

T ) = u | u = U |R

2T

: U ∈ Hr,s, U(·, t) = 0, t < 0, (0 < T <∞).

Thus, for example Hr,s00 (R2

∞) is the space of those functions in Hr,s(R2∞) for which

the zero extension with respect to the time variable remains in Hr,s. Finally, weneed the negative order space H−r,−s00 (R2

T ) for 0 < r < 1, 0 < s < 12 , which is

defined by duality H−r,−s00 (R2T ) = (Hr,s

00 (R2T ))′.

Next, we consider some sets of continuous (respectively smooth) functions,which are one-periodic with respect to the first argument. Let C1(R2) (resp.C∞1 (R2)) be the space of continuous (resp. infinitely smooth) functions. The spaceC10(R2) (resp. C∞10(R2)) consists of continuous (resp. infinitely smooth) functionsu, which are defined on R2, such that the support supp u(θ, ·) ⊂ K, where K ⊂ Ris a compact set. The spaces of restrictions C1(R2

T ) (resp. C∞1 (R2T )) consist of con-

tinuous (resp. infinitely smooth) functions such that u = U |R

2T

. The space C10(R2T )

(resp. C∞10(R2T )) consists of continuous (resp. infinitely smooth) functions u, which

are defined on R2T , such that the support supp u(θ, ·) ⊂ K, where K ⊂ R is a

compact set. In order to describe the initial condition, we introduce the subspace

C∞00(R2T ) = u | u = U |

R2T

: U ∈ C∞1 (R2), U(·, t) = 0, t < 0, (0 < T ≤ ∞).

This subspace enjoys the property that the embedding C∞00(R2T ) ⊂ Hr,s

00 (R2T ) is

dense for all r, s ∈ R, if s ≥ 0.A comprehensive proof of the following Sobolev embedding theorem is given in

[31] or [32]. For the convenience of the reader the proof is repeated here.

Theorem 1 In the anisotropic spaces the embeddings

Hr, r2 (R2) ⊂ C(R2), Hr, r2 (R2T ) ⊂ C(R2

T ), Hr, r200 (R2

T ) ⊂ C00(R2

T )

are continuous provided that r > 32 .

21

Proof. We present the proof for Hr, r2 = Hr, r2 (R2). Note that the anisotropicnorm (1.33) is given by

‖U‖2Hr,

r2

=∑n∈Z

∫R

(1 + |n|+ |η| 12 )2r|U(n, η)|2 dη, (U ∈ Hr, r2 ).

Since C∞10(R2) is dense in Hr, r2 , r ≥ 0, it is enough to present the proof for U ∈C∞10(R2). The Cauchy-Schwarz inequality applied to the Fourier representation

U(θ, t) =1

2π

∑n∈Z

∫R

U(n, η)ein2πθ+iηt dη

gives

|U(θ, t)| ≤ 12π

∑n∈Z

∫R

|U(n, η)|dη

=1

2π

∑n∈Z

∫R

(1 + |n|+ |η| 12 )−r(1 + |n|+ |η| 12 )r|U(n, η)|dη

≤ 12π

√√√√∑n∈Z

∫R

(1 + |n|+ |η| 12 )−2r

√√√√∑n∈Z

∫R

(1 + |n|+ |η| 12 )2r|U(n, η)|2 dη.

The result follows, since the function I(r) =∑n∈Z

∫R

(1 + |n|+ |η| 12 )−2r is bounded,

if r > 32 . The other embedding results follow by applying the infimum characteri-

zation of the norm for the subspaces of restrictions.

The anisotropic Sobolev spaces enjoy a similar compactness property as ordi-nary Sobolev spaces.

Theorem 2 Assume that 0 ≤ r1 ≤ r and 0 ≤ s1 ≤ s. Then the embeddingHr,s

00 (R2T ) ⊂ Hr1,s1

00 (R2T ) is compact for any finite T > 0.

The proof is given in article [29], Theorem 3.1 p. 93.

1.5 Mapping properties

The convergence analysis and error estimates are based on the mapping propertiesof the heat operators in the anisotropic Sobolev spaces. In this work, we justquote the results. The proofs are lengthy and technical, and can be found in thegiven articles. Throughout this work c, c′, ci etc. are generic constants, whichmay change in size, but are always independent of the mesh parameters. In thefollowing, the duality pairing extends the L2(R2

T )-inner product.

22

Theorem 3 The single layer heat operator S : Hr, r200 (R2

T ) → Hr+1, r+1

200 (R2

T ) is anisomorphism for all r ≥ − 1

2 . Furthermore, it is coercive such that

(Sw , w) ≥ c ‖w‖2− 12 ,−

14

for all w ∈ H−12 ,−

14

00 (R2T ). (1.37)

For the proof see [3], [37], [14], [28]. The mapping properties for the double layerheat operator and its spatial adjoint are given in

Theorem 4 The mapping properties of the operators D and D′ are:

(i) D,D′ : Hr, r2 (R2T )→ Hr+1, r+1

2 (R2T ) are bounded for all r ≥ − 1

2 .

(ii) D,D′ : Hr, r2 (R2T )→ Hr, r2 (R2

T ) are compact for all r ≥ − 12 .

(iii) For any a 6= 0, r ≥ − 12 the operators aI+D, aI+D′ : Hr, r2 (R2

T )→ Hr, r2 (R2T )

are isomorphisms.

This result is proved in paper [29], Theorem 4.2. See also [14], Corollary 3.14.

Theorem 5 The hypersingular heat operator H : Hr, r200 (R2

T ) → Hr−1, r−1

200 (R2

T ) isan isomorphism for r ≥ 1

2 . Furthermore, it is coercive such that(Hw , w

)≥ c ‖w‖21

2 ,14

for all w ∈ H12 ,

14

00 (R2T ). (1.38)

The proof is given in [29], Theorem 4.4 or [14], Corollary 3.13. The followingcoercivity estimate is used for the analysis of the nonlinear boundary integralequation. The proof can be found in [30], Theorem 4.1.

Theorem 6 The operator S−1Γ ( 1

2I + DΓ) : H12 ,

14

00 (ΣT ) → H− 1

2 ,−14

00 (ΣT ) is an iso-morphism. Further, it is strongly coercive, i.e. there exists a positive constant csuch that(

S−1Γ ( 1

2I +DΓ)w , w)

ΣT≥ c ‖w‖21

2 ,14

for all w ∈ H12 ,

14

00 (ΣT ). (1.39)

The kernel functions of the heat operators are, in the case of circular bound-aries, functions of the difference of the coordinates. Therefore, the heat operatorsformally commute with the partial differentation operator. Also, the mean valuefunctional (J w)(t) :=

∫ 1

0w(θ, t) dθ commutes with the heat operators.

Theorem 7 Let w ∈ Hr, r200 (R2

T ).

For r ≥ s− 12 , ∂

sθ(Sρw)(θ, t) = (Sρ(∂sϕw))(θ, t), s = 1, 2, . . . (1.40)

For r ≥ s+ 12 , ∂

sθ(Hρw)(θ, t) = (Hρ(∂sϕw))(θ, t), s = 1, 2, . . . (1.41)

Moreover, J (Sρw)(t) = Sρ(J w)(t) and J (Hρw)(t) = Hρ(J w)(t).(1.42)

For the moment, we adopt some notations and results from [31] and [32]. LetS be the single layer heat operator for the circle with radius ρ

2π . We define

∂θ := ∂θ + J,

Kw := (S − S)w, Kw = ∂θK∂−1θ ,

Sw := Sw +Kw.

23

Theorem 8 Assume that the one-periodic parametric representation of the bound-ary curve is chosen such that |x′(θ)| = ρ = constant > 0. Let 0 < T < ∞. Thenthe operator S admits the representation S = ∂θS∂

−1θ , and we have the mapping

properties

S : H−12 ,−

14

00 (R2T )→ H

12 ,

14

00 (R2T ) is an isomorphism,

K : Hr, r200 (R2

T )→ Hr+2, r+2

200 (R2

T ) continuous, if r ≥ − 12 .

The proof of these results can be found in [31], Section 3. It is based on explicitestimation of the symbol of the perturbation operator K. The continuity propertiesof the operator K combined with the compactness result given in Theorem 2, showthat the operator K can be interpreted as a compact perturbation of the principalpart S. Thus, a standard theory of compact perturbations is available.

We emphasize that this technique generalizes the analysis of our Petrov-Galerkinmethods to general, smooth boundary curves with the restriction that the para-metric representation is with respect to the arc length parameter. However, theseaspects are not explicitly shown here.

1.6 Spline spaces

Next, we discuss the space of approximations. Let 0 = θ0 < θ1 < . . . < θN = 1 bea one-periodic mesh, with h = maxθn+1− θn. The mesh is called quasi-uniformif

maxθn+1 − θnminθn+1 − θn

≤ C,

where C ≥ 1 is a constant. We denote by Sdθh ([0, 1]) the space of one-periodic,smoothest splines of degree dθ. Note that Sdθh ([0, 1]) is a N -dimensional linearspace with basis functions ψdθn Nn=1. Analogously, we define the quasi-uniformgrid 0 = t0 < t1 < . . . < tM = T , k = maxtm+1 − tm. With respect to thetime variable, we consider the lowest order spaces Sdtk,[0,T ]. In practice, these arethe spaces of piecewise constant splines generated by the basis functions (m =1, . . . ,M)

φ0m(t) =

1, if tm−1 < t < tm,0, otherwise,

or the spaces of piecewise linear continuous splines generated by the basis functions(m = 1, . . . ,M)

φ1m(t) =

1 + t−tm

tm−tm−1, if tm−1 < t < tm,

1− t−tmtm+1−tm , if tm ≤ t < tm+1,

0, otherwise.

We draw the attention of the reader to the fact that the elements of the spaceϕ ∈ S1

k,[0,T ] satisfy the initial condition ϕ(0) = 0. The trial functions on the

24

space-time boundary are tensor products

Sdθ,dth,k (R2T ) := Sdθh ([0, 1])× Sdtk,[0,T ] =

v∣∣ v(θ, t) =

M∑m=1

N∑n=1

αm,nψdθn (θ)φdtm(t)

,

where ψdθn Nn=1 and φdtmMm=1 are the one-dimensional basis functions. We noticethat the inclusion

Sdθ,dth,k (R2T ) ⊂ Hr, r2

00 (R2T ), (r < mindθ + 1

2 , 2dt + 1),

is generally valid for the tensor product spline spaces. Our methods use pointwisevalues. Therefore, it is essential that the images Su and Hu are continuous. In thelight of the mapping properties combined with the Sobolev embedding theorem,we obtain appropriate ranges for Sobolev indices

12 < r < mindθ + 1

2 , 2dt + 1 for the single layer heat operator,52 < r < mindθ + 1

2 , 2dt + 1 for the hypersingular heat operator.

1.6.1 Inverse properties

We now recall some basic inverse and approximation results. For quasi-uniformmeshes θn and tm, the one-dimensional inverse estimates

‖ψ‖r ≤ c h−(r−s)‖ψ‖s, (ψ ∈ Sdθh , s ≤ r < dθ + 12 ),

‖φ‖r ≤ c k−(r−s)‖φ‖s, (φ ∈ S0k,[0,T ], s ≤ r < 1

2 ),‖φ‖r ≤ c k−(r−s)‖φ‖s, (φ ∈ S1

k,[0,T ], s ≤ r ≤ 1),

are available [21]. In the anisotropic spaces, the following inverse estimates arevalid

‖v‖r, r2 ≤ c max(h−r, k−r/2

)‖v‖0,0, (v ∈ S0,0

h,k, 0 ≤ r < 12 ), (1.43)

‖v‖0,0 ≤ c max(h−r, k−r/2

)‖v‖−r,− r2 , (v ∈ S0,0

h,k, 0 ≤ r ≤ 12 ), (1.44)

‖v‖r, r2 ≤ c max(h−r, k−r/2

)‖v‖0,0, (v ∈ S1,0

h,k, 0 ≤ r < 1), (1.45)

‖v‖0,0 ≤ c max(h−r, k−r/2

)‖v‖−r,− r2 , (v ∈ S1,0

h,k, 0 ≤ r < 1), (1.46)

‖v‖r, r2 ≤ c max(h−r, k−r/2

)‖v‖0,0, (v ∈ S1,1

h,k, 0 ≤ r < 1), (1.47)

‖v‖0,0 ≤ c max(h−r, k−r/2

)‖v‖−r,− r2 . (v ∈ S1,1

h,k, 0 ≤ r < 1). (1.48)

Note that estimates (1.43) and (1.44), as well as the other two pairs, can becombined to a single estimate. For the sake of simplicity, the use of two estimatesis preferred. We recall here the proof given in paper I, Lemma 5.2 p. 56-57.

25

Proof. Let v ∈ S1,1h,k and 0 ≤ r < 1. The representation formula (1.36) of the

anisotropic norm, together with one-dimensional inverse estimates yields (1.47)

‖v‖2r, r2 ≤ cT∫0

‖v(·, t)‖2Hr dt+ c1∫0

‖v(θ, ·)‖2Hr2 (0,T )

dθ

≤T∫0

c h−2r‖v(·, t)‖2H0 dt+1∫0

c k−r‖v(θ, ·)‖2H0(0,T ) dθ

≤ c(h−2r + k−r

)‖v‖20,0 ≤ c

(max(h−r, k−

r2 ))2‖v‖20,0.

We use the duality H−r,−r2

00 (R2T ) =

(Hr, r200 (R2

T ))′ for 0 ≤ r < 1. Then, the Schwarz

inequality implies

‖v‖20,0 ≤ ‖v‖r, r2 ‖v‖−r,− r2 ≤ c max(h−r, k−r2 )‖v‖−r,− r2 ‖v‖0,0,

which yields (1.48). The inverse estimates (1.45), (1.46) for 0 ≤ r < 1 and (1.43),(1.44) for 0 ≤ r < 1

2 follow analogously. To see (1.44) for r = 12 we use

‖v‖0,0 ≤ cmax(h−14 , k−

18 )‖v‖− 1

4 ,−18, (v ∈ S0,0

h,k), (1.49)

which further gives

‖v‖2− 14 ,−

18≤ ‖v‖0,0‖v‖− 1

2 ,−14

≤ c max(h−14 , k−

18 )‖v‖− 1

4 ,−18‖v‖− 1

2 ,−14,

and therefore‖v‖− 1

4 ,−18≤ c max(h−

14 , k−

18 )‖v‖− 1

2 ,−14.

Using this with (1.49) we obtain (1.44) with r = 12 .

1.6.2 Approximation properties

The associated one-dimensional orthogonal projection operator P dθh : H0 → Sdθh isdefined by (P dθh u , v) = (u , v) for all v ∈ Sdθh . The following approximation resultis true for periodic splines, [21],

‖u− P dθh u‖Hr ≤ c hs−r‖u‖Hs , (u ∈ Hs, 0 ≤ r ≤ s ≤ dθ + 1, r < dθ + 12 ).

With respect to the time variable, the orthogonal projection P dtk : H0(0, T ) →Sdtk,[0,T ] is defined analogously. Now, we restrict ourselves to the approximationproperties

‖u− P 0ku‖Hr(0,T ) ≤ c ks−r‖u‖Hs(0,T ), (u ∈ Hs(0, T ), 0 ≤ r ≤ s ≤ 2, r < 1

2 )‖u− P 1

ku‖Hr(0,T ) ≤ c ks−r‖u‖Hs(0,T ), (u(0) = 0, u ∈ Hs(0, T ), 0 ≤ r ≤ 1).

26

In the piecewise linear case, u satisfies the initial condition u(0) = 0 and s = 1or s = 2. For r = 0 or r = 1, the result is a standard error bound for thepiecewise linear case. The interpolation inequality yields the given estimate for0 < r < 1. Note that in the piecewise constant case, no initial condition isrequired. In the following, we use the same notation for two-dimensional andone-dimensional interpolation and projection operators e.g. P dθh = P dθh ⊗ I.

The two-dimensional L2-projection P dθ,dth, k : L2(R2T ) → Sdθ,dth,k (R2

T ) is definedby requiring (P dθ,dth, k u , v) = (u , v) for all v ∈ Sdθ,dth,k (R2

T ). We have also therepresentation P dθ,dth, k = P dθh ⊗ P

dtk = P dtk ⊗ P

dθh . The low order projections have

the approximation property

‖u− P 1,0h, ku‖r, r2 ≤ c

(hs−r + k(s−r)/2)‖u‖s, s2 , (u ∈ Hs, s2

00 (R2T )), (1.50)

where 0 ≤ r ≤ s ≤ 2, r < 1. For the higher order cases, we restrict ourselves tothe following basic bounds

‖u− P 1,1h, ku‖r, r2 ≤ c

(hs−r + k(s−r)/2)‖u‖s, s2 , (0 ≤ r ≤ 2, s = 1 or s = 2),(1.51)

‖u− P 3,1h, ku‖r, r2 ≤ c

(hs−r + k(s−r)/2)‖u‖s, s2 , (0 ≤ r ≤ 2, s = 2 or s = 4),(1.52)

for u ∈ Hs, s200 (R2

T ). Compare also [14], and [37].The spline interpolation makes sense only when the associated functions are

continuous. According to the Sobolev embedding theorem this is guaranteed ifu ∈ Hr, r > 1

2 . The interpolation operator Idθh : Hr → Sdθh , r > 12 , is defined as

Idθh u ∈ Sdθh(Idθh u)(θi) = u(θi), i = 1, 2, . . . , N,

where θi are the usual interpolation points

θi =θi for all i, when dθ is odd12 (θi−1 + θi) for all i, when dθ is even.

In this work, we apply only odd degree interpolation. In the periodic case, we havefor quasi-uniform meshes the approximation property [21]

‖u− Idθh u‖Hr ≤ c hs−r‖u‖Hs , (0 ≤ r ≤ s ≤ 2, r < dθ + 12 , s >

12 ). (1.53)

Our Petrov-Galerkin approach is to use interpolation with respect to the spacevariable and L2-projection with respect to the time variable. For this, we define thetwo-dimensional projection operator Qdθ,dth,k := Idθh ⊗P

dtk = P dtk ⊗I

dθh . The following

approximation property is used for the analysis of the nonlinear problems.

Lemma 1 Let the meshes be quasi-uniform. Then we have the estimates

‖u−Q1,0h,ku‖0,0 ≤ c

(hs + k

s2)‖u‖s, s2 , (u ∈ Hs, s2

00 (R2T )), 1

2 < s ≤ 2),(1.54)

‖u−Q1,0h,ku‖ 1

2 ,14≤ c

(h

12 + k

14)‖u‖1, 12 , (u ∈ H1, 12

00 (R2T )). (1.55)

27

Proof. We use the decomposition

(I −Q1,0h,k)u = (I − P 0

k)u− P 0k(I − I1

h)u (1.56)

and bound the H0,0-norm of the terms separately. For the first term, we have

‖(I − P 0k)u‖20,0 ≤

1∫0

c k2r‖u(θ, ·)‖2Hr(0,T ) dθ = c k2r‖u‖20,r, (0 ≤ r ≤ 2). (1.57)

A similar estimate combined with the L2-stability of the P 0k-operator gives

‖P 0k(I − I1

h)u‖20,0 ≤ cT∫

0

‖(I − I1h)u(·, t)‖2H0 dt = c h2r‖u‖2r,0, ( 1

2 < r ≤ 2). (1.58)

Estimate (1.54) for 12 < s ≤ 2 follows by putting together (1.56) with (1.57) and

(1.58)

‖u−Q1,0h,ku‖0,0 ≤ c k

s2 ‖u‖0, s2 + c hs‖u‖s,0 ≤ c (hs + k

s2 )‖u‖s, s2 .

Another decomposition, namely

u−Q1,0h,ku = (u− P 1,0

h, ku) + (P 1,0h, ku−Q

1,0h,ku),

inverse estimate (1.45) and the approximation properties of the projection opera-tors yield (1.55).

1.7 Some Petrov-Galerkin approximations

For the single layer heat operator equation Su = f , we have analysed severalmethods. These include collocation methods, as well as Petrov-Galerkin typemethods, where collocation is used with respect to the space variable, and theGalerkin method with respect to the time variable. In the case of the hypersingularheat operator equation Hu = f , only the latter approach is used. The purpose ofthis chapter is to describe a unified approach which covers the treatment given inarticles I, II, and III in the light of Theorem 11.

An essential device is the equivalence of the proposed projection method to acertain Galerkin method. In order to establish this equivalence, we need the meanvalue functional (J w)(t) :=

∫ 1

0w(θ, t) dθ, as well as the corresponding trapezoidal

rule approximation

(J∆w)(t) :=N∑n=1

θn+1 − θn−1

2w(θn, t). (1.59)

28

We define the bilinear forms

Bjθ,jt(w, v) =(∂jtt (∂jθθ + J )w , ∂jtt (∂jθθ + J )v

)(1.60)

Bjθ,jt∆ (w, v) =(∂jtt (∂jθθ + J∆)w , ∂jtt (∂jθθ + J )v

). (1.61)

In particular, we interpret here ∂0t u = u. Throughout this work, the index jθ is

related to the degree of the spline approximation by the formula dθ = 2jθ − 1.With respect to the time index, this convention is not applicable. In practice, jttakes the values jt = 0 or jt = 1. Using the orthogonality conditions

(∂sθu , J u) = 0, (∂sθu , J∆u) = 0, s = 1, 2, . . . ,

we may rewrite the bilinear forms

Bjθ,jt(w, v) =(∂jtt ∂

jθθ w , ∂

jtt ∂

jθθ v)

+(∂jtt J w , ∂

jtt J v

)Bjθ,jt∆ (w, v) =

(∂jtt ∂

jθθ w , ∂

jtt ∂

jθθ v)

+(∂jtt J∆w , ∂

jtt J v

).

The next step is to define a family of seminorms. For brevity, we introduce thenotation

|||v|||jθ,jt;r = ‖∂jtt ∂jθθ v‖r, r2 + ‖∂jtt J v‖r, r2 . (1.62)

The indices jθ and jt depend on the degree of the approximating tensor productsplines. The third index r depends on the order of the operator under considera-tion. In the cases of the single layer heat operator equation and the hypersingularheat operator equation, the index r takes the values r = − 1

2 , r = + 12 , respectively.

In any space of functions, where the expression ||| · |||jθ,jt;r is well defined and finite,it gives a seminorm. However, in the subspace of the approximating functions itdefines a norm. This is proved in the following Lemma.

Lemma 2 Let r ∈ R, dθ = 2jθ − 1, and jt = 0 or jt = 1. Then the mappingv 7→ |||v|||jθ,jt;r defines a norm in the space Sdθ,dth,k (R2

T ).

Proof. It suffices to show that the condition

|||v|||jθ,jt;r = 0, for v ∈ Sdθ,dth,k (R2T ) (1.63)

implies v(θ, t) ≡ 0. We consider only the case jt = 1 (dt = 1), since the other casejt = 0 (dt = 0) is a simplified version of the proof. Assuming (1.63), we obtainfrom the basis representation

(∂t∂jθθ v)(θ, t) = ∂t

( M∑m=1

N∑n=1

αm,n(∂jθθ ψdθn )(θ)φdtm(t)

)(1.64)

and

(∂tJ v)(t) = ∂t

( M∑m=1

N∑n=1

αm,n(J ψdθn )φdtm(t)). (1.65)

29

According to (1.63) and (1.64) we haveM∑m=1

(N∑n=1

αm,n(∂jθθ ψdθn )(θ)

)φdtm(t) ≡ C(θ),

which, because of the initial condition φ1m(t) = 0 for all m = 1, . . . ,M , yields

N∑n=1

αm,n(∂jθθ ψdθn )(θ) ≡ 0, (m = 1, . . . ,M). (1.66)

Analogously, from (1.63) and (1.65), we deduce that

N∑n=1

αm,n(J ψdθn ) = 0, (m = 1, . . . ,M). (1.67)

Adding (1.66) and (1.67) together gives

N∑n=1

αm,n(∂jθθ + J )ψdθn = (∂jθθ + J )(N∑n=1

αm,nψdθn ) ≡ 0, (m = 1, . . . ,M).

The operator (∂jθθ + J ) is an isomorphism from Sdθh to the space S(dθ−1)/2h . Thus

N∑n=1

αm,nψdθn = 0, (m = 1, . . . ,M),

which implies that all the coefficients αm,n vanish, proving our statement.

After these preparations, we are able to reformulate the proposed projectionmethods as certain Galerkin methods. This equivalence is carried by the approxi-mating bilinear form defined in (1.61). Our next aim is to establish the coercivityof the approximating bilinear form with respect to the associated energy norm||| · |||jθ,jt;r. For our method, it is enough that this property is valid in the subspaceSdθ,dth,k (R2

T ).

1.7.1 Reformulation of the collocation problem

The lowest order spline collocation equations corresponding to the single layer heatoperator equation are the following:

find u∆ ∈ Sdθ,1h,k such that Idθ,1h,k Su∆ = Idθ,1h,k f, dθ = 1 or dθ = 3. (1.68)

For a sufficiently smooth solution to the equation Su = f , the collocation problemis equivalent to the Galerkin problem:

find u∆ ∈ Sdθ,1h,k such that Bjθ,1∆ (Su∆, v) = Bjθ,1∆ (Su, v). (1.69)

The case dθ = 1 was analysed by Hamina & Saranen in paper I for a circularboundary. Their regularity requirements were described in terms of the spaces of

30

continuous functions. Later Hamalainen & Saranen [32] and Hamalainen [31] re-duced the regularity requirements by introducing appropriate spaces of the Sobolevtype. They also generalized the analysis to cover the case of a general smoothboundary curve, provided that the parametric representation is with respect tothe arc length parameter. The case dθ = 3 has not been explicitly treated in thearticles.

1.7.2 Reformulation of the Petrov-Galerkin method

Another possibility is to use the projection operator Qdθ,dth,k . Then the lowest orderapproximations to the single layer heat operator equation are the following.

Find u∆ ∈ S1,0h,k(R2

T ) such that Q1,0h,kSu∆ = Q1,0

h,kf. (1.70)

Find u∆ ∈ S1,1h,k(R2

T ) such that Q1,1h,kSu∆ = Q1,1

h,kf. (1.71)

Find u∆ ∈ S3,1h,k(R2

T ) such that Q3,1h,kSu∆ = Q3,1

h,kf. (1.72)

The equivalent Galerkin method is now available.

Theorem 9 Let u ∈ Hs, s200 (R2

T ), s > 12 be the solution of the equation Su = f .

The function u∆ ∈ S1,0h,k(R2

T ) is a solution of problem (1.70) if and only if

B1,0∆ (Su∆, v) = B1,0

∆ (Su, v), (v ∈ S1,0h,k(R2

T )). (1.73)

The function u∆ ∈ S1,1h,k(R2

T ) is a solution of problem (1.71) if and only if

B1,0∆ (Su∆, v) = B1,0

∆ (Su, v), (v ∈ S1,1h,k(R2

T )). (1.74)

Let u ∈ Hs, s200 (R2

T ), s > 32 be the solution of the equation Su = f . Then the

function u∆ ∈ S3,1h,k(R2

T ) is a solution of problem (1.72) if and only if

B2,0∆ (Su∆, v) = B2,0

∆ (Su, v), (v ∈ S3,1h,k(R2

T )). (1.75)

The proof is similar to that of Theorem 10, and is omitted. It is remarkablehere that the required regularity for the equivalence is described in terms of theanisotropic Sobolev spaces contrary to article II, where the spaces of essentiallybounded functions were used. See also Theorem 18, formula (3.4).

The hypersingular heat operator decreases smoothness. Therefore, we haveconsidered only the lowest order method corresponding to the projection operatorQdθ,dth,k . The discrete approximation to the hypersingular heat operator equationis:

find u∆ ∈ S3,1h,k(R2

T ) such that Q3,1h,kHu∆ = Q3,1

h,kf. (1.76)

This operator equation is equivalent to the following system of equations:

31

find u∆ ∈ S3,1h,k(R2

T ) such that

T∫0

φ1m(t)(Hu∆)(θn, t) dt=

T∫0

φ1m(t)f(θn, t) dt, (n = 1, . . . , N, m = 1, . . . ,M).

(1.77)As an example, we prove that problem setting (1.76) is equivalent to a certainGalerkin problem. In this particular case, the bilinear forms are

B2,0(w, v) =(∂2θw , ∂

2θv)

+(J w , J v

)(1.78)

B2,0∆ (w, v) =

(∂2θw , ∂

2θv)

+(J∆w , J v

). (1.79)

The proof is quoted from paper III.

Theorem 10 Let u ∈ Hs, s200 (R2

T ), s > 52 be the solution of the equation Hu = f .

Then the function u∆ ∈ S3,1h,k(R2

T ) is a solution of problem (1.76) if and only if

B2,0∆ (Hu∆, v) = B2,0

∆ (Hu, v), (v ∈ S3,1h,k(R2

T )). (1.80)

Proof. We denote w = Hu∆ − f, and consider the condition

B2,0∆ (w, v) = 0, for all v ∈ S3,1

h,k(R2T ). (1.81)

It is known that the operator ∂2θ + J : S3

h → S1h is an isomorphism between

the one-dimensional periodic spline spaces. Therefore, we can choose the basisfunctions vm,n of the trial space S3,1

h,k(R2T ) such that

(∂2θ + J )vm,n(θ, t) = φ1

m(t)ψ1n(θ),

where ψ1n is the 1-periodic Courant basis function, such that for θ ∈ [0, 1]

ψ1n(θ) =

1 + (θ − θn)/(θn − θn−1), θn−1 < θ < θn,1− (θ − θn)/(θn+1 − θn), θn < θ < θn+1,0, otherwise.

Then we obtain by partial integration

B2,0∆ (w, vm,n) =

((∂2θ + J∆)w , (∂2

θ + J )vm,n)

=((∂2θ + J∆)w , φ1

mψ1n

)= −

(∂θw , φ

1m∂θψ

1n

)+(J∆w , φ

1mψ

1n

)= −

T∫0

φ1m(t)

[w(θn, t)−w(θn−1, t)

θn − θn−1− w(θn+1, t)−w(θn, t)

θn+1 − θn

]dt

+ J ψ1n

∑Nn=1

θn+1−θn−12

T∫0

φ1m(t)w(θn, t) dt.

(1.82)

32

Suppose first that u∆ satisfies the discrete equations, which means that

T∫0

φ1m(t)w(θn, t) dt = 0, for all n = 1, . . . , N,m = 1, . . . ,M. (1.83)

According to (1.82), the bilinear form B2,0∆ (w, v) reduces to a linear combination

of integrals of type (1.83) and consequently B2,0∆ (w, v) = 0 for all v ∈ S3,1

h,k(R2T ).

Conversely, assume that (1.81) is valid. For all v ∈ S3,1h,k(R2

T ) also J v ∈ S3,1h,k(R2

T ).

The identity B2,0∆ (w, Jv) =

∫ T0

(J∆w)(t)(J v)(t) dt together with (1.79) implies

T∫0

(J∆w)(t)(J v)(t) dt = 0, (v ∈ S3,1h,k(R2

T )), (1.84)

(∂2θ w , ∂

2θv) = 0, (v ∈ S3,1

h,k(R2T )). (1.85)

As in (1.82), formula (1.85) yields∫ T

0

φ1m(t)

[w(θn+1, t)− w(θn, t)

θn+1 − θn− w(θn, t)− w(θn−1, t)

θn − θn−1

]dt = 0 (1.86)

for 1 ≤ n ≤ N, 1 ≤ m ≤M . Let m be fixed. Then we have for all n = 1, ..., N∫ T

0

φ1m(t)

w(θn+1, t)− w(θn, t)θn+1 − θn

dt=∫ T

0

φ1m(t)

w(θn, t)− w(θn−1, t)θn − θn−1

dt = Cm.

(1.87)Using the periodicity of w with respect to θ, we obtain

N∑n=1

(w(θn, t)− w(θn−1, t)) = w(θN , t)− w(θ0, t) = 0. (1.88)

On the other hand, (1.87) implies

Cm = Cm

N∑n=1

(θn − θn−1) =N∑n=1

T∫0

φ1m(t)(w(θn, t)− w(θn−1, t)) dt

=

T∫0

φ1m(t)

N∑n=1

(w(θn, t)− w(θn−1, t)) dt,

which, with (1.88), gives Cm = 0. Inserting this back to (1.87), we have

T∫0

φ1m(t)w(θn, t) dt =

T∫0

φ1m(t)w(θ0, t) dt, (1 ≤ n ≤ N). (1.89)

33

Finally, according to (1.84) and (1.89)

0 = J ψ3n

N∑n=1

θn+1 − θn−1

2

T∫0

φ1m(t)w(θn, t) dt = J ψ3

n

T∫0

φ1m(t)w(θ0, t) dt. (1.90)

Since J ψ3n is nonzero, the value of the integral

T∫0

φ1m(t)w(θ0, t) dt vanishes and, due

to formula (1.89), all integrals of type (1.83) vanish. Thus, u∆ satisfies Petrov-Galerkin equations (1.76).

1.8 Description of the nonlinearity

Now, we specify the assumptions on the nonlinearity. The real valued nonlinearfunction F (x, t, ξ) : ΣT×R→ R is assumed to satisfy the Caratheodory conditions:

The mapping (x, t) 7→ F (x, t, ξ) : ΣT → R is measurable for all fixed ξ ∈ R.(1.91)The mapping ξ 7→ F (x, t, ξ) : R→ R is continuous for almost all (x, t) ∈ ΣT .(1.92)

The associated Nemitsky operator or superposition operator

u 7→ F(u) : L2(ΣT )→ L2(ΣT ); F(u)(x, t) = F (x, t, u(x, t))

is well defined, provided that the Caratheodory conditions and the following growthcondition

|F (x, t, ξ)| ≤ b(x, t) + c |ξ|, ((x, t, ξ) ∈ ΣT × R) (1.93)

are valid (see [38]). Here, c is a constant and b(x, t) ∈ L2(ΣT ). For the analysis ofthe numerical approximation scheme, we assume that the mapping ξ 7→ F (x, t, ξ)is nondecreasing for each (x, t) ∈ ΣT

(F (x, t, ξ)− F (x, t, ξ′))(ξ − ξ′) ≥ 0 for all ξ, ξ′ ∈ R. (1.94)

Note that the Caratheodory conditions, together with (1.93) and (1.94) imply thatthe Nemitsky operator F : L2(ΣT ) → L2(ΣT ) is monotone, which by definitionmeans that

(F(u)−F(w) , u− w) ≥ 0.

34

In order to obtain regularity results and error estimates, we need the Lipschitzand Holder conditions.There exists a constant L > 0 such that

|F (x, t, ξ)− F (x, t, ξ′)| ≤ L|ξ − ξ′| for all ξ, ξ′ ∈ R, (x, t) ∈ ΣT . (1.95)

There are constants N,M > 0 and 0 < λ ≤ 1 such that

|F (x, t, ξ)− F (y, τ, ξ)| ≤ N |x− y|λ +M |t− τ |λ2 (1.96)

for all ξ ∈ R, (x, t), (y, τ) ∈ ΣT .We recall some basic properties concerning the Nemitsky operator in anisotropic

Sobolev spaces. The first one is a special case of the results given for example inbook [38]. The latter is presented in [30]. For the convenience of the reader, theproofs are repeated here.

Lemma 3 Assume that the Caratheodory conditions are valid.

(i) If (1.93) is valid, then F : L2(ΣT )→ L2(ΣT ) is bounded and continuous.

(ii) If, in addition (1.94) is valid, then the Nemitsky operator F : L2(ΣT ) →L2(ΣT ) is monotone.

Proof. Assuming growth condition (1.93), we obtain

‖F(u)‖2L2(ΣT ) =

T∫0

∫Γ

|(F(u))(x, t)|2dΓx dt =

T∫0

∫Γ

|F (x, t, u(x, t))|2dΓx dt

≤T∫

0

∫Γ

(b(x, t) + c|u(x, t)|)2dΓx dt

≤ ‖b‖2L2(ΣT ) + c ‖u‖2L2(ΣT ).

The monotonicity of the Nemitsky operator follows because

(F(u)−F(w) , u−w) =

T∫0

∫Γ

(F (x, t, u(x, t))−F (x, t, w(x, t)))(u(x, t)−w(x, t))dΓx dt

is non-negative.

Lemma 4 Assume that Caratheodory conditions are valid.

(i) If (1.93) and (1.95) are valid, then F : L2(ΣT ) → L2(ΣT ) is Lipschitzcontinuous.

(ii) If (1.93),(1.95) and (1.96) are valid, then F : Hs, s200 (ΣT ) → H

s, s200 (ΣT ) is

bounded for all 0 ≤ s < λ.

35

Proof. The norm of the space Hr, r200 (ΣT ) is defined by

‖u‖2s, s2 = ‖u‖2L2(ΣT ) + |u|2s,0 + |u|20, s2 ,

where the seminorms are defined by

|u|2s,0 =

T∫0

∫Γ

∫Γ

|u(x, t)− u(y, t)|2

|x− y|1+2sdΓx dΓy dt,

|u|20, s2 =

T∫0

T∫0

∫Γ

|u(x, t)− u(x, τ)|2

|t− τ |1+sdΓx dt dτ .

(i) The Lipschitz continuity of the Nemitsky operator follows from condition (1.95)

‖F(u)−F(w)‖2L2(ΣT ) =T∫0

∫Γ

|(F(u))(x, t)− (F(w))(x, t)|2dΓx dt

=T∫0

∫Γ

|F (x, t, u(x, t))− F (x, t, w(x, t))|2dΓx dt

≤T∫0

∫Γ

L2|u(x, t)− w(x, t)|2dΓx dt

= L2‖u− w‖2L2(ΣT ).

(ii) The Lipschitz and Holder conditions for the function F = F (x, t, ξ) yield forthe seminorms the estimates

|F(u)|2s,0 =T∫0

∫Γ

∫Γ

|(F(u))(x,t)−(F(u))(y,t)|2|x−y|1+2s dΓx dΓy dt

≤T∫0

∫Γ

∫Γ

[ |F (x,t,u(x,t))−F (y,t,u(x,t))|2|x−y|1+2s + |F (y,t,u(x,t))−F (y,t,u(y,t))|2

|x−y|1+2s

]dΓxdΓy dt

≤T∫0

∫Γ

∫Γ

N2|x−y|2λ|x−y|1+2s dΓx dΓy dt+ L2|u|2s,0

≤ N2T∫Γ

∫Γ

|x− y|2λ−2s−1dΓx dΓy + L2|u|2s,0 ≤ c1 + L2|u|2s,0

|F(u)|20, s2 =T∫0

T∫0

∫Γ

|(F(u))(x,t)−(F(u))(x,τ)|2|t−τ |1+s dΓx dt dτ

≤T∫0

T∫0

∫Γ

[ |F (x,t,u(x,t))−F (x,τ,u(x,t))|2|t−τ |1+s + L2|u(x,t)−u(x,τ)|2

|t−τ |1+s

]dΓx dtdτ

=T∫0

T∫0

∫Γ

M2|t−τ |λ|t−τ |1+s dΓx dtdτ + L2|u|20, s2

= M2m(Γ)T∫0

T∫0

|t− τ |λ−s−1 dt dτ + L2|u|20, s2 = c2 + L2|u|20, s2 .

The assertion follows since‖F(u)‖2s, s2 = ‖F(u)‖2L2(ΣT ) + |F(u)|2s,0 + |F(u)|20, s2

≤ ‖b‖2L2(ΣT ) + c1 + c2 + c ‖u‖2L2(ΣT ) + L2|u|2s,0 + L2|u|20, s2≤ c3 + c4 ‖u‖2s, s2 .

2 Abstract theory

The theory of the Finite Element Method, as well as that of the Boundary ElementMethod, is closely related to the abstract functional analysis. The so-called Lax-Milgram theorem, together with the underlying Riesz representation theorem, playcrucial roles in this context. In this chapter, we introduce some fundamentalelements of the linear and nonlinear theory.

In the nonlinear case, the essential references are Petryshyn’s articles [41], [42]and [43], Minty’s work [36], and Browder’s results [11], [12]. The reader may findthe textbooks by Pascali & Sburlan [38] and Deimling [19] useful.

2.1 Linear equations

Let X be a Hilbert space. Let ` : X → R be a bounded linear functional. LetB(· , ·) : X ×X → R be a bilinear form. The bilinear form is called

bounded, if |B(w, v)| ≤ CB ‖w‖X‖v‖X , for all w, v ∈ X,strongly coercive, if B(w,w) ≥ c0 ‖w‖2X , for all w ∈ X, with c0 > 0.

We consider the variational problem of finding u ∈ X, such that

B(u, v) = `(v) for all v ∈ X. (2.1)

The solvability is given in the following so-called Lax-Milgram Theorem. For theproof, see [10] or [13].

Theorem 11 Let B(·, ·) be a real valued, strongly coercive and bounded bilinearform in a Hilbert space X. Let ` be a bounded real valued linear form on X. Thenthe equation

B(u, v) = `(v) for all v ∈ X

is uniquely solvable, and

‖u‖X ≤1c0‖`‖.

37

In order to approximate the solution of the variational problem (2.1), we consideranother variational problem of finding un ∈ Xn such that

B(un, vn) = `(vn) for all vn ∈ Xn, (2.2)

where Xn ⊂ X is a closed subspace. We recall the so-called Cea Lemma, whichdescribes the solvability of problem (2.2).

Theorem 12 Let B(·, ·) be a real valued and bounded bilinear form in a Hilbertspace X. Let Xn be a closed subspace of X. Let B(·, ·) be strongly coercive in Xn.Let ` be a bounded real valued linear form on X. Then equation (2.2) admits aunique solution such that

‖un‖X ≤cBc0‖u‖X .

For the approximation error in un, we have the quasi-optimal bound

‖u− un‖X ≤(1 +

cBc0

)infψ∈Xn

‖u− ψ‖X .

Proof. The solvability of the variational problem (2.2) follows from the Lax-Milgram theorem, since closed subspaces of Hilbert spaces are themselves Hilbertspaces. The coercivity and boundedness of the bilinear form combined to theequations

B(u, vn) = `(vn) for all vn ∈ Xn

B(un, vn) = `(vn) for all vn ∈ Xn

give

c0 ‖un‖2X0≤ B(un, un) = B(u, un) ≤ cB ‖u‖X0‖un‖X .

This yields the stability‖un‖X ≤

cBc0‖u‖X .

In particular, we have for all ψ ∈ Xn

c0 ‖un − ψ‖2X ≤ B(un − ψ, un − ψ)= B(u− ψ, un − ψ) +B(un − u, un − ψ)= B(u− ψ, un − ψ)≤ cB ‖u− ψ‖X‖un − ψ‖X

implying‖un − ψ‖X ≤

cBc0

infψ∈Xn

‖u− ψ‖X .

Now, the triangle inequality gives for all ψ ∈ Xn

‖u− un‖X ≤ ‖u− ψ‖X + ‖ψ − un‖X ≤(1 +

cBc0

)infψ∈Xn

‖u− ψ‖X

proving the quasi-optimality.

38

In practice, spaces Xn are finite dimensional. The elements un ∈ Xn are calledshape functions. The importance of the quasi-optimal error estimate is that theasymptotic convergence depends only on the approximation properties of the shapefunctions.

In this study we apply these fundamental tools to the case of small perturbationsof coercive operators. The formulation of Theorems 11 and 12 can be generalizedin several ways. It is possible to replace the coercivity by the Babuska stabilitycondition [10]. The theorems can also be modified to cover the case of compactperturbations of isomorphic operators [13], Theorem 10.1.3.

2.2 Nonlinear equations

The purpose of this section is to give the general theoretical framework which hasbeen applied to articles IV and V. The approach is based on Saranen’s article [55]concerning projection methods applied to the equation

A(u) = g,

where A : X → X is a nonlinear Hammerstein operator

A(u) = Iu+Du+ S(F (u)).

Here, S and D are given linear operators and F describes the nonlinearity. LetX0 be a Hilbert space equipped with the norm ‖ · ‖X0 such that the embeddingX ⊂ X0 is dense and continuous. We denote the dual space of X by X∗. Byidentifying X0 with its dual, we have X ⊂ X0 ⊂ X∗. Let (· , ·) be the continuousbilinear form defined in X ×X∗ or X∗×X such that it extends the inner productof the pivot space X0. We use another Hilbert space Z such that Z ⊂ X withcontinuous injection.

We recall some concepts concerning nonlinear behaviour. The nonlinearity F :X0 → X0 is called

monotone if (F (u)− F (w) , u− w) ≥ 0,strongly monotone if (F (u)− F (w) , u− w) ≥ cF ‖u− w‖2X0

.

Let (S−1)∗ : X → X∗ be the adjoint of the operator S−1 : X∗ → X defined bythe relation (S−1u∗ , u) = (u∗ , (S−1)∗u), u ∈ X. We assume that the operator Ais strongly (S−1)∗-monotone in the sense that the inequality

(A(u)−A(w) , (S−1)∗(u− w)) ≥ cA‖u− w‖2X (2.3)

is valid for all u, v ∈ X with a positive constant cA. The monotonicity (2.3) impliesthat the operator A : X → X is a homeomorphism.

Let Xn ⊂ X, n ∈ N be a sequence of finite dimensional subspaces with givenprojection Qn : X → Xn for each n ∈ N. We look for an approximation un for thesolution u such that

QnA(un) = QnA(u). (2.4)

39

The following abstract result describes the solvability of the approximate equation.

Theorem 13 We assume that the following conditions are fulfilled

D : X → Z is bounded, (2.5)S : X∗ → X is an isomorphism and S : X0 → Z is bounded, (2.6)F : X0 → X0 is Lipschitz continuous, (2.7)A : X → X is strongly (S−1)∗ − monotone. (2.8)

Assume that the projection operators Qn have the asymptotic approximation prop-erty

‖u−Qnu‖X ≤ ε(n)‖u‖Z , n ∈ N (2.9)

such that ε(n) → 0 when n → ∞. Then there exists an integer n0 ∈ N and apositive constant c1 such that

‖QnA(v)−QnA(v′)‖X ≥ c1 ‖v − v′‖X for all v, v′ ∈ Xn, (2.10)

when n ≥ n0. Moreover, equation (2.4) is uniquely solvable for n ≥ n0.

Proof. Since Qn is a projection Qnv = v for all v ∈ Xn, and we can write

(I −Qn)(A(v)−A(v′)) = (I −Qn)D(v − v′) + (I −Qn)S(F (v)− F (v′)),

for all v, v′ ∈ Xn. A lower bound is obtained by decomposing and using (S−1)∗

-monotonicity(QnA(v)−QnA(v′), (S−1)∗(v − v′)

)=(A(v)−A(v′), (S−1)∗(v − v′)

)−((I −Qn)

(A(v)−A(v′)

), (S−1)∗(v − v′)

)≥ c ‖v − v′‖2X− c′‖(I −Qn)

(A(v)−A(v′)

)‖X‖v − v′‖X .

On the other hand, the Schwarz inequality yields the upper bound(QnA(v)−QnA(v′), (S−1)∗(v−v′)

)≤ c ‖QnA(v)−QnA(v′)‖X‖(S−1)∗(v−v′)‖X∗≤ c ‖QnA(v)−QnA(v′)‖X‖v − v′‖X .

Therefore, we obtain the estimate

‖QnA(v)−QnA(v′)‖X ≥ c ‖v − v′‖X − c′ ‖(I −Qn)(A(v)−A(v′)

)‖X . (2.11)

According to approximation property (2.9) combined with assumptions (2.5) -(2.8), we find

‖(I−Qn)(A(v)−A(v′)

)‖X ≤ ‖(I−Qn)D(v−v′)‖X+ ‖(I−Qn)S(F (v)−F (v′))‖X≤ c ε(n)‖D(v − v′)‖Z + c ε(n)‖S(F (v)−F (v′))‖Z≤ c ε(n)‖v − v′‖X + c ε(n)‖F (v)−F (v′)‖X0

≤ c ε(n)‖v − v′‖X + c ε(n)‖v − v′‖X0

≤ c ε(n)‖v − v′‖X .

Thus, (2.11) yields the stability

‖QnA(v)−QnA(v′)‖X ≥ c1 ‖v − v′‖X , for all v, v′ and n ≥ n0.

40

Due to stability, the operator QnA : Xn → Xn is an injection. According toBrouwer’s theorem on invariance of the domain, the range R(QnA) is open. Be-cause of the stability and continuity of QnA in Xn, the range R(QnA) is alsoclosed in Xn. Therefore, R(QnA) = Xn, and QnA is a homeomorphism.

Theorem 14 We assume that conditions (2.5) - (2.8) and approximation property(2.9) are valid. Then, we have the asymptotic error estimate

‖u− un‖ ≤ c ‖u−Qnu‖X . (2.12)

Furthermore, if the mapping D : X0 → Z is bounded, then

‖u− un‖ ≤ c ‖u−Qnu‖X0 . (2.13)

Proof. For all v ∈ Xn, because of equation (2.4) it holds that

QnA(un)−QnA(v) = QnA(u)−QnA(v) (2.14)= Qn(u− v) +QnD(u− v) +QnV (F (u)−F (v)).

Since Qn is a projection operator in Xn, the choice v = Qnu gives

QnA(un)−QnA(Qnu) = QnD(u−Qnu) +QnV (F (u)− F (Qnu)). (2.15)

Now, it follows by using (2.5) and approximation property (2.9)

‖QnD(u−Qnu)‖X ≤ ‖D(u−Qnu)‖X + ‖(I −Qn)D(u−Qnu)‖X(2.16)≤ c ‖u−Qnu‖X + c ε(n)‖D(u−Qnu)‖Z≤ c ‖u−Qnu‖X + c ε(n)‖u−Qnu)‖X≤ c ‖u−Qnu‖X .

An analogous estimate follows by using (2.6) and (2.7)

‖QnV (F (u) − F (Qnu))‖X ≤ ‖V (F (u)− F (Qnu))‖X (2.17)+ ‖(I −Qn)V (F (u)− F (Qnu))‖X

≤ c ‖F (u)− F (Qnu)‖X0 + c ε(n)‖V (F (u)− F (Qnu))‖Z≤ c ‖u−Qnu‖X0 + c ε(n)‖F (u)− F (Qnu)‖X0

≤ c ‖u−Qnu‖X0 .

Combining stability (2.10) with (2.15), (2.16) and (2.17) we have

c1 ‖un −Qnu‖X ≤ ‖QnA(un)−QnA(Qnu)‖X≤ c ‖u−Qnu‖X + c ‖Qnu− un‖X .

Finally, the triangle inequality yields the convergence

‖u− un‖X ≤ ‖u−Qnu‖X + ‖Qnu− un‖X ≤ c ‖u−Qnu‖X .

41

If mapping property (2.13) is valid then we may replace (2.16) by the strongerestimate

‖QnD(u−Qnu)‖X ≤ ‖u−Qnu‖X0 ,

which together with (2.10), (2.15) and (2.17) implies

‖un −Qnu‖X ≤ c ‖u−Qnu‖X0 .

The theorem is proved.

3 Summary of the original articles

The thesis consists of two lines of research. The first deals with questions of thestability and convergence of the spline collocation method applied to the singlelayer heat operator equation. These results are generalized to some spline Petrov-Galerkin methods applied to the single layer heat operator equation, as well as tothe case of the hypersingular heat operator equation. The analysis is based on theassumption that the spatial domain is a disk.

The rest of the thesis consists of the analysis of the boundary element approxi-mation of some problems, where the boundary condition is nonlinear. The effect ofthe numerical integration is analysed in the stationary case. Also some numericalresults are given. Later, the theory is extended to the nonlinear time-dependentcase.

3.1 Article I: On the spline collocation method for thesingle layer heat operator equation

As discussed in the introduction, the direct boundary element method yields thesingle layer heat operator equation for the unknown boundary density. We con-struct and analyse a spline collocation scheme for solution of the single layer heatoperator equation, assuming that the spatial domain is two-dimensional and hasa smooth boundary. We apply piecewise linear approximation both in space andtime together with nodal point collocation. For the proof of the stability and theconvergence we restrict ourselves here to the case of the circle, but the methodcan be applied to all smooth closed curves, as was shown in [31], [32].

The collocation equations corresponding to the single layer heat operator equa-tion Su = f are: find u∆ ∈ S1,1

h,k such that

(Su∆)(θn, tm) = f(θn, tm), (1 ≤ n ≤ N, 1 ≤ m ≤M). (3.1)

For the equivalent characterization of the collocation equations, we need someregularity assumptions. Let C1(R

2

T ) be the space of continuous functions f(θ, t)

43

on the closure R2

T of R2T such that f is 1-periodic with respect to the variable θ.

Moreover, we define the spaces

C00(R2

T ) = f ∈ C1(R2

T ) | f(θ, 0) ≡ 0,Hc00(R

2

T ) = f ∈ C00(R2

T ) | ∂θf, ∂tf, ∂t∂θf = ∂θ∂tf ∈ C1(R2

T ).

Next, we give sufficient conditions on the function u to guarantee the propertySu ∈ Hc00(R

2

T ). Recall that the measurable function u is essentially bounded onR

2

T if there exists C > 0 such that |u(θ, t)| ≤ C for almost all (θ, t) ∈ R2T . We

introduce the space

Ct00(R2T ) = f | f(θ, ·) is continuous for almost all θ and f(θ, 0) = 0.

The space H00(R2T ) consists of functions f ∈ Ct00(R2

T ) such that ∂θf ∈ Ct00(R2T )

and the partial derivatives ∂tf, ∂t∂θf = ∂θ∂tf are measurable, essentially boundedfunctions on R

2

T . We emphasize that S1,1h,k is a subspace of H00(R2

T ).Now, we are ready to establish the equivalence between the collocation problem

and a certain Galerkin problem. For this, we define the bilinear form

B1,1∆ (w, v) =

(∂t∂θw , ∂t∂θv

)+((∂t ⊗ J∆)w , (∂t ⊗ J)v

)=

1∫0

T∫0

∂t∂θw(θ, t)∂t∂θv(θ, t) dt dθ+T∫0

((∂t ⊗ J∆)w)(t)((∂t ⊗ J)v)(t) dt.

Theorem 15 (I, Theorem 3.1) Let u ∈ H00(R2T ) be the solution of the equation

Su = f . Then the function u∆ ∈ S1,1h,k is a solution of the collocation problem (3.1)

if and only ifB1,1

∆ (Su∆, v) = B1,1∆ (Su, v), (v ∈ S1,1

h,k).

The unique solvability of collocation equations (3.1) is a consequence of thecoercivity and continuity of the bilinear form B1,1

∆ (Su, v). This yields stabilityand convergence of the method. To prove the coercivity, we consider the formB1,1

∆ (w, v) as a small perturbation of the bilinear form

B1,1(w, v) =(∂t∂θw , ∂t∂θv

)+((∂t ⊗ J)w , (∂t ⊗ J)v

).

The energy norm induced by the bilinear form is

|||v|||− 12 ,−

14

= ‖∂t∂θv‖− 12 ,−

14

+ ‖(∂t ⊗ J)v‖− 12 ,−

14.

For the analysis, it is essential that ||| · |||− 12 ,−

14

is a norm in the subspace S1,1h,k.

The proof of continuity and coercivity of the bilinear form B1,1(S·, ·) is based oncertain commutation properties of the single layer heat operator. The bilinear formB1,1

∆ (S·, ·) is a small perturbation of the form B1,1(S·, ·). Therefore, we obtain thecontinuity estimate

|B1,1∆ (Su, v)| ≤ c1 |||u|||− 1

2 ,−14|||v|||− 1

2 ,−14, (u, v ∈ H00(R2

T )),

44

as well as the coercivity

B1,1∆ (Sv, v) ≥ c2 |||v|||2− 1

2 ,−14, (v ∈ H00(R2

T )),

for a sufficiently small mesh parameter h. Now, the general theory gives the exis-tence of the unique solution of collocation equations (3.1), as well as the stability

|||u∆|||− 12 ,−

14≤ c1c2|||u|||− 1

2 ,−14

and the quasi-optimal approximation result

|||u− u∆|||− 12 ,−

14≤(1 +

c1c2

)inf

v∈S1,1h,k

|||u− v|||− 12 ,−

14. (3.2)

The convergence analysis of the collocation approximation defined by (3.1) isbased on the quasi-optimal error estimate. It is natural first to discuss the asymp-totic accuracy of the approximation when the error is measured by means of theenergy norm ||| · |||− 1

2 ,−14. For this, we need error estimates for the L2-orthogonal

projection P 1,1h, k : L2(R2

T ) → S1,1h,k as well as some additional regularity assump-

tions on the solution of the single layer heat operator equation.

Lemma 5 Assume that u ∈ H00(R2T ) such that u(θ, ·), (∂θu)(θ, ·) ∈ H2(0, T ) for

almost all θ, and (∂tu)(·, t) ∈ H2 for almost all t. Then we have the approximationresults

|||u− P 1,1h, ku|||− 1

2 ,−14≤ c h

32 ‖∂tu‖2,0 + c k

(h

12 + k

14)(‖u‖0,2 + ‖∂θu‖0,2

),

|||u− P 1,1h, ku|||0,0 ≤ c h‖∂tu‖2,0 + c k

(‖u‖0,2 + ‖∂θu‖0,2

).

Theorem 16 (I, Theorem 5.1) Assume that the solution of the equation Su =f satisfies u ∈ H00(R2

T ) such that u(θ, ·), (∂θu)(θ, ·) ∈ H2(0, T ) for almost all θ,and (∂tu)(·, t) ∈ H2 for almost all t. Then the collocation approximation u∆ ∈ S1,1

h,k

defined by (3.1) furnishes the asymptotic error estimate

|||u− u∆|||− 12 ,−

14≤ c h

32 ‖∂tu‖2,0 + c k

(h

12 + k

14)(‖u‖0,2 + ‖∂θu‖0,2

).

In the estimate the time step dominates the order of the convergence. This effectcan be compensated for by letting the time-discretization be finer than the dis-cretization in the space variable. We also consider the convergence by using thenorm |||v|||0,0 = ‖∂t∂θv‖0,0 + ‖(∂t ⊗ J)v‖0,0. For this we need the inverse estimate

‖∂t∂θv‖0,0 + ‖(∂t ⊗ J)v‖0,0 ≤ c max(h−12 , k−

14 )|||v|||− 1

2 ,−14, (v ∈ S1,1

h,k).

These aspects are summarized in the following theorem.

Theorem 17 (I, Theorems 5.2, 5.3 and 5.4) Let the assumptions of Theorem16 be valid. Moreover, suppose that k ≤ c hν , where h ≤ h0 is sufficiently small.Then the collocation approximation u∆ satisfies

|||u− u∆|||− 12 ,−

14≤ c h

32 ‖∂tu‖2,0 + c hmin( 5

4 ν,ν+ 12 )(‖u‖0,2 + ‖∂θu‖0,2

).

45

In particular, for ν ≥ 65

|||u− u∆|||− 12 ,−

14≤ c h

32(‖u‖0,2 + ‖∂θu‖0,2 + ‖∂tu‖2,0

).

If in addition, c0 h ≤ h ≤ h and c0 h2 ≤ k ≤ c h2, where h ≤ h0 is sufficiently

small. Then, we have for ν ≥ 65

|||u− u∆|||0,0 ≤ c h(‖u‖0,2 + ‖∂θu‖0,2 + ‖∂tu‖2,0

).

The final results are the pointwise and L2-convergence

maxθ∈[0,1]

|(u− u∆)(θ, t)| ≤ c t 12h(‖u‖0,2 + ‖∂θu‖0,2 + ‖∂tu‖2,0

), (0 ≤ t ≤ T ),

‖u− u∆‖0,R2T≤ c h

(‖u‖0,2 + ‖∂θu‖0,2 + ‖∂tu‖2,0

).

We have carried out some numerical experiments which confirm the convergenceindicating, even the quadratic rate of the convergence. The numerical implemen-tation of the scheme, covering also the general case, has been explained in somedetail in article I.

3.2 Article II: A collocation type projection method for thesingle layer heat operator equation

In order to simplify convergence analysis, and reduce regularity requirements, wepropose another projection method for the approximation. Our approach is to useinterpolation with respect to the space variable and L2-projection with respect tothe time variable. With this aim in view, we define Q1,0

h,k := I1h ⊗ P 0

k = P 0k ⊗ I1

h,

where I1h : Hr → S1

h, r > 12 , is the interpolation operator. Discrete equations

corresponding to the equation Su = f are: find u∆ ∈ S1,0h,k(R2

T ) such that

Q1,0h,kSu∆ = Q1,0

h,kf. (3.3)

In order to describe the required regularity, we define some spaces of functions.Let C1(R

2

T ) be the space of continuous functions f(θ, t) on the closure R2

T of R2T

such that f is 1-periodic with respect to the variable θ. The measurable function uis essentially bounded on R

2

T if there exists a constant C > 0 such that |u(θ, t)| ≤ Cfor almost all (θ, t) ∈ R2

T . The space of essentially bounded functions on R2T is

denoted by L∞(R2T ). Moreover, we define the spaces

C00(R2

T ) = f ∈ C1(R2

T ) | f(θ, 0) ≡ 0,

Hc00(R2

T ) = f ∈ C00(R2

T ) | ∂θf ∈ C1(R2

T ),H00(R2

T ) = f ∈ L∞(R2T ) | ∂θf ∈ L∞(R2

T ).

46

We notice that the inclusions S1,0h,k(R2

T ) ⊂ Hr, r2 (R2T ), r < 1, S1,0

h,k(R2T ) ⊂ L∞(R2

T )are valid for the trial spaces.

Problem setting (3.3) is also equivalent to a certain Galerkin problem. For this,we define the bilinear form

B1,0∆ (w, v) = (∂θw , ∂θv) + (J∆w , J v)

=1∫0

T∫0

∂θw(θ, t)∂θv(θ, t) dt dθ +T∫0

(J∆w)(t)(J v)(t) dt,

which is a small perturbation of the bilinear form

B1,0(w, v) = (∂θw , ∂θv) + (J w , J v).

Theorem 18 (II, Theorem 2) Let u ∈ H00(R2T ) be the solution of the equation

Su = f . Then the function u∆ ∈ S1,0h,k(R2

T ) is a solution of discrete problem (3.3)if and only if

B1,0∆ (Su∆, v) = B1,0

∆ (Su, v), (v ∈ S1,0h,k(R2

T )). (3.4)

Here, the energy norm |||v|||− 12 ,−

14

= ‖∂θv‖− 12 ,−

14

+ ‖J v‖− 12 ,−

14

is a norm in the

subspace S1,0h,k(R2

T ). The continuity and coercivity of the form B1,0(S·, ·) followsfrom certain commutation properties of the single layer heat operator. The bilinearform B1,0

∆ (S ·, ·) is a small perturbation. Therefore, we have, for sufficiently smallh, the continuity

|B1,0∆ (Su, v)| ≤ c1 |||u|||− 1

2 ,−14|||v|||− 1

2 ,−14, (u, v ∈ H00(R2

T )). (3.5)

and the coercivity

B1,0∆ (Sv, v) ≥ c2 |||v|||2− 1

2 ,−14, (v ∈ H00(R2

T )). (3.6)

We assume that the solution of the equation Su = f satisfies u ∈ H00(R2T ).

Then for all 0 < h ≤ h0 there exists a unique solution u∆ of equations (3.3).Moreover, we have the stability

|||u∆|||− 12 ,−

14≤ c1c2|||u|||− 1

2 ,−14

(3.7)

and the quasi-optimal error estimate

|||u− u∆|||− 12 ,−

14≤(1 +

c1c2

)inf

v∈S1,0h,k(R2

T )|||u− v|||− 1

2 ,−14. (3.8)

The quasi-optimal error estimate (3.8) yields the asymptotic accuracy of the ap-proximation when the error is measured by means of the norm ||| · |||− 1

2 ,−14.

For the proof of our convergence results, we need additional regularity assump-tions on the solution of the equation Su = f . We also need approximation prop-erties with respect to the norm |||v|||0,0 = ‖∂θv‖0,0 + ‖J v‖0,0. These are describedin the following lemma.

47

Lemma 6 Assume that u ∈ H00(R2T ) such that u ∈ Hs, s2

00 (R2T ), 1 ≤ s ≤ 2. Then

we have the approximation results

|||u− P 1,0h, ku|||− 1

2 ,−14≤ c (hs−

12 + k(s− 1

2 )/2)‖u‖s, s2 , (1 ≤ s ≤ 2),

|||u− P 1,0h, ku|||0,0 ≤ c

(hs−1 + k(s−1)/2

)‖u‖s, s2 , (1 ≤ s ≤ 2).

¿From the quasi-optimal error estimate and Lemma 6, we obtain

Theorem 19 (II, Theorem 4) Let Su = f , u ∈ H00(R2T ) and u ∈ Hs, s2

00 (R2T ),

1 ≤ s ≤ 2. Then the approximation u∆ ∈ S1,0h,k(R2

T ) defined by (3.3) furnishes theasymptotic error estimate

|||u− u∆|||− 12 ,−

14≤ c (hs−

12 + k(s− 1

2 )/2)‖u‖s, s2 . (3.9)

In Theorem 19, the time step dominates the order of the convergence. Thiseffect can be compensated for by letting the time-discretization be finer than thediscretization in the space variable.

Theorem 20 (II, Theorem 5 and Theorem 6) Let Su = f , u ∈ H00(R2T )

and u ∈ Hs, s200 (R2

T ), 1 ≤ s ≤ 2. Moreover, suppose that h ≤ h and k ≤ c h2,where h ≤ h0 is sufficiently small. Then the collocation type approximation u∆

satisfies|||u− u∆|||− 1

2 ,−14≤ c h

32 ‖u‖2,1. (3.10)

Further, let c0 h ≤ h ≤ h and c0 h2 ≤ k ≤ c h2, where h ≤ h0 is sufficiently

small. Then we have for the collocation type approximation u∆

|||u− u∆|||0,0 ≤ c h‖u‖s, s2 . (3.11)

In particular, we obtain the L2-convergence ‖u− u∆‖0,0 ≤ c h‖u‖2,1.We have proved that the Petrov-Galerkin approximation is stable and convergent.Compared to the collocation method, the regularity requirements are significantlyreduced.

3.3 Article III: An approximation method for thehypersingular heat operator equation

Here, we consider the case of a pseudodifferential operator of positive order. Themodel problem is the hypersingular heat operator equation. Our Petrov-Galerkinapproach uses interpolation with respect to the space variable and L2-projectionwith respect to the time variable. For this, we define Q3,1

h,k := I3h ⊗ P 1

k = P 1k ⊗ I3

h,

where I3h : Hr → S3

h, r > 12 , is the interpolation operator. Since our method uses

pointwise values, it is essential that the image Hu is continuous. According tothe Sobolev embedding theorem this condition is true if u ∈ H

r, r200 (R2

T ), r > 52 .

48

The discrete equation corresponding to the hypersingular heat operator equationHu = f is: find u∆ ∈ S3,1

h,k(R2T ) such that

Q3,1h,kHu∆ = Q3,1

h,kf. (3.12)

This operator equation is equivalent to the following system of equations:find u∆ ∈ S3,1

h,k(R2T ) such that

T∫0

φ1m(t)(Hu∆)(θn, t) dt =

T∫0

φ1m(t)f(θn, t) dt, (n = 1, . . . , N,m = 1, . . . ,M).

Again, problem setting (3.12) is equivalent to a certain Galerkin problem. Forthis, we define the bilinear forms

B2,0(w, v) =(∂2θw , ∂

2θv)

+(J w , J v

)B2,0

∆ (w, v) =(∂2θw , ∂

2θv)

+(J∆w , J v

).

Theorem 21 (III, Theorem 2) Let u ∈ Hs, s200 (R2

T ), s > 52 be the solution of

the equation Hu = f . Then the function u∆ ∈ S3,1h,k(R2

T ) is a solution of problem(3.12) if and only if

B2,0∆ (Hu∆, v) = B2,0

∆ (Hu, v), (v ∈ S3,1h,k(R2

T )). (3.13)

Discrete equations (3.12) are uniquely solvable if the spatial discretization pa-rameter is small enough. This result is a consequence of the coercivity esti-mate for the bilinear form B2,0

∆ (Hu, v), which yields stability and convergenceof the method. For the sake of brevity, we introduce the notation |||v|||r,s =‖∂2θv‖r,s + ‖J v‖r,s. In any space of functions, where this expression is well defined

and finite, it gives a seminorm. In particular, in the subspace of the approximat-ing functions ||| · |||s, s2 defines a norm. Our next aim is to establish the requiredcoercivity of the approximating bilinear form with respect to the norm ||| · ||| 1

2 ,14.

For our method, it is enough that this property is valid in the subspace S3,1h,k(R2

T ).

Lemma 7 Let u,w ∈ Hs, s200 (R2

T ), s > 52 . Then we have the continuity

|B2,0(Hu,w)| ≤ c1 |||u||| 12 ,

14|||w||| 1

2 ,14, (u,w ∈ Hs, s2

00 (R2T )) (3.14)

and the coercivity

B2,0(Hw,w) ≥ c2 |||w|||212 ,

14, (w ∈ Hs, s2

00 (R2T )). (3.15)

The perturbation due to the trapezoidal rule approximation is small in the sensethat

|B2,0(Hu,w)−B2,0∆ (Hu,w)| ≤ c h

(|||u|||0,0 + |||u|||0, 12

)|||w|||0,0, (3.16)

49

for all u,w ∈ Hs, s200 (R2

T ). In particular,

|B2,0(Hv, v)−B2,0∆ (Hv, v)| ≤ c h

(1 + k−

14)|||v|||21

2 ,14, (v ∈ S3,1

h,k(R2T )). (3.17)

Therefore, also the approximating bilinear form is coercive

B2,0∆ (Hv, v) ≥ c3 |||v|||21

2 ,14, (v ∈ S3,1

h,k(R2T )) (3.18)

for 0 < h ≤ h0, h4−ε ≤ c0 k, ε > 0 and h0 is sufficiently small.

The main result is the following error bound.

Theorem 22 (III, Theorem 3) Assume that u ∈ Hs, s200 (R2

T ), s > 52 and Hu =

f . Then, for all 0 < h ≤ h0, h4−ε ≤ c0 k, ε > 0 and h0 is sufficiently small,there exists a unique solution u∆ of Petrov-Galerkin equations (3.12). Moreover,we have the approximation result

|||u− u∆||| 12 ,

14≤ C inf

v∈S3,1h,k(R2

T )

(|||u− v||| 1

2 ,14

+ h|||u− v|||0, 12). (3.19)

The final aim is to establish L2-convergence for the discrete solution. Based onthe error estimate, we first discuss the asymptotic accuracy of the approximationwhen the error is measured by means of the norm ||| · ||| 1

2 ,14. For the proof of the

convergence results, we make additional regularity assumptions on the solution ofthe equation Hu = f .

Lemma 8 Assume that u ∈ H4,200 (R2

T ). Then we have the approximation results

|||u− P 3,1h, ku|||0,0 ≤ c

(h2 + k

)‖u‖4,2,

|||u− P 3,1h, ku||| 12 , 14 ≤ c (h

32 + k

34 )‖u‖4,2.

¿From Theorem 22 and Lemma 8, we obtain

Theorem 23 (III, Theorem 4 and Theorem 5) Let Hu = f , such that u ∈H4,2

00 (R2T ). Then the Petrov-Galerkin approximation u∆ ∈ S3,1

h,k(R2T ) defined by

(3.12) furnishes the asymptotic error estimate

|||u− u∆||| 12 ,

14≤ c

(h

32 + k

34)‖u‖4,2. (3.20)

Moreover, suppose that h4−ε ≤ c0 k ≤ c h2, where h ≤ h0, and h0 is sufficientlysmall. Then we have the asymptotic estimates

|||u− u∆||| 12 ,

14≤ c h

32 ‖u‖4,2, (3.21)

|||u− u∆|||0,0 = ‖∂2θ (u− u∆)‖0,0 + ‖J (u− u∆)‖0,0 ≤ c h

32 ‖u‖4,2. (3.22)

In particular, we have the L2-convergence ‖u− u∆‖0,0 ≤ c h32 ‖u‖4,2.

Note that the asymptotic order of convergence with respect to the |||·|||0,0 remainsthe same. It is remarkable here that the required regularity of the solution isdescribed in terms of the anisotropic Sobolev spaces. This technique can be appliedalso to the case of the single layer heat operator equation.

50

3.4 Article IV: The numerical approximation of the solutionof a nonlinear boundary integral equation with the

collocation method

In the stationary case, the nonlinear heat equation reduces to the potential equa-tion (1.2) with a given nonlinear Neumann type boundary condition. The directboundary integral approach yields the nonlinear boundary integral equation, [53]

( 12I +DΓ)µ+ SΓF(µ) = SΓgNΓ. (3.23)

We denote the associated nonlinear integral operator A(u) = ( 12I+DΓ)u+SΓF(u).

Here, F describes the nonlinearity, and the linear boundary integral operators arethe single layer and double layer operators, respectively

SΓu(x) = − 12π

∫Γ

u(y) ln |x− y|dΓy, DΓu(x) =1

2π

∫Γ

u(y)∂

∂nyln |x− y|dΓy.

The purpose of our paper is to introduce an approximation scheme for (3.23)by using an easily computable L2-orthogonal projection of the nonlinear function.This approach applies to general projection methods, but for simplicity we discussonly collocation.

Now, we specify the assumptions on the nonlinearity. The real valued nonlinearfunction F (x, ξ) : Γ× R→ R is assumed to satisfy the Caratheodory conditions:

The mapping x 7→ F (x, ξ) : Γ→ R is measurable for all fixed ξ ∈ R.The mapping ξ 7→ F (x, ξ) : R→ R is continuous for almost all x ∈ Γ.

The Nemitsky operator u 7→ F(u) : L2(Γ) → L2(Γ); F(u)(x) = F (x, u(x)) iswell defined, provided that the Caratheodory conditions and the growth condition|F (x, ξ)| ≤ b(x) + c |ξ|, (x, ξ) ∈ Γ × R are valid (see [38]). Here, c is a constantand b(x) ∈ L2(Γ). For the analysis of the numerical approximation scheme, weassume that(A1) the Nemitsky operator F : L2(Γ)→ L2(Γ) is strongly monotone,(A2) the Nemitsky operator F : L2(Γ)→ L2(Γ) is Lipschitz continuous,(A3) the Nemitsky operator F : Hs(Γ)→ Hs(Γ) is bounded for 0 ≤ s < 1.We point out that the Lipschitz continuity of the Caratheodory function F guar-antees (A2) and (A3). These properties are necessary for regularity results anderror estimates.

Theorem 24 Let the capacity of the boundary curve differ from unity.

(i) For every g ∈ H−12 (Γ) integral equation (3.23) has a unique solution u ∈

H12 (Γ).

(ii) For the solution u, the following regularity result is true: If gNΓ ∈ Hs−1(Γ),12 ≤ s < 2, and assumptions (A1), (A2) and (A3) are valid, then the solutionsatisfies u ∈ Hs(Γ).

51

The proof is presented in [53]. It is based on the fact that integral operator A isstrongly S−1

Γ monotone.Next, we consider the collocation method for finding an approximate solution

of equation (3.23). We require that gNΓ ∈ Hs−1(Γ), s > 12 . Then the function gNΓ

is continuous, and the collocation equations are given by: find uh ∈ Sdh such that

(A(uh))(xi) = (SΓgNΓ)(xi), (i = 0, . . . , N − 1),

Nodal point collocation is applied for odd degree splines, and midpoint collocationis applied for even degree splines. An equivalent formulation of the collocationproblem, given in terms of the interpolation operator is: find uh ∈ Sdh such that

IdhA(uh) = IdhSΓgNΓ, (3.24)

For numerical purposes, we define an approximate collocation equation as follows:find uh ∈ Sdh such that

Ah(uh) := 12 uh − I

dhDΓuh + IdhSΓP

dhF(uh) = IdhSΓgNΓ. (3.25)

The following theorem describes the convergence properties.

Theorem 25 (IV, Theorem 3.1 and Theorem 3.3) Assume d > 0. Let u ∈Hs(Γ), 1

2 < s ≤ d + 1, be the solution of (3.23) and suppose that (A1) and (A2)are valid. Then, for sufficiently small h, the collocation problem (3.24) as wellas the approximate collocation problem (3.25) admit unique solutions uh, and uh,respectively. Furthermore, we have the asymptotic error estimates

‖u− uh‖t ≤ c hs−t||u||s, (3.26)

‖uh − uh‖ 12≤ c hs+

12 ‖u‖s + c hτ+ 1

2 ‖F(u)‖τ , (3.27)

‖u− uh‖t ≤ c hs−t‖u‖s + c hτ+1−max(t, 12 )‖F(u)‖τ , (3.28)

for 0 ≤ t ≤ s, t < d+ 12 , provided that F(u) ∈ Hτ (Γ), 0 ≤ τ ≤ d+ 1.

The proof for (3.26) presented in [53] covers indices 12 ≤ t ≤ s, and the results in

[55] give (3.26) when 0 ≤ t < 12 . Here, the crucial idea is to choose an appropriate

Hilbert space, and show that the assumptions of Theorem 13 are fulfilled.Another source of error is the effect of the numerical integration. We decompose

the single layer operator to a singular part with logarithmic singularity, and to asmooth part. The singular part is integrated exactly. The smooth part as wellas integrals concerning the double layer operator are computed using numericalquadratures. We suppose that the quadrature error can be bounded above byc hσ. For the right hand side of the equation, we use the orthogonal projectionapproximation. The resulting nonlinear system defines the mapping A : Sdh → Sdhsuch that

A(vh) = IhPhgNΓ. (3.29)

The solution vh is related to the approximate solution of the original problem bythe formula vh = |x′|uh. We summarize the final result in the following theorem.The details are written in the original article.

52

Theorem 26 (IV, Theorem 5.2) Assume d > 0. Let g ∈ Hs−1(Γ) and letv ∈ Hs(Γ), 1

2 < s ≤ d + 1 be the solution of (3.29) and suppose that assumptions(A1), (A2) are valid. Then we have the estimate

‖u− uh‖t ≤ c hs−t(‖v‖s + ‖g‖s−1) + c hτ+1−max(t, 12 ) + c hσ−1−max(t, 12 ),

for 0 ≤ t ≤ s, t < d+ 12 , provided that the data is sufficiently smooth.

It evolves that our method retains the optimal convergence order of the collo-cation method. Numerical experiments confirm our theoretical results.

3.5 Article V: On the numerical solution of a non-linearheat conduction problem

In this article, we analyse a projection method scheme for solution of the secondkind non-linear heat operator equation (1.1), assuming that the spatial domain Ωis two-dimensional and has a smooth boundary Γ. The direct boundary integralapproach yields the nonlinear boundary integral equation (1.21). We denote theassociated nonlinear integral operator with

A(u) = ( 12I +DΓ)u+ SΓF(u) (3.30)

and consider the nonlinear operator equation A(u) = SΓgNΓ. Concerning thenonlinearity, we assume that Caratheodory conditions (1.91) and (1.92), growthcondition (1.93) and Lipschitz condition (1.95) are true. These imply that theNemitsky operator F : L2(ΣT )→ L2(ΣT ) is a well-defined, bounded and Lipschitzcontinuous operator. If, in addition, the mapping ξ 7→ F (x, t, ξ) is non-decreasingfor each (x, t) ∈ ΣT then the Nemitsky operator is monotone. See Lemma 3 andLemma 4. According to the following theorem, equation (3.30) is uniquely solvable

in H12 ,

14

00 (ΣT ), [30].

Theorem 27 The operator A : H12 ,

14

00 (ΣT ) → H12 ,

14

00 (ΣT ) is a homeomorphism.Furthermore, we have(

A(u)−A(w) , S−1Γ (u− w)

)ΣT≥ c ‖u− w‖21

2 ,14, (u,w ∈ H

12 ,

14

00 (ΣT )).

We propose a method where we use interpolation with respect to the spacevariable, and orthogonal projection with respect to the time variable. The trialfunctions are tensor products of piecewise linear and piecewise constant splines.We define the projection operator Q1,0

h,k := I1h⊗P 0

k = P 0k⊗I1

h. The discrete problemasks us to find u∆ ∈ S1,0

h,k such that

Q1,0h,kA(u∆) = Q1,0

h,kSΓgNΓ. (3.31)

The method requires evaluation of pointwise values. Thus, we need a certainregularity concerning A(u∆) and SΓgNΓ. For the right-hand side, it is enough to

53

assume that gNΓ ∈ Hs, s200 (ΣT ), s > 1

2 , since, according to the mapping property ofthe single layer operator, SΓgNΓ ∈ H

s, s200 (ΣT ), s > 3

2 . Furthermore, it is known thatthe embedding Hr, r2

00 (ΣT ) ⊂ C(ΣT ) is continuous for r > 32 . Sufficient regularity

for A(u∆) follows easily from the special structure of the operator A. The identityoperator preserves splines and therefore the required pointwise values exist. Onthe other hand, the trial function space satisfies S1,0

h,k ⊂ Hr, r200 (ΣT ), r < 1, and the

mapping properties of the associated operators yield DΓu∆ + SΓF(u∆) ∈ C(ΣT ).

Lemma 9 If the meshes are quasi-uniform and v, v′ ∈ S1,0h,k, then

‖(I −Q1,0h,k)DΓ(v − v′)‖ 1

2 ,14≤ c

(h

12 + k

14)‖v − v′‖ 1

2 ,14, (3.32)

‖(I −Q1,0h,k)SΓ(F(v)−F(v′))‖ 1

2 ,14≤ c

(h+ k

12)‖v − v′‖ 1

2 ,14. (3.33)

Proof. Estimate (3.33) follows from Lemma 1, with s = 1 by means of the map-ping property of the single layer heat operator and the Lipschitz continuity of theNemitsky operator

‖(I −Q1,0h,k)SΓ(F(v)−F(v′))‖ 1

2 ,14≤ c

(h

12 + k

14)‖SΓ(F(v)−F(v′))‖1, 12

≤ c(h

12 + k

14)‖F(v)−F(v′)‖0,0

≤ c(h

12 + k

14)‖v − v′‖0,0

≤ c(h

12 + k

14)‖v − v′‖ 1

2 ,14.

Estimate (3.33) is proved analogously.

Theorem 28 (V, Theorem 3) For sufficiently small h and k, we have the sta-bility inequality

‖Q1,0h,kA(v)−Q1,0

h,kA(v′)‖ 12 ,

14≥ c0 ‖v − v′‖ 1

2 ,14, (v, v′ ∈ S1,0

h,k).

In addition, equation (3.31) is uniquely solvable and we have the error estimates

‖u− u∆‖s, s2 ≤ c infv∈S1,0

h,k

‖u− v‖s, s2 ≤ c(h2−s + k1− s2

)‖u‖2,1, (s = 0, s = 1

2 ).

Proof. Since Q1,0h,k is a projection, we can write

(I −Q1,0h,k)(A(v)−A(v′)) = (I −Q1,0

h,k)DΓ(v − v′) (3.34)

+ (I −Q1,0h,k)SΓ(F(v)−F(v′)).

The Schwarz inequality(Q1,0h,kA(v)−Q1,0

h,kA(v′) , S−1Γ (v−v′)

)ΣT≤ c ‖Q1,0

h,kA(v)−Q1,0h,kA(v′)‖ 1

2 ,14‖v−v′‖ 1

2 ,14

combined with the decomposition(Q1,0h,kA(v)−Q1,0

h,kA(v′) , S−1Γ (v − v′)

)ΣT

=(A(v)−A(v′) , S−1

Γ (v − v′))

ΣT−((I −Q1,0

h,k)(A(v)−A(v′)

), S−1

Γ (v − v′))

ΣT

≥ c ‖v − v′‖212 ,

14− c′ ‖(I −Q1,0

h,k)(A(v)−A(v′)

)‖ 1

2 ,14‖v − v′‖ 1

2 ,14

54

leads to the estimate

‖Q1,0h,kA(v)−Q1,0

h,kA(v′)‖ 12 ,

14≥ c ‖v−v′‖ 1

2 ,14− c′ ‖(I−Q1,0

h,k)(A(v)−A(v′)

)‖ 1

2 ,14.

This implies stability for 0 < h ≤ h0 and 0 < k ≤ k0, since Lemma 9 togetherwith (3.34) yields

‖(I −Q1,0h,k)

(A(v)−A(v′)

)‖ 1

2 ,14≤ c (h

12 + k

14 )‖v − v′‖ 1

2 ,14.

Due to stability, the operator Q1,0h,kA : S1,0

h,k → S1,0h,k is an injection. According to

Brouwer’s theorem on domain invariance, the range R(Q1,0h,kA) is open. Because

of the stability and continuity of Q1,0h,kA in S1,0

h,k, the range R(Q1,0h,kA) is also closed

in S1,0h,k. Therefore, R(Q1,0

h,kA) = S1,0h,k, and Q1,0

h,kA is a homeomorphism. Thequasi-optimality with respect to ‖ · ‖ 1

2 ,14

is a direct consequence of stability.

The L2(ΣT )-estimate is proved by means of the technique in [55]. In fact, theresult follows from Theorem 14, since conditions (2.5) - (2.8), and approximation

property (2.9) are true when we choose the spaces X0 = L2(ΣT ), X = H12 ,

14

00 (ΣT ),

Z = H1, 1200 (ΣT ) and the operators S = SΓ, D = DΓ, F = F , Qn = Q1,0

h,k andXn = S1,0

h,k. Also, the mapping property D : X0 → Z is bounded, is fulfilled.

In the numerical implementation, the resulting time-integrals can be calculatedexactly. Compared to the Galerkin method, this approach simplifies the realiza-tion of the matrix equations. It seems that straightforward application of thepreceeding analysis does not extend to the collocation method.

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