A Posteriori Error Estimator for the Multi-Level hp-Finite ...Angella.pdf · Namely, the multi-level Finite Element Method and the Finite Cell’s Method (FCM). The multi-level FEM

Faculty of Civil Engineering and Geodesy

Chair for Computation in Engineering

Prof. Dr.rer.nat. Ernst Rank

A Posteriori Error Estimator for the

Multi-Level hp-Finite Element Method

Davide D’Angella

Master’s Thesis for the Master of Science program

Computational Science and Engineering

Author: Davide D’Angella

Matriculation number:

Supervisors: Prof. Dr.rer.nat. Ernst Rank

Prof. Dr. Andreas Schroder

Advisors: Dr.-Ing. Stefan Kollmannsberger

Nils Zander, M.Sc.

Date of issue: 18th May 2015

Date of submission: 26th October 2015

Involved Organisations

Chair for Computation in EngineeringFaculty of Civil Engineering and GeodesyTechnische Universitat MunchenArcisstrasse 21D-80333 Munchen

Declaration

With this statement I declare, that I have independently completed this Master’s thesis. Thethoughts taken directly or indirectly from external sources are properly marked as such. Thisthesis was not previously submitted to another academic institution and has also not yetbeen published.

Munchen, October 27, 2015

Davide D’Angella

Davide D’AngellaChristoph Probst Str. 16\102D-80805 Munchene-Mail: [email protected]

III

Abstract

This works aims to investigate the applicability of residual-based error estimators to twovariants of the Finite Element Method: the multi-level Finite Element Method and the FiniteCell’s Method (FCM). The study is performed both theoretically and numerically, showingexcellent results for the Multi-Level Finite Element Method in the context of Poisson’s andlinear elastic problems. Furthermore, the analysis carried out in the context of the FCMfurnishes a starting point for further developments and investigations on how to estimate theerror in case of a non-conforming discretization.

IV

Acknowledgements

I would like to express my gratitude to Dr.-Ing. Stefan Kollmannsberger for the accurate andsuccessful leading of this work. Moreover, his very warm personality created a comfortableand productive working atmosphere. I really appreciated it.

My sincere thanks also go to Nils Zander, who dedicated a lot of energies and time in thiswork. His help was definitely indispensable for completing the implementational tasks. Fur-thermore, his ideas gave a great contribution to this work. I feel sorry for his wife, for havingkept him repeatedly in the office until late.

I would like to thank Prof. Ernst Rank and Prof. Andreas Schroder for the discussions andfor examining this work.

I am grateful to the whole team of the chair for Computation in Engineering for the variousform of support and help. I am looking forward to becoming a colleague of yours.

A really special thank you goes to my bachelor professor Prof. Massimo Tarallo for havingshaped my mind and for having instilled so much passion into me. This gave me the strengthand the will to achieve many results, including having successfully concluded this work.

My thanks go also to Massimo Carraturo and Luca Coradello for the constant mutual helpin the work-room.

Finally, my special thanks go to my best friends here in Munich, namely Sasha, Nusa and Eni.They supported my work by many different recreational activities indispensable to “rechargethe battery”.

VI

Contents

1 Introduction and Motivation 21.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Refinement Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Error Estimator Driven Adaptivity . . . . . . . . . . . . . . . . . . . . 31.3 Goal of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Error Estimators 52.1 Classification of Error Estimators . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Explicit Residual-Based Error Estimator . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Model Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.2 Finite Element Solution Space . . . . . . . . . . . . . . . . . . . . . . 72.2.3 Error Upper Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.4 Error Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Effectivity Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4 Smoothness Indicator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.5 Adaptivity Driven by Error Estimator and Smoothness Indicator . . . . . . . 142.6 Error Estimator for Linear Elasticity . . . . . . . . . . . . . . . . . . . . . . . 14

3 Error Estimation for Multi-Level hp-Finite Element Method 173.1 General Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.1.2 The Multi-Level hp-Finite Element Method . . . . . . . . . . . . . . . 17

3.2 Error Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2.1 The Concept of Element in the Multi-Level Basis . . . . . . . . . . . . 193.2.2 Applicability of Error Estimation to Multi-Level hp-Finite Element

Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2.3 The Span of the Nodal Multi-Level Basis . . . . . . . . . . . . . . . . 213.2.4 The Span of the Multi-Level Basis . . . . . . . . . . . . . . . . . . . . 213.2.5 Error Estimation for One-Dimensional and Regular Meshes . . . . . . 22

3.3 Implementational Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.4.1 Results in 1D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.4.2 Results in 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4 Error Estimation for the Finite Cell Method 444.1 General Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.1.2 The Finite Cell Method . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.1.3 Adaptive Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2 Error Upper Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2.1 A Possible Strategy to Prove the Error Upper Bound . . . . . . . . . 484.2.2 Another Possible Strategy to Prove the Error Upper Bound . . . . . . 514.2.3 Error Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.3.1 Results in 1D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5 Summary and Conclusions 595.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

1

2

Chapter 1

Introduction and Motivation

1.1 Overview

In the last few decades, the rapid growth of computational power and number of availabledifferent methods made the Computer-Aided Engineering (CAE) an industry standard. Forexample, the Finite Element Method (FEM) is the nowadays most-established method forsolving complex mechanical problems using computers. The growth in size and complex-ity of the engineering design requires continuous advances in the efficiency and reliabilityof the adopted computational methods. Therefore, many innovative ideas in this directionsare constantly developed. For instance, the chair of Computation in Engineering (CiE)at the Technische Universitat Munchen (TUM) developed two powerful variants of the fi-nite element method. Namely, the multi-level Finite Element Method and the Finite Cell’sMethod (FCM). The multi-level FEM aims to reduce the implementational difficulties repre-sented by the hanging-nodes, while allowing the full approximation capabilities of the FEM.Instead, the Finite Cell’s Method tackles the time-consuming and error-prone problem ofgenerating meshes conforming to complex geometries. Indeed, this method allow to performFinite Element Analysis (FEA) to structures of almost any shape, while adopting a simplenon-conforming regular mesh. Both these methods are shown to achieve full accuracy withoptimal rate of convergence [ZBK+15, PDR07, DPYR08, DDR14].

1.2 Refinement Process

Given a problem to solve, the FEM is based on a discretization into elements of the domainand it produces a numerical approximation of the actual solution. This, of course, needssome precautions to ensure that the solution is approximated precisely enough to provide theinformation of interest. For example, in practical problems the accuracy of the approximationcan be easily deteriorated by extremely steep gradients or singularities arising by re-entrantcorners.

The accuracy of a numerical solution can be improved by a refinement process. In this work,refinements in space (h-refinement) and in polynomial order of the basis shape functions(p-refinement) are considered. Namely, the discretization can be made finer in some criti-

1.2. Refinement Process 3

cal areas, or the approximation capabilities of some elements can be increased. Since thecombination of h- and p-refinements enlarge considerably the computational complexity, it ispractically necessary to perform these operations efficiently. It is of common knowledge thatareas close to singularities can be better approximated by h-refinements, while areas wherethe true solution is smooth can be estimated more efficiently by p-refinements. Therefore, anatural problem that arises is how to identify these regions, and how to perform refinementsoptimally. Namely, how to obtain the maximum increase in the accuracy, with the leastincrease in computational complexity.

1.2.1 Error Estimator Driven Adaptivity

Efficient refinements can be performed by using some so-called a-priori knowledge about theproblem. For example, some refinements can be employed near a re-entrant corner of thegeometry, or near the imposition of a surface load or a source of heat. Indeed, in these areassingularities and steep gradients in the solution might be encountered. However, the produc-tion of discretizations from a-priori knowledge might not be easily performed automatically.Furthermore, it is not straightforward to asset the quality of such a discretization, in orderto evaluate if spatial- and order-level of refinement are satisfactory in one area. Finally, alsocases in which the amount of a-priori knowledge is insufficient or absent have to be takeninto account.

These considerations motivate the employment of an error estimator, a tool able to auto-matically estimate the accuracy of the numerical approximation and the local quality of thediscretization. Error estimators make typically use of a-posteriori information, such as thenumerical solution obtained. An error estimator is required to satisfy an error upper- andlower-bound. While, the first ensures that the numerical solution has accuracy below a cer-tain threshold, also the lower bound is needed to assure sharpness of the estimate. As clarifiedlater, this is a crucial property that will ensure efficiency when designing meshes based onerror estimates. Indeed, it assures to avoid unnecessary over-refinements and increase in thecomputational costs.

The capability of producing local estimates, enables the error estimator to drive adaptivity :an automatic refinement process. In the following the computational complexity will be mea-sured by the number of Degrees of Freedom (DOFs) that have to be computed to identifythe numerical approximations. Hence, given a problem with a solution function u, we ideallyseek an approximate solution uT that yields an error estimate η below a certain thresholdCerror, while minimizing the number of DOFs. Approximate solutions to this problem can beattained by the Algorithm 1. Such an algorithm is quite generic, as it is not practically spec-ified how to identify the elements to be refined and how to choose the refinement strategies.These operations can be performed following different approaches. For a complete surveyand comparison of the available adaptive techniques we refer to [MM11]. The methodologyadopted in this work will be introduced later in Chapter 2.

1.3. Goal of the Thesis 4

Algorithm 1 Adaptivity

1: procedure Adaptivity(Problem P, Tolerance Cerror, Maximum Steps Csteps)2: j ← 03: η ← Cerror + 14: Produce an initial coarse discretization mesh T0 for P5: while η > Cerror and j ≤ Csteps do6: Solve P on Tj7: Compute error estimate η8: Identify the elements to refine9: Choose the refinement strategy for each element to be refined

10: Construct Tj+1

11: j ← j + 112: end while13: end procedure

1.3 Goal of the Thesis

While error estimator and adaptivity have been extensively investigated both theoreticallyand numerically for the Finite Element Method, this has not been done yet for the FEMvariants proposed by the chair of CiE.

Therefore, the goal of this thesis is to conduct conceptual and numerical investigations about:

• Error estimation for the Multi-Level Finite Element Method• Adaptivity steered by an error estimator for the Multi-Level Finite Element Method

Finally, also a small investigation will be conducted on error estimation for the Finite Cell’sMethod.

1.4 Structure of the Thesis

This thesis has the following structure. Chapter 2 will provide a general classification of errorestimators, formulate the type of estimator considered in this work and prove an error upperbound. Furthermore, the effectivity index will be introduced as a measure of the quality ofan error estimator. Moreover, a smoothness indicator will be introduced as decider betweenh- and p-refinements.Next, Chapter 3 will introduce the Multi-Level Finite Element Method, discuss the applica-bility of the error estimator to this technique, and present various numerical results.Finally, Chapter 4 will present the Finite Cell’s Method and briefly investigate the errorestimation in this context with some numerical experiments.

5

Chapter 2

Error Estimators

In this chapter, a brief classification of the popular error estimators found in literature willbe presented. Then, more detailed explanations are directed toward the particular class ofestimator treated in this work: the residual-based error estimators. Indeed, an error upper-bound is proved for the model problem, and a formal definition is provided. Furthermore, theeffectivity index is introduced as a measure of the quality of the considered error estimator.Next, a smoothness indicator based on the decay rate of the Legendre coefficients is presentedas decider between h or p refinement for an element with an associated high local error. Thiswill accompany the error estimator in a fully automatic hp-adaptive algorithm. Finally, theresidual-based error estimators is defined for linear elastic problems.

2.1 Classification of Error Estimators

Over the last decades, the practical benefits of reliable error estimators became apparent,making them a popular tool. Therefore, many different techniques and idea have been de-veloped in this context. In this section a brief list of the most common approaches in errorestimation on linear second-order elliptic problem is presented. For a complete survey, werefer to [Ver99, AO97].

The first obvious distinction is on the information used by the estimator:

• a priori error estimators are based on the analytical solution of the boundary valueproblem and its qualitative properties. The estimates are obtained by theoretical inves-tigation and, often, strong assumptions are made. Unfortunately, these are not alwayssatisfied in practical applications. Moreover, a priori estimates typically furnish infor-mation just on the asymptotic behaviour of the error, giving them a limited practicaluse. However, this type of estimator can be useful to choose the discretization schemeand tune some parameters. They also provide information about possible improvementsof a solution strategy, as they can furnish the best theoretical convergence behaviourthat one can possibly reach.

• a posteriori error estimators are based on the numerical solution of the boundaryvalue problem. They are due to the pioneer work [BR78a] and they rapidly expanded

2.1. Classification of Error Estimators 6

due to their huge practical implications as adaptivity [BR78b, BR78a].

The a posteriori estimators have been developed with many different techniques, deservingan own classification. We will present a rough survey following [Ver99, AO97].

• Residual-based (explicit) estimators: bounds to the approximation error of a nu-merical solution are obtained by computing some norm of its residuals with respect tothe strong form of the partial differential equation [BR78b]. This will be the type oferror estimator considered in this work, and more detailed explanations follow in thischapter.

• Local problem-based (implicit) estimators: discrete auxiliary problems, whichare similar, but simpler than the original problem are solved locally. Then, estimatesare computed by suitable norms of the obtained solutions. The auxiliary problemscan be defined on small sub-domains and different kind of boundary conditions can bespecified. For example, it was proposed to take as sub-domains one element and itsdirect neighbours, or the set of elements sharing a node. Then, Dirichlet boundaryconditions can be imposed taking the value of the current numerical solution. Alter-natively, Neumann boundary conditions can be specified based on the inter-elementresidual of the numerical solution. Of course, the Neumann and Dirichlet boundaryconditions can be enforced on the sub-boundary of the local problem that conforms theNeumann and Dirichlet boundaries of the original domain. See [Ver99] for details. Thisapproach was initially proposed [BR78b, BR78a, BR79] and expanded for example in[BW85, Ban86, DOS84, DOS85].

• Equilibrated residual method: this technique represents a variant of the localproblem-based estimator. Indeed, a sub-problem is defined for each element and itsdirect neighbours. The particularity of this approach is that Neumann boundary con-ditions are imposed with values obtained by the actual numerical fluxes. Moreover,an equilibrium condition is enforced. This strategy was first proposed by [LL83]. Fur-thermore, the equilibrium condition has been proved in [AO97, AO93] to assure thewell-posedness of the local problem. This was not necessarily true for the sub-problemsconsidered by implicit estimators.

• Averaging-based (recovery methods) estimators: this method is based on con-structing a smoothed gradient from the (in general discontinuous along inter-elementinterfaces) gradient of the numerical solution. Then, error estimates are obtained bycomparing the smoothed gradient to the original one. This method was introduced in[ZZ88, ZZ87], further investigated for example in [AZCZ89, RZ87, CF01] and improvedfor important classes of problems in [ZZ92a, ZZ92b].

• Hierarchical-basis estimators: Error estimates are obtained from the residual ofthe current numerical approximation with respect to the solution obtained from a finiteelement space enhanced by a nearly orthogonal complement space. In this way, theresidual is represented just by the contribution due to the complement. Moreover, theorthogonality properties allow efficient computations, as a small problem can be setup just for the complement contribution alone. Such a technique was introduced in[Ban96, BS93] and [DLY89].

2.2. Explicit Residual-Based Error Estimator 7

• Goal-oriented estimators: typically estimates of the the energy- or Lp-norm of theerror are delivered. Nevertheless, it is often of practical relevance to estimate the error inother quantities of direct interest, such as stress or temperature. Such estimates wouldallow also to assess if a particular discretization produces a satisfactory resolution ofthese quantities. This approach was initially discussed in [BM82a, BM82b, BM84,Lou79]. Modern approaches and applications can be found in [PO99, OP01, SR11,BRS12]

2.2 Explicit Residual-Based Error Estimator

2.2.1 Model Problem

We will consider as second-order elliptic linear model problem the Poisson’s Equation on ad-dimensional bounded domain Ω ⊂ Rd. We assume the boundary Γ of Ω to be Lipschitzand consisting of two disjoint parts ΓD and ΓN , representing the Dirichlet- and Neumann-Boundary, respectively.

−∆u = f on Ω

∇u · n = g on ΓN (2.1)

u = 0 on ΓD.

Then, its weak formulation reads

Find u ∈ H1D(Ω) such that

B(u, v) = F(v) ∀v ∈ H1D (Ω), (2.2)

where

B(u, v) =

∫Ω∇u · ∇v dΩ

F(v) =

∫Ωf · v dΩ +

∫ΓN

g · v dΓ ∀v ∈ H1D (Ω)

H1D (Ω) =

φ ∈ H1(Ω), φ = 0 on ΓD

.

Here, H1(Ω) ⊂ L2(Ω) denotes the classic Sobolev space of functions in L2(Ω) with weak-derivatives up to order one.

2.2.2 Finite Element Solution Space

To approximate a solution of the model problem (2.1), we consider a high-order Finite Ele-ment Discretization of (2.2).

Let n ∈ N, n > 0, and T (Ω) = Ωene=1 be a suitable partition or mesh of Ω ⊂ Rd into setsΩe called elements. Considering high-order approximations, we associate to each element Ωe

a polynomial degree pe and a diameter he. We consider meshes satisfying the conditions:


• ⋃Ωe∈T (Ω)

Ωe = Ω

• Affine equivalence: each Ωe is a rectangle or a triangle when d = 2, or a tetrahedronon parallelepiped when d = 3• Admissibility : any two elements in T (Ω) are either disjoint or share a vertex or a

complete edge, when d = 2, or a complete face, when d = 3• Shape regularity : for any element Ωe, the ratio of its diameter he to the diameter ρe of

the largest ball inscribed into Ωe is bounded independently of Ωe

The shape regularity assumption assures in two-dimensions that the smallest angle of allelements stays bounded away from zero. Furthermore, when considering sequences of meshesobtained by refinement procedures, we will asume that the ratio he/ρe is bounded uniformlywith respect to all elements of all meshes. We will denote the polynomial degree distributionas p = pene=1. Furthermore, let S (T , p) denote the corresponding Finite Element solutionspace

S (T , p) = φ ∈ H1D (Ω) ∩ C0 (Ω) : φ|Ωe ∈ Ppe∀Ωe ∈ T (Ω),

where Ppe is the space of polynomials of degrees at most pe, i.e. Ppe = spanxα11 · . . . · xαdd :

α1, . . . , αd ∈ R,max(α1, . . . , αd) ≤ pe. Then, the Finite Element Method for the modelproblem (2.1) reads

Find uT ∈ S (T , p) such that

B (uT , vT ) = F (vT ) ∀vT ∈ S (T , p). (2.3)

In the following we will denote by ∂Ωe the boundary of Ωe and by ∂T int(Ω) the set of internalinter-element faces of the finite element partition T (Ω).

2.2.3 Error Upper Bound

In this section we derive an a-posteriori error estimate following closely the derivation in[TG05] and [CJ92].

Let e = u − uT denote the approximation error. Then, our aim is to approximate the errorin energy norm

‖e‖E(Ω) =√B(e, e). (2.4)

Such an estimate can be obtained as follows.

B(e, v) = B(u, v)− B(uT , v)

= F(v)− B(uT , v)

=

∫Ωf · v dΩ +

∫ΓN

g · v dΓ−∫

Ω∇uT · ∇v dΩ

=∑

Ωe∈T (Ω)

∫Ωe

f · v dΩ +

∫∂Ωe∩ΓN

g · v dΓ−∫

Ωe

∇uT · ∇v dΩ

. (2.5)


Using the Gauss theorem on the last integral, we obtain

B(e, v) =∑

Ωe∈T (Ω)

∫Ωe

f · v dΩ +

∫∂Ωe∩ΓN

g · v dΓ−∫∂Ωe

v · ∇uT · n dΓ +

∫Ωe

∆uT · v dΩ

(2.6)

=∑

Ωe∈T (Ω)

∫Ωe

[∆uT + f ] · v dΩ +

∫∂Ωe∩ΓN

[g −∇uT · n] · v dΓ−∫∂Ωe\ΓN

v · ∇uT · n dΓ

,

(2.7)

where n(x) is the boundary outward normal at the point x. To express the integrand on∂Ωe \ ΓN as inter-element boundary residual, we rearrange the sums in terms of the set ofinternal inter-element interfaces ∂T int. Since uT = 0 on ΓD, we can write

B(e, v) =∑

Ωe∈T (Ω)

∫Ωe

[∆uT + f ] · v dΩ +

∫∂Ωe∩ΓN

[g −∇uT · n] · v dΓ

+

+∑

γ∈∂T int

∫γv ·[∇uT ,Ωe1 −∇uT ,Ωe2

]· n dΓ, (2.8)

where Ωe1 and Ωe2 are the two elements sharing the internal interface γ, and uT ,Ωe1 , uT ,Ωe2are associated local numerical solution, respectively. Namely,[

∇uT ,Ωe1 −∇uT ,Ωe2]· n = lim

t→0(∇uT · n) (x+ tn)− (∇uT · n) (x− tn) ∀x ∈ γ (2.9)

Therefore, we obtain

B(e, v) =∑

Ωe∈T (Ω)

∫Ωe

RΩe(uT ) · v dΩ +

∫∂Ωe

R∂Ωe(uT ) · v dΓ

,

with

RΩe(uT ) = ∆uT + f (2.10)

R∂Ωe(uT ) =

12

[∇uT ,Ωe1 −∇uT ,Ωe2

]· n on ∂Ωe \ Γ

g −∇uT · n on ΓN

0 on ΓD,

(2.11)

where the factor 1/2 accounts for the fact that every inter-element internal interface is con-sidered twice in Equation (2.8): once for each adjacent element. The quantities RΩe(uT ) andR∂Ωe(uT ) are called the interior and boundary residuals, respectively.

The classic derivation of the residual-based error estimator makes use of the following esti-mates from interpolation theory [CJ92]:

‖v − IS(T ,p)v‖L2(Ωe) ≤ C1,T he‖v‖H1(Ωe)∀Ωe ∈ T , v ∈ H1

D (Ω) (2.12)

‖v − IS(T ,p)v‖L2(∂Ωe) ≤ C2,T√he‖v‖H1(Ωe)

∀Ωe ∈ T , v ∈ H1D (Ω), (2.13)

where IS(T ,p) : H1D (Ω)→ S (T , p) is the Clement interpolation operator [Cle75] and Ωe is the


set of all elements in that share at least a vertex with Ωe. The operator IS(T ,p) is a general-ization of the standard nodal interpolation operator that can be used also for non-continuousfunctions and higher-spatial dimensions [Car06, Cle75].

The Galerkin Orthogonality

B(e, v) = B(e, v − vT ) ∀v ∈ H1D (Ω),∀vT ∈ S (T , p)

enables us to use the estimates (2.12) and (2.13), by choosing for every v ∈ H1D (Ω) the

function vT ∈ S (T , p), vT = IS(T ,p)v. Indeed, by using the Cauchy-Schwarz inequality twice,we obtain ∀v ∈ H1

D (Ω)

B(e, v) = B(e, v − IS(T ,p)v) (2.14)

=∑

Ωe∈T (Ω)

∫Ωe

RΩe(uT ) ·(v − IS(T ,p)v

)dΩ +

∫∂Ωe

R∂Ωe(uT ) ·(v − IS(T ,p)v

)dΓ

≤∑

Ωe∈T (Ω)

‖RΩe(uT )‖L2(Ωe)‖v − IS(T ,p)v‖L2(Ωe) + ‖R∂Ωe(uT )‖L2(∂Ωe)‖v − IS(T ,p)v‖L2(∂Ωe)

≤∑

Ωe∈T (Ω)

C1,T he‖RΩe(uT )‖L2(Ωe)‖v‖H1(Ωe)+ C2,T

√he‖R∂Ωe(uT )‖L2(∂Ωe)‖v‖H1(Ωe)

≤

∑Ωe∈T (Ω)

C21,T h

2e‖RΩe(uT )‖2L2(Ωe)

+ C22,T he‖R∂Ωe(uT )‖2L2(∂Ωe)

12

·

∑Ωe∈T (Ω)

‖v‖2H1(Ωe)

12

≤ K1‖v‖H1(Ω)

∑Ωe∈T (Ω)

C21,T h



12

,

for some constant K1 that depends just on T which takes into account that every elementΩe is counted several times through Ωe. The last step is allowed by the shape-regularityassumption on the mesh. Finally, assuming B(·, ·) to be coercive in H1(Ω), i.e.

∃K2 > 0 : ‖φ‖H1(Ω) ≤ K2‖φ‖E(Ω)∀φ ∈ H1(Ω),

we can conclude

‖e‖2E(Ω) = B(e, e) (2.15)

≤ K1‖e‖H1(Ω)

∑Ωe∈T (Ω)

C21,T h



12

≤ K1K2‖e‖E(Ω)

∑Ωe∈T (Ω)

C21,T h



12

≤ C

∑Ωe∈T (Ω)

C21,T h



,


where C = K21K

22 .

2.2.4 Error Estimator

Motivated by the results of the Section 2.2.3, we define the error estimator in the energynorm

‖e‖2E(Ω) ≈ η2 =∑

Ωe∈Tλ2

Ωe(2.16)

λ2Ωe

= η2Ωe

+ η2∂Ωe

η2Ωe

= C21,T h


η2∂Ωe

= C22,T he‖R∂Ωe(uT )‖2L2(∂Ωe)

,

where the quantity λΩeis called element error indicator, and it is expressed in terms of ele-

ment interior residual ηΩeand element boundary residual η∂Ωe

.

As shown by Equation (2.15), such an estimator will provide an upper bound to the analyticalerror in the energy norm, up to a constant factor C. This property is usually called reliability,see for example [Ver98] [CJ92] [Mel05]. In case of low-order finite element approximations, theerror estimator (2.16) will also satisfy an error lower bound satisfying the so-called efficiencyproperty [Ver98] [CJ92]. This suggests sharpness of the error estimates.

Lately, a new interpolation operator has been proposed in [MW01] and [Mel05] that gener-alizes the Clement interpolation operator used in Equation (2.12) and (2.13) to the contextof high-order finite element approximations. The error estimator proposed in [MW01] reads

‖e‖2E(Ω) ≈ η2 =∑

Ωe∈Tλ2

Ωe(2.17)

λ2Ωe

= η2Ωe

+ η2∂Ωe

η2Ωe

= C21,T

h2e

p2e

‖RΩe(uT )‖2L2(Ωe)

η2∂Ωe

= C22,T

hepe‖R∂Ωe(uT )‖2L2(∂Ωe)

,

where the essential difference is the substitution of he by he/pe. In the paper [MW01] and[Mel05] an error upper-bound is given with a derivation similar to Section 2.2.3. However, incontrast to the case of low-order FEM, the error lower-bound given in [MW01] depends onthe polynomial order. It seems to be still an open question whether it is possible to obtainuniform efficiency. Namely, an error lower-bound independent of mesh size and polynomialdegree. Furthermore, note that in [MW01] and [Mel05] only homogeneous Dirichlet bound-ary conditions are considered, while here also Neumann boundary conditions are taken intoaccount. Since the latter seems more suitable to the context of hpFEM dealt in this work, inthe following we will make use of Definition (2.17) as error estimator.

Remark 2.2.1 The error estimator η = η (T , p, uT , f, g) and the element error indicatorλe = λe (T , p, uT , f, g) depend just on

2.3. Effectivity Index 12

• The mesh T (and therefore element size he)• The polynomial degree distribution p• The numerical solution uT• Right hand side function f• Boundary conditions g

2.3 Effectivity Index

To asset the quality of an explicit error estimator, it is common practice to use an effectivityindex, see for example [Ver98] [CJ92]. It is defined as the ratio between the error estimateand the actual true analytical error:

θ =η

‖e‖E(Ω). (2.18)

It is a measure of under- or over-estimation and its optimal value is θ = 1.

2.4 Smoothness Indicator

To decide whether the elements with an high error-level should be refined by an h- or p-refinement, a smoothness indicator based on the decay rate of the Legendre coefficients isconsidered in this work. This approach was first developed in [Mav94] and inspired similartechniques based on more sophisticated theoretical investigations, see for example [HSS03][HS05] [EM07].

This approach is based on the expansion of a function u ∈ L2([−1, 1],R) into the basis ofLegendre polynomials

u(x) =

∞∑i=0

αiPi(x), (2.19)

for some αi ∈ R. The coefficients αi are called Legendre coefficients and, due to the orthog-onality property of the Legendre polynomials∫ 1

−1Pi(x)Pj(x) dx =

2

2i+ 1δij ,

where δij denotes the Kronecker delta, they can be computed as

αi =2i+ 1

2

∫ 1

−1Pi(x)u(x) dx. (2.20)

This expression can be easily be obtained by multiplying Equation (2.19) by Pi(x) and inte-grating over [−1, 1].

The key idea in estimating the smoothness properties of u is that for smooth functions, |αi|

2.4. Smoothness Indicator 13

is expected to decay exponentially to zero as i→∞, while we expect this to be false in caseof non-smooth functions [Mav90] [Mav94].

Hence, in [Mav94] it is proposed to estimate the smoothness of the analytical solution u of aboundary value problem, by computing the decay rate of Legendre coefficients of the availablenumerical solution uT . Consider, for example, an element Ωe with an associated polynomialdegree pe and a local numerical solution uΩe

. Since uΩe∈ Ppe , the finite dimensional space of

polynomials of degree at most pe, the expansion (2.19) of uΩebecomes

uΩe(x) =

pe∑i=0

αiPi(x).

The coefficients αi are computed by formula (2.20) and the decay rate τ by a best-fit of

|αi| = ceτi. (2.21)

This can be done by linear least square fitting of

ln |αi| = ln c+ τi,

yielding the formula

τ =

(pe + 1)pe∑i=0

i ln |αi| −pe∑i=0

i ·pe∑i=0

ln |αi|

(pe + 1)pe∑i=0

i2 −(pe∑i=0

i

)2 . (2.22)

It is well known that areas in which the solution is smooth can be approximated effectivelyby a p-refinement, while for areas close to singular points or containing steep gradients it ismore convenient to perform h-refinement [SB91]. Hence, τ can be used as decider betweenthese two refinement strategies, by performing a p-refinement if τ ≤ Cdecay and h-refinementwhen τ > Cdecay. Here, Cdecay ∈ R is a chosen parameter. In [Mav94] it was suggested todefine Cdecay = 1.

In case of multi-dimensional problems, we followed the approach presented in [HSS03, Wan11]for the anisotropic finite element discretizations. Namely, where element can have differentpolynomial orders in each spatial direction. Here, we consider quadrilateral meshes. In thiscontext, a function u can be expanded into the basis obtained by the tensor product of thestandard Legendre polynomials. For example, in a three-dimensional problem

u(x, y, z) =∞∑

i,j,k=0

αi,j,kPi(x)Pj(x)Pk(x),

with

αi,j,k =2i+ 1

2

2j + 1

2

2k + 1

2

1∫−1

1∫−1

1∫−1

u(x, y, z)Pi(x)Pj(x)Pk(x) dx dy dz.

2.5. Adaptivity Driven by Error Estimator and Smoothness Indicator 14

Therefore, the numerical solution uΩedefined locally to an element Ωe, can be expanded as

u(x, y, z) =

pe,x∑i=0

pe,y∑j=0

pe,z∑k=0

αi,j,kPi(x)Pj(x)Pk(x),

where pe,x, pe,y, pe,z are the polynomial degree in the x-, y- and z-directions. From the third-order tensor αi,j,k of Legendre coefficients, it is possible to obtain a one-dimensional sequenceof coefficients representative of each spatial direction by accumulating the coefficients of otherdirections. For example, for the x-direction

|αi| =pe,y∑j=0

pe,z∑k=0

|αi,j,k|√

2

2j + 1

√2

2k + 1.

At this point, it is possible to compute the decay rate τx in x-direction by performing thefit (2.21) to the representative coefficients |αi| as it was done for functions in one-dimension.Therefore, it is possible to use Formula (2.22) for |αi| and pe,x. Analogously can be done forthe other spatial dimensions.

Note that this method does not depend on the choice of the finite element basis used, butjust on the numerical solution and on the discretization.

2.5 Adaptivity Driven by Error Estimator and SmoothnessIndicator

In this section we present a specialization to the vague adaptivity Algorithm 1 that make useof the tools introduced so far: the explicit-residual based error estimator and the smoothnessindicator based on the Legendre coefficients decay rate. See Algorithm 2.

In particular, elements are marked for refinement based on their element error indicatornormalized by the maximum element error in the mesh. Therefore, it make sense to defineCelementError ∈ [0, 1]. Moreover, the decision between h- and p-refinements is made by com-paring a parameter Cdecay to the decay rate of the Legendre coefficients of the numericalsolution local to one element, as described in Section 2.4. In [Mav94], it was suggested tochoose Cdecay = −1.

2.6 Error Estimator for Linear Elasticity

In the previous sections, error estimators were discussed only in the context of the Poissonmodel problem (2.1). However, the same residual based estimators can be defined also for

2.6. Error Estimator for Linear Elasticity 15

Algorithm 2 Adaptivity

1: procedure Adaptivity(Problem P, Error Tolerance Cerror, Element Refinement Toler-ance CelementError, Smoothness Tolerance Cdecay, Maximum Steps Csteps)

2: j ← 03: η ← Cerror + 14: Produce an initial coarse discretization mesh T for P5: while η > Cerror and j ≤ Csteps do6: Solve P on T7: Compute element error indicator λe (T , p, uT , f, g)8: Compute error estimate η (T , p, uT , f, g)9: Compute maximum element error level λmax among all the elements

10: for each element Ωe ∈ Tj do

11: ifλΩe

λmax> CelementError then

12: Compute Legendre coefficients decay rate τΩe

13: if τΩe< Cdecay then

14: Perform p-refinement on Ωe

15: else16: Perform h-refinement on Ωe

17: end if18: end if19: end for20: j ← j + 121: end while22: end procedure

2.6. Error Estimator for Linear Elasticity 16

linear elastic problems reading:

−∇ · σ(u) = f on Ω (2.23)

σ(u) = C : ε(u) on Ω

ε(u) =1

2(∇u+∇uT ) on Ω

u = 0 on ΓD

σ(u) · n = g on ΓN .

Here, Γ is the boundary of the problem domain Ω, consisting of two disjoint parts ΓD andΓN , denoting the Dirichlet and Neumann boundary, respectively. On the Dirichlet bound-ary homogenous displacement are enforced, while in the Neumann boudnary a traction g isimposed and n denotes the unit vector normal to the boundary. Moreover, σ denotes thestress tensor, ε the strain tensor, u the displacement function, f is the right-hand-side of theproblem that can be interpreted as a body force and, finally, C is the material tensor.

The weak formulation of (2.23) then reads

Find u ∈ H1D(Ω) such that

B(u, v) = F(v) ∀v ∈ H1D (Ω),

where

B(u, v) =

∫Ω

ε(v) : σ(u) dΩ

F(v) =

∫Ω

v · f dΩ +

∫ΓN

v · g dΓ.

A derivation of an error upper bound can be performed in a completely analogous way as itwas done in Section 2.2.3 with the newly defined bi-linear form B(·, ·). This way, an errorestimator for linear elasticity can be defined using the same Definition (2.17), but with thefollowing residual functions

RΩe(uT ) = ∇ · σ(uT ) + f

R∂Ωe(uT ) =

12

[σ(uT ,Ωe1)− σ(uT ,Ωe2)

]· n on ∂Ωe \ Γ

g − σ(uT ) · n on ΓN

0 on ΓD

17

Chapter 3

Error Estimation for Multi-Levelhp-Finite Element Method

In this chapter, the multi-level finite element method is motivated and introduced. Then,the applicability of error estimation to this context is discussed. Finally, numerous numer-ical results are presented for simple one- and three-dimensional linear elastic and Poisson’sproblems.

3.1 General Idea

3.1.1 Motivation

The aim of a hp-adaptive process is to improve the quality of the approximated solution atan optimal computational cost. One of the features used to reach this goal is to perform h-refinements locally on areas of interest, while keeping other areas unrefined. Obviously, thiscreates interfaces between elements with different level of refinement. Therefore, the meshcontains some nodes, edges and faces whose degrees of freedom are not free to assume anyvalue, but they have to be constrained to ensure C0-continuity of the numerical solution (see[Dem06], [SSD03]). They are called hanging-nodes, -edges and -faces and their treatmentincreases considerably the implementational difficulties of hp-adaptive procedures.

The multi-level Finite Element Method aims to fully avoid the problem of hanging nodes,while allowing local h-refinements and the full potential of hp-adaptivity.

3.1.2 The Multi-Level hp-Finite Element Method

The conceptual idea underlying the multi-level hp-Finite Element Method is based on theadopted mesh of elements. As introduced in Section 2.2.2, the standard FEM is based ona mesh T (Ω) = Ωene=1. The multi-level FEM generalises this framework by adopting an

overlay of meshes. Starting from a base mesh T 0(Ω) =

Ωe0n0

e0=1, additional meshes can

be superimposed on some base elements. For example, the element Ωa ∈ T 0(Ω) can be over-

3.1. General Idea 18

laid by T 1(Ωa) =

Ωa,e1n1

e1=1. One element Ωa,b ∈ T 1(Ωa), in turn, can be further refined

by superimposing an additional mesh T 2(Ωa,b). Superimposed meshes of the same level kare considered to form a unique mesh T k(Ω) on Ω. This procedure of mesh-overlaying canbe iterated an arbitrary number of times for each element of each overlay mesh. Finally, apolynomial degree has to be assigned to every element of each level to define its local basis.This can be done in many different way, as long as linear independence of the global shapefunctions is preserved. In this manner, a multi-level hierarchical structure of elements indefined.

We will call the elements in T k(Ωa) sub-elements of Ωa, Ωa is the parent of elements T k(Ωa)and elements without sub-elements will be called leaf or refined-most elements.

0 0.2 0.4 0.6 0.8 1−1

0

1

y

x

0 0.2 0.4 0.6 0.8 1−1

0

1

y

0 0.2 0.4 0.6 0.8 1−1

0

1

y

0 0.2 0.4 0.6 0.8 1−1

0

1

y

Figure 3.1: Example of multi level mesh with 4 levels

To clarify this concept, the multi-level basis depicted in Figure 3.1 has four hierarchicalmeshes

T 0(Ω) = Ω1 = [0, 1] T 1(Ω1) = Ω1,1 = [0, 0.5] ,Ω1,2 = [0.5, 1] T 2(Ω1,1) = Ω1,1,1 = [0, 0.25] ,Ω1,1,2 = [0.25, 0.5] T 3(Ω1,1,2) = Ω1,1,2,1 = [0.25, 0.375] ,Ω1,1,2,2 = [0.375, 0.5]

where the leaf elements Ω1,1,1,Ω1,1,2,1,Ω1,1,2,2,Ω1,2 have degree distribution from left toright 3, 2, 1, 4, and all other elements have degree 1.

In the following, when referring to multi-level mesh T (Ω), we will denote the set of leafelements as T leaf(Ω) and their degree distribution as pleaf. In case Ω is a one-dimensionaldomain, the spatial partition defined by T leaf(Ω) will be represented as a sequence of meshnodes belonging to the refined-most elements ordered from left to right. Analogously, this willbe done for pleaf. For example, for Figure 3.1 we will write T leaf(Ω) = [0, 0.25, 0.375, 0.5, 1]and pleaf = [3, 2, 1, 4].

The mesh overlay procedure requires some precautions in order to ensure

• Linear independence of the shape functions• C0-continuity of the numerical solution

3.2. Error Estimation 19

Intuitively, C0-continuity is obtained by imposing homogeneous Dirichlet boundary condi-tions on the meshes overlaying the base mesh. Namely, in each overlay mesh, the degrees offreedom on the interface with other meshes are set to zero (c.f. Figure 3.1). Practically, allthe basis function defined on one level are required to vanish on the nodes of the underlyinglevels. Note that this way difficulties of handling hanging components are completely elimi-nated by construction, while local refinements are allowed by mesh superimposition.Furthermore, linear dependence can be obtained with different combination of polynomialdegree distribution, for example by keeping all high-order modes on the leaf elements (c.f.Figure 3.1).From an implementational point of view, continuity and linear independence can be attainedby deactivating some topological components (nodes, edges or faces), i.e. by setting theirdegrees of freedom to zero. The full explanation on how to perform a correct deactivationis explained in the original paper [ZBK+15]. Here, we just point out that, from a practicalpoint of view, it is enough to implement two simple rules. Therefore, the multi-level finiteelement method is algorithmically well-suited to be implemented in a computer.

In conclusion, the multi-level version of the finite element method allows to perform hp-adaptivity with arbitrary refinement level, without the complication of handling hangingnodes edges or faces.

In the following, we will often consider multi-level basis where the high-order modes areplaced on the leaf-elements and all other elements are linear. We will call such a polynomialdegree distribution admissible. Moreover, each element will be refined with a number ofsub-elements equal to 2d, where d is the number of spatial dimensions.

3.2 Error Estimation

3.2.1 The Concept of Element in the Multi-Level Basis

The key idea on formulating an error estimator for the multi-level hp-Finite Element Methodrelies on the concept of “element”. As described above, a multi-level mesh is obtained bysuperimposition of standard meshes, each formed by usual elements . However, the union ofelements of all the levels yields overlapping subsets of the domain. Therefore, this choice doesnot produce a suitable definition of elements. Indeed, the admissibility property of Section2.2.2 is violated. On the other hand, there are more possible choices of subsets satisfying allthe properties of a mesh given in Section 2.2.2. Hence, a natural question that can arise iswhat is a meaningful choice for the element entities in case of a multi-level mesh Tml(Ω).

In the standard FEM, elements are used to define an integration domain and a local basisof shape functions. A natural requirement in the definition of the basis functions consistsof assuring convergence of the Galerkin approximations to the exact solution as the mesh isrefined. According to [Hug12], a sufficient set of conditions to guarantee this property onsecond-order problems is

• Smoothness: the shape functions shall be C1-continuous within each element,• Continuity : the shape functions shall be C0-continuous across element boundaries,


• Completeness: the shape functions shall represent exactly all polynomials of order upto one.

Despite the fact that these are just sufficient conditions, most of the finite element basissatisfy these requirements. In this work, we will consider shape functions fulfilling these req-uisites.

Therefore, not all the definition of “elements” of a multi-level basis are equally appropriate.Indeed, the smoothness property could be affected from this choice. For example, considerFigure 3.2a. The local basis on an subset Ωj ⊂ Ω of the domain can be defined as theunion of functions that are non-zero on Ωj . Then, choosing the interval [0, 1] as uniqueelement of the whole multi-level mesh, i.e. considering just the partition of the base overlaymesh, would not be a good option. Indeed, this would define an element with local shapefunctions not belonging to C1. In particular, the piecewise linear function have discontinuousfirst-derivatives at x ∈ 0.25, 0.375, 0.5. Analogously, considering the interval [0.25, 0.5] as

an element would lead to a similar situation. However, it easy to see that T leafml defines a

partition in which each element has a local basis of functions that are C∞-continuous in itsinterior. Moreover, the required mesh properties of Section 2.2.2 are also satisfied.

Consequently, the most natural choice of the element entities of a multi-level mesh Tml(Ω)

are the leaf elements T leafml (Ω). Indeed, the refined-most elements T leaf

ml (Ω) = Ωleafe are the

coarser entities satisfying properties of Section 2.2.2 and that define sub-domains in whichall the local basis functions within each leaf element Ωleaf

e are smooth. Loosely speaking,the refined-most elements define a “good” integration domain and a “good” local basis asin the standard finite element analysis. More precisely, the leaf elements T leaf

ml (Ω) form thecoarser partition of Ω that can define shape functions satisfying the above requirements ofsmoothness, continuity and completeness.

3.2.2 Applicability of Error Estimation to Multi-Level hp-Finite ElementMethod

The concept of “elements” of a multi-level basis is not of relevance just for convergenceproperties, as discussed in Section 3.2.1. It is also of fundamental importance for the errorestimation. In particular, in the derivation of an error-upper bound, Equation (2.5) makesuse of a sum over all the elements Ωe ∈ T (Ω) of a standard mesh T (Ω) was employed.

In the context of multi-level meshes, the definition of elements discussed in Section 3.2.1 canbe adopted. Indeed, it is possible to define a standard mesh Tstd(Ω) as Tstd(Ω) = T leaf

ml (Ω).Moreover, for each element Ωe ∈ Tstd(Ω), a set of local basis functions can be defined as theset of non-zero functions on Ωe in Tml. With this framework, the whole derivation of the er-ror upper-bound presented in Section 2.2.3 holds. As a result, the same the error estimators(2.16) and (2.17) can be defined on the elements T leaf

ml . Therefore, an element error indicatorλe is assigned just to the leaf elements, and not to other elements in the multi-level hierarchy.

This is somehow intuitive, indeed as it was said in Section 3.2.1, it is somehow “natural”to consider the leaf element in a multi-level basis as regular elements in a standard basis.


Here, we are just using the traditional error estimator for standard meshes, on the “natural”elements of the multi-level basis.

It should be noted that different definitions of elements would not yield the same result.Indeed, a choice different than T leaf(Ω) could lower the smoothness properties of the localshape functions. This, in turn, would not allow to use the derivation of Section 2.2.3. Forexample, the use of the Gauss theorem in (2.6) could not be allowed, as a continuouslydifferentiable vector field is required as an integrand.

3.2.3 The Span of the Nodal Multi-Level Basis

In the Section 3.2.5, the argument of the previous section is specialized to the case of one-dimensional domains or to the case of regular meshes. To this end, in this section it is statedthat, for each nodal multi-level basis, i.e. without high-order modes, there exists a standardbasis that spans the same space.

Let Φ (T ) denote the set of global nodal basis functions defined on the nodes of the mesh T .Then, we state that for each multi-level nodal mesh Tml(Ω) on a domain Ω, there exists astandard mesh Tstd(Ω) such that spanΦ (Tml) = spanΦ (Tstd).

This is intuitively possible by choosing Tstd = T leafml , as motivated in Section 3.2.1. Indeed,

this way each mesh defines a different set of basis functions that is, in both cases, formedby multi-linear functions on the same subsets Ωe : Ωe ∈ T leaf

ml (Ω) of Ω. Therefore, the

same space of continuous piecewise multi-linear functions on Ωe : Ωe ∈ T leafml (Ω) is spanned.

For example, for one-dimensional domains Ω ⊂ R, the mesh T leafml = [x0, . . . , xn] identifies

a basis that spans the set of continuous function that are piecewise linear on [xi, xi+1], fori = 0 . . . n− 1. A similar argumentation holds for the space of bi-linear functions on a set ofquadrilateral elements, or the space of tri-linear function on parallelepipeds.

The equivalence of the standard and multi-level basis was also treated informally in [Yse86]by furnishing an isomorphism S : spanΦ (Tml) → spanΦ (Tstd) without a proof.

3.2.4 The Span of the Multi-Level Basis

In this section we informally state that, for each multi-level basis on a one-dimensional do-main Ω ⊂ R with high-order modes on the leaf elements, there exists a standard basis thatspans the same space. Let Φ (T , p) denote the set of global basis functions defined by a meshT and a polynomial degree distribution p. Then, in particular, for every multi-level meshTml(Ω) = [x0, . . . , xn] and admissible degree distribution pml, there exists a standard meshTstd(Ω) and a degree distribution pstd such that spanΦ (Tml, pml) = spanΦ (Tstd, pstd).Such a mesh Tstd can be constructed by considering the refined-most elements in the multi-level mesh T leaf

ml (Ω) as elements in a standard finite element mesh and use pleaf as degree

distribution. Namely, by taking Tstd(Ω) = T leafml (Ω) and pstd = pleaf

ml . See Figure 3.2 as anexample.


0 0.2 0.4 0.6 0.8 1−1

0

1

y

x

0 0.2 0.4 0.6 0.8 1−1

0

1

y

0 0.2 0.4 0.6 0.8 1−1

0

1

y

0 0.2 0.4 0.6 0.8 1−1

0

1y

(a) Multi-Level Basis with T leaf =[0, 0.25, 0.375, 0.5, 1] and pleaf = [3, 2, 1, 4]

0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

y

x

(b) Standard Basis with mesh T =[0, 0.25, 0.375, 0.5, 1] and p = [3, 2, 1, 4]

Figure 3.2: Example of Multi-Level basis and standard basis that spans the same space

This concept becomes obvious by adopting an approach similar to [Yse86]. Specifically,let Φnodal (Tml, pml), Φhigh (Tml, pml) be the sets of nodal and the high-order functions inΦ (Tml, pml), respectively. Then, the function space spanΦ (Tml, pml) can be written asdirect sum

spanΦ (Tml, pml) = spanΦnodal (Tml, pml) ⊕ spanΦhigh (Tml, pml).

Now, as stated in the previous section it holds that spanΦnodal (Tml, pml) = spanΦnodal (Tstd, pstd).Furthermore, since the high-order modes Φhigh (Tml, pml) are defined just on the leaf elements

T leafml , taking Tstd(Ω) = T leaf

ml (Ω) and pstd = pleafml defines the same set of functions. Namely,

for each high-order function in Φ (Tml, pml), the same function of identical order and supportis also present in Φ (Tstd, pstd). Compare, for example, the high-order modes in Figure 3.2aand 3.2b. Hence, we obtain that also spanΦhigh (Tml, pml) = spanΦhigh (Tstd, pstd) and,finally, spanΦ (Tml, pml) = spanΦ (Tstd, pstd). In conclusion, both basis span the same

finite element solution space S(T leaf

ml , pleaf)

.

While the argumentation for one-dimensional domains holds for general meshes, in higherdimensional spaces this can be directly extended just to regular meshes. Indeed, takingTstd(Ω) = T leaf

ml (Ω) and pstd = pleafml yields again spanΦnodal (Tml, pml) = spanΦnodal (Tstd, pstd)

and Φhigh (Tml, pml) = Φhigh (Tstd, pstd). Unfortunately, in case of irregular meshes, i.e withhanging-nodes, -edges and -faces, it is not clear if, in general, it holds that spanΦhigh (Tml, pml) =spanΦhigh (Tstd, pstd). Indeed, it might happen that some high-order modes are active onnon-leaf elements. As a result, there are high-order shape functions whose support coversmore than one element. This case cannot be produced directly on standard meshes.

3.2.5 Error Estimation for One-Dimensional and Regular Meshes

The argument of Section 3.2.2 to the applicability of the error estimation to the multi-levelhp-FEM is generalized in this section. In particular, here the case of one-dimensional domains

3.3. Implementational Aspects 23

and the case of regular meshes are considered. In this context, we state that the proposederror estimator will give the same exact estimate of an error estimator applied to a standardmesh. Here, the following argumentation is based on the Remark 2.2.1 on Section 2.2.4 andon Section 3.2.4.

Indeed, consider a domain Ω and a multi-level mesh Tml(Ω) with an admissible degree dis-tribution pml. We assume Tml to be regular in case of two- and three- dimensional do-mains. Then, as discussed in Section 3.2.4, it is possible to construct a standard meshTstd(Ω) = T leaf

ml (Ω) with degree distribution pstd = pleafml , such that spanΦ (Tml(Ω), pml) =

spanΦ (Tstd(Ω), pstd). This means that, fixed a right-hand-side f and the boundary condi-tions g, the solution uTstd ∈ spanΦ (Tstd(Ω), pstd) to (2.2) will be the same as the solutionuTml

∈ spanΦ (Tml(Ω), pml) to the same problem obtained by using the multi-level basis.

Namely, uTml= uTstd and they both belong to S

(T leaf

ml , pleafml

).

Therefore, following the notation of Remark 2.2.1, for the error estimator and element errorindicator defined in (2.17), it holds

η(T leaf

ml , pleafml , uTml

, f, g)

= η (Tstd, pstd, uTstd , f, g) (3.1)

λe

(T leaf

ml , pleafml , uTml

, f, g)

= λe (Tstd, pstd, uTstd , f, g) ∀ Ωe ∈ T leafml

simply because Tstd = T leafml , pstd = pleaf

ml and uTml= uTstd .

Therefore, it does not just hold that the error estimator (2.17) can be applied to multi-level

settings by regarding T leafml as elements. In addition, the proposed estimator will give exactly

the same estimates as an error estimator on an equivalent standard mesh.

3.3 Implementational Aspects

In this work, the error estimator (2.17) and the smoothness indicator described in Section2.4 were implemented and tested in a object-oriented MATLAB finite element solver for the1D Poisson equation. The program can work with both standard and multi-level high-orderelements.

After obtaining promising results in one dimension, the same error and smoothness estimatorswere implemented in AdhoC++, a object-oriented C++ high-order finite element solverdeveloped by the chair of Computation in Engineering.

The core implementation of the error estimator consists of the integration of the residuals.The interior residual can be implemented straightforwardly with a numerical integration pro-cedure on the volume of each leaf element. In particular, some derivative of the numericalsolution has to be evaluated on each leaf element Ωe. To this extent, in addition to the shapefunctions local to Ωe, all the shape functions of the parent element have to be consideredrecursively. Moreover, also the integration of the Neumann-boundary residuals can be per-formed analogously.

3.3. Implementational Aspects 24

Ω1

Ω1,1 Ω1,2

D

Ω1,1,2 Ω1,2,1

AB

C

Ω1,2,1,1

Boundary EdgeInternal Edge

Figure 3.3: Example of multi-level element hierarchy

On the contrary, the integration of the jumps in the inter-element flux (Equation (2.11)) ismore challenging. In particular, a suitable sequence of local-coordinates conversion throughthe multi-level element hierarchy is necessary. A simple strategy is based on the followingobservation. Consider Figure 3.3 and a point x on the edge AB. To the extent of evaluatingthe flux difference at x, it is sufficient to evaluate the difference in the contribution of Ω1,1 andΩ1,2 and of their respective sub-elements recursively, ignoring the contribution of Ω1. Thiscan be clarified as follows. Let Φ (Ωe) = φΩe, j denote the local basis defined by the elementΩe. Then, the shape functions Φ

(Ω1)

are C∞-continuous at x, giving the same contributionto the numerical solutions uΩ1,1,2 and uΩ1,2,1,1 local to elements Ω1,1,2 and Ω1,2,1,1, respectively.This can be expressed as

uΩ1,2,1,1(x) =∑

Ωe∈Ω1,Ω1,2,Ω1,2,1,Ω1,2,1,1

∑φΩe,j∈Φ(Ωe)

αΩe,j φΩe,j(x)

uΩ1,1,2(x) =∑

Ωe∈Ω1,Ω1,1,Ω1,1,2


αΩe,j φΩe,j(x),

where the αΩe,j represents the degree of freedom associated to the local basis function φΩe,j .

3.4. Results 25

Therefore, we obtain ∀x ∈ AB

[R∂Ωe(uT )] (x) = [∇uΩ1,2,1,1(x)−∇uΩ1,1,2(x)] · nAB(x)

=

∑Ωe∈Ω1, Ω1,2, Ω1,2,1,Ω1,2,1,1


αΩe,j ∇φΩe,j(x)−

∑Ωe∈Ω1,Ω1,1,Ω1,1,2


αΩe,j ∇φΩe,j(x)

· nAB(x)

=

∑Ωe∈Ω1,2,Ω1,2,1,Ω1,2,1,1


αΩe,j ∇φΩe,j(x)−

∑Ωe∈Ω1,1,Ω1,1,2


αΩe,j ∇φΩe,j(x)

· nAB(x),

In general, when integrating the inter-element flux difference over an interface component(node, edge or face), the contribution of all the elements on coarser levels can be ignored.Indeed, they would give the same contribution to the solution local to both elements adjacentto the interface. Therefore, the difference would cancel out this contribution.

Following this idea, the pseudo-code Algorithm 4 highlights the steps needed to integrate overinterface component in the multi-level hierarchy. The basic idea consists of looping over allthe elements of all the levels and to consider the internal interfaces of each element. Namely,the interfaces between two sub-elements, i.e. the dashed edges in Figure 3.3. For each ofthem, the integration is performed on each refined-most sub-interface. For example, consid-ering the edge AD internal to element Ω1 in Figure 3.3, the integration is performed on thesub interfaces AB, BC and CD. Finally, the error level of the two leaf elements adjacent toeach leaf sub-interface is updated. Note that each integral is computed only once. Indeed,one interface is internal just to one element and its refined sub-interfaces are belonging tothe boundary of the refined elements and, therefore, they are not internal.

Moreover, Algorithm 3 shows a mapping chain suitable for AdHoC++ data structures, thatconverts the coordinates from the index space of an interface γ internal to an element Ωe

parent

to the index space of a element Ωe child of Ωeparent. This strategy is used in Algorithm 4.

The notation xγ and xΩe represents the coordinates of a point x expressed in the index spaceof the interface γ and element Ωe, respectively. Furthermore, we will denote as ancestor ofa component the component itself, if it does not posses a parent component, or (recursively)the ancestor of its parent element.

3.4 Results

This section will showcase some numerical experiments for the Poisson and Linear-Elasticityequations. The results obtained confirm the applicability of the proposed error estimator tothe multi-level hpFEM. Moreover, it is here empirically confirmed that the error estimates

3.4. Results 26

Algorithm 3 Convert from Refined Sub-Face Coordinates to Element Coordinates

1: function conv(Interface γ, Coordinates xγ , Element Ωe)2: Get the interface γanc ancestor of γ3: Let Ωparent

e be the parent element of Ωe

4: Map xγ to xγanc5: Map xγanc to xΩparent

e

6: Map xΩparente

to xΩe

7: return xΩe

8: end function

Algorithm 4 Flux Difference Integration Over Multi-Level Internal Interfaces

1: procedure Integrate flux difference on internal intefaces2: for each element Ωe ∈ T (Ω) do3: for each interface γ ∈ T (Ωe) internal to Ωe do4: Obtain elements Ωe,1, Ωe,2 adjacent to γ.5: for each refined sub-interface γsub of γ do6: Obtain elements Ωa, Ωb adjacent to γsub

7: Let FΩe(xΩe) be the flux obtained considering just the contributions of theelement Ωe and its sub-elements recursively

8: Let GΩe(xγsub) = FΩe(conv(γsub, xγsub ,Ωe))

9: Compute J =∫γsub‖GΩe,1(xγsub)−GΩe,1(xγsub)‖

2

2dxγsub

10: λΩa ← λΩa + 12J

11: λΩb ← λΩb + 12J

12: end for13: end for14: end for15: end procedure

3.4. Results 27

on a multi-level mesh Tml equal the estimates of a traditional error estimator applied on thestandard mesh Tstd = T leaf

ml , ( considering just uniform meshes in two- and three- dimensionaldomains). This supports Section 3.2.5 . Furthermore, the results show that also the analyt-

ical errors associated to Tml match the errors associated to Tstd = T leafml , confirming Section

3.2.4.

To asset quality of the error estimator, all the considered problems have a known analyticalsolution. This enables the computation of the energy norm of the analytical absolute error(2.4) and of the efficiency index (2.18).

3.4.1 Results in 1D

The following results have been obtained with the object-oriented MATLAB implementationof a finite element solver for the Poisson equation introduced in Section 3.3. The interior- andboundary- residual constants used are simply C2

1,T = C22,T = 1/24, as introduced heuristi-

cally in [KGZ+83]. The adaptive examples follow Algorithm 2 by using the considered errorestimator and the smoothness indicator presented in Section 2.4 .

In this section, the analytical absolute error is computed by numerical integration of Equation(2.4). Moreover, the vertical dotted lines in the following plots denote an element boundary.Finally, in the legend the results obtained by multi-level meshes are indicated by mlFEM,while results obtained by standard meshes are indicated by stdFEM.

Smooth Poisson Problem: Multi-Level hFEM

The multi-level error estimator has been applied to the problem

u′′(x) = − sin(8x) on Ω = [0, 1] (3.2)

u(0) = 0

u′(1) = 0,

that has a smooth analytical solution u(x) = − sin(8x)/64 + x cos(8)/8. Starting from oneelement of degree one, as illustrated in Figure 3.4d, multi-level h-refinements have been per-formed. In Figure 3.4 the first three steps of this process are shown along with the multi-levelbasis, the associated numerical solution and the analytical solution.

As can be seen from Figure 3.5a, the expected algebraic convergence behaviour is trackedquite closely by the estimated error. Moreover, in the figure it is possible to see that theestimates obtained for multi-level FEM match perfectly the ones computed for the standardhFEM, confirming the concepts expressed in Section 3.2.5. Analogously, it can be observedthat also the analytical errors associated to multi-level hFEM and classic hFEM match, sup-porting Section 3.2.4. Finally, Figure 3.5b shows that the effectivity index converges towarda constant. This should correspond to the constant C of the Equation (2.15), indicatingsharpness of the error upper bound.

3.4. Results 28

0 0.5 1

−3

−2

−1

0

·10−2

X

Y

Analytical Solution

Numerical Solution

(a) Linear Solution

0 0.5 1

−3

−2

−1

0

·10−2

X

Y

Analytical Solution

Numerical Solution

(b) Solution obtained by 1 uni-form multi-level h-refinement

0 0.5 1

−3

−2

−1

0

·10−2

X

Y

Analytical Solution

Numerical Solution

(c) Solution obtained by 2 uni-form multi-level h-refinements

0 0.5 1

−1

0

1

X

Y

(d) Linear Basis

−1

0

1

Y

0 0.5 1

−1

0

1

X

Y

(e) Multi-level basis obtainedby 1 uniform multi-level h-refinement

−1

0

1

Y

−1

0

1

Y0 0.5 1

−1

0

1

XY

(f) Multi-level basis obtainedby 2 uniform multi-level h-refinements

Figure 3.4: First three uniform multi-level h-refinements of a smooth Poisson problem

101 102

10−3

10−2

10−1

−1

DOFs

Global

Error

inEnergy

Norm stdFEM Analytical

stdFEM Estimated

mlFEM Analytical

mlFEM Estimated

(a) Analytical and estimated error obtained withmulti-level hFEM and comparison to standardhFEM

101 102

1

1.2

1.4

1.6

DOFs

Eff

ecti

vit

yIn

dex

stdFEM

mlFEM

Ideal Value: One

(b) Efficiency index for multi-level hFEM andcomparison to standard hFEM

Figure 3.5: Smooth Poisson problem: error estimates and effectivity index obtained by uniformmulti-level h-refinements

3.4. Results 29

Smooth Poisson Problem: pFEM on a Multi-Level Mesh

To assess the quality of the error estimator on multi-level meshes with high-order elements, wesolved Problem (3.2) by performing p-refinements on a fixed mesh with two uniform multi-levels (see Figure 3.6d). As before, in Figure 3.6 are illustrated the first three refinementsteps together with the multi-level basis, the associated numerical solution and the analyticalsolution.

Figure 3.7a compares analytical and estimated error, while 3.7b shows the effectivity index.In this case, the effectivity index does not show a clear converging behaviour. Nevertheless,it varies within a very narrow range until a very accurate solution is achieved. As before,the error estimates and the analytical error for the multi-level hpFEM match their standardhpFEM counterpart.

0 0.5 1

−3

−2

−1

0

·10−2

X

Y

Analytical Solution

Numerical Solution

(a) Solution obtained for p = 1

0 0.5 1

−3

−2

−1

0

·10−2

X

Y

Analytical Solution

Numerical Solution

(b) Solution obtained for p = 2

0 0.5 1

−3

−2

−1

0

·10−2

X

Y

Analytical Solution

Numerical Solution

(c) Solution obtained for p = 3

−1

0

1

Y

−1

0

1

Y

0 0.5 1

−1

0

1

X

Y

(d) Multi-level basis obtainedby 2 uniform multi-level h-refinements, p = 1

c

−1

0

1

Y

−1

0

1

Y

0 0.5 1

−1

0

1

X

Y

(e) Multi-level basis obtainedby 2 uniform multi-level h-refinements, p = 2

−1

0

1

Y

−1

0

1

Y

0 0.5 1

−1

0

1

X

Y

(f) Multi-level basis obtainedby 2 uniform multi-level h-refinements, p = 3

Figure 3.6: Smooth Poisson problem: first three uniform multi-level p-refinements on a fixed meshobtained by two uniform multi-level refinements

3.4. Results 30

10 20 30 40 50

10−14

10−10

10−6

10−2

102

−0.77

DOFs

GlobalError

inEnergyNorm

stdFEM Analytical

stdFEM Estimated

mlFEM Analytical

mlFEM Estimated

(a) Analytical and estimated error obtainedwith multi-level pFEM and comparison to clas-sic pFEM

101 101.5

1

1.2

1.4

1.6

DOFs

Eff

ecti

vit

yIn

dex

stdFEM

mlFEM

Ideal Value: One

(b) Efficiency index for multi-level pFEM andcomparison to classic pFEM

Figure 3.7: Smooth Poisson problem: error estimates and effectivity index obtained by p-refinementson a fixed mesh obtained by two uniform multi-level refinements

Non-Smooth Poisson Problem: pFEM on a Multi-Level Mesh

A non-smooth Poisson problem can be formulated as

u′′(x) = λ(λ− 1)xλ−2 on Ω = [0, 1] (3.3)

u(0) = 0

u′(1) = 0,

The analytical solution of such a boundary value problem is u(x) = −xλ + λx. A commonnon-smooth example consist in taking 1/2 < λ < 1, yielding a singularity at x = 0 in thefirst derivative u′(x) = −λxλ−1 +λ. Unfortunately, residual-based error estimators cannot beapplied to this problem, as u′′(x) is not square integrable for λ < 3/2. Therefore, we chose avalue 3/2 < λ < 2, obtaining a solution with a singularity at x = 0 on the second derivativeinstead of the first one, but yielding a square integrable interior residual.

Figure 3.8a shows the typical algebraic convergence for non-smooth problem of the standardh- and p-FEM obtained for the proposed problem. Moreover, in Figure 3.8b the known effectof heavy error overestimation as the polynomial degree is increased can be seen, indicating atotally different convergence behaviour. Exponential pre-asymptotic convergence and betterestimates can be achieved with meshes that are geometrically refined toward the singularity[SB91]. Because of Section 3.2.4 and 3.2.5 this is obviously true also for multi-level basis andthe proposed error estimator. Here, we confirm this numerically.

We discretized the domain Ω = [0, 1] with a multi-level mesh geometrically refined towardthe singular point x = 0, giving T leaf =

[0, qk, qk−1, . . . , q1, q0

]. An example of such a mesh

can be visualized in Figure 3.9a obtained with a non-optimal progression factor q = 0.2 (fora better visualization) and k = 3. Figure 3.9b shows the numerical solution obtained using

3.4. Results 31

101 102

10−3

10−2

10−1

−0.99

−2.33

DOFs

Global

Errorin

Energy

Norm hFEM Analytical

pFEM Analytical

(a) Algebraic convergence of both the standardhFEM and pFEM

100.5 101

10−3

10−2

10−1

DOFs

GlobalErrorin

EnergyNorm

Analytical Error

Estimated Error

(b) Heavy overestimation of the error for pFEM

Figure 3.8: Non-smooth Poisson Problem (λ = 1.75): convergence and error estimation of thestandard hFEM and pFEM

the optimal factor q = 0.17 (see [SB91]) and k = 8.

Comparisons of analytical to estimated error and effectivity index can be found in Figures3.10a and 3.10b, respectively. The effectivity index does not show a converging behaviour.Nevertheless, in the given example the effectivity varies within a acceptably narrow rangeuntil a accurate solution is achieved. Again, error estimates and analytical error for themulti-level hpFEM match their standard hpFEM counterpart.

−1

0

1

Y

−1

0

1

Y

−1

0

1

Y

0 0.5 1

−1

0

1

X

Y

(a) Multi-level mesh geometrically refined to-ward x = 0 with geometrical progression factorq = 0.2 and k = 3.

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

X

Y

Analytical Solution

Numerical Solution

(b) Numerical solution obtained with multi-level mesh geometrically refined toward x = 0with optimal geometrical progression factor q =1.7 and k = 8

Figure 3.9: Non-smooth Poisson Problem (λ = 1.75): example of geometrically refined mesh andnumerical solution

3.4. Results 32

0 50 100 150

10−11

10−8

10−5

10−2

−0.11

DOFs

GlobalErrorin

EnergyNorm

stdFEM Analytical

stdFEM Estimated

mlFEM Analytical

mlFEM Estimated

(a) Analytical and estimated error

101 102

1

1.5

2

2.5

DOFs

Eff

ecti

vit

yIn

dex

stdFEM

mlFEM

Ideal Value: One

(b) Efficiency index

Figure 3.10: Non-smooth Poisson Problem (λ = 1.75): error estimates and effectivity index obtainedfor p-refinements on a multi-level mesh geometrically refined toward x = 0 with optimal geometricalprogression factor q = 1.7 and k = 8 and comparison to p-refinements on equivalent standard mesh

Non-Smooth Poisson Problem: Multi-Level hp-Adaptivity

Automatic multi-level hp-adaptivity driven by the error estimator and the smoothness esti-mator has been applied to Problem (3.3) with λ = 1.75. The numerical solution obtainedafter 40 multi level hp-adaptive steps is depicted in Figure 3.11a. The associated polynomialdegree distribution is shown in Figure 3.11b.

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

X

Y

Analytical Solution

Numerical Solution

(a) Numerical Solution

0 0.2 0.4 0.6 0.8 1

2

3

4

5

6

7

X

ElementPolynomialDegree

(b) Polynomial degree distribution

Figure 3.11: Adaptivity on non-smooth Poisson problem (λ = 0.75): numerical solution and polyno-mial degree distribution obtained after 40 multi-level hp-adaptive steps (CelementError = 0.7, Cdecay =−1)

Analytical and estimated error are plotted in Figure 3.12a, while the effectivity index is showin Figure 3.12b. The effectivity index is oscillatory around one and roughly contained in therange [0.6, 1.5]. Note that, while in Figure 3.10b the effectivity index tend to diverge, here itdoes not show a diverging behaviour. This is probably due to the better mesh produced bythe adaptive procedure that eases the error estimation.

3.4. Results 33

0 20 40 60

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

−0.13

DOFs

GlobalError

inEnergyNorm

stdFEM Analytical

stdFEM Estimated

mlFEM Analytical

mlFEM Estimated

e−0.13x−6.69


100.5 101 101.5

0.6

0.8

1

1.2

1.4

DOFs

Eff

ecti

vit

yIn

dex

stdFEM

mlFEM

Ideal Value: One


Figure 3.12: Adaptivity on non-smooth Poisson problem (λ = 0.75): error estimates and effectiv-ity index in 40 multi-level hp-adaptive steps (CelementError = 0.7, Cdecay = −1) and comparison toadaptivity using standard mesh

Shock Problem: Multi-Level hp-Adaptivity

Another benchmark problem for the hp-Adaptivity consists of the so-called ”shock problem“

u′′(x) = 2α3

(x− 1

π

) [1 + α2

(x− 1

π

)2]2

on Ω = [0, 1] (3.4)

u(0) = arctan(α (0− 1

π) )

u(1) = arctan(α (1− 1

π) ),

which has an analytical solution u(x) = arctan(α(x − 1/π)) that is smooth, but presents agradient depending on α that can be made really steep in very small region. This can beobserved in Figure 3.13 along with the numerical solution and associated basis function ofthe first four multi-level hp-adaptive steps. As a comparison, in Figure 3.14 can be observedthe same adaptive steps obtained by using a standard basis. Note the equivalence of thesetwo processes.

In Figure 3.15a the numerical solution obtained after 40 multi-level hp-adaptive steps andCelementError = 0.7, Cdecay = −1 is shown. The associated polynomial degree distributioncan be visualized in 3.15b. Finally, the usual comparison of error estimates to analyticalerror are plotted in Figure 3.16a, while the associated effectivity index in depicted in Figure3.16b. Note the equivalence in the standard and multi-level hp-adaptivity and error estimates.Furthermore, the effectivity index shows a good converging behaviour around the optimalvalue one.

3.4. Results 34

0 0.5 1

−1

0

1

X

Y

An. Sol.

Num. Sol.

(a) Solution after 1 step

0 0.5 1

−1

0

1

X

Y

An. Sol.

Num. Sol.

(b) Solution after 2 step

0 0.5 1

−1

0

1

X

Y

An. Sol.

Num. Sol.

(c) Solution after 3 step

0 0.5 1

−1

0

1

X

Y

An. Sol.

Num. Sol.

(d) Solution after 4 step

0 0.5 1

−1

0

1

X

Y

(e) Basis after 1 step

0 0.5 1

−1

0

1

X

Y

(f) Basis after 2 step

−1

0

1

Y

0 0.5 1

−1

0

1

X

Y

(g) Basis after 3 step

−1

0

1

Y

−1

0

1

Y

0 0.25 0.5 1

−1

0

1

X

Y

(h) Basis after 4 step

Figure 3.13: Shock Problem: basis and solution of first four multi-level hp-adaptive steps

0 0.5 1

−1

0

1

X

Y

An. Sol.

Num. Sol.

(a) Solution after 1 step

0 0.5 1

−1

0

1

X

Y

An. Sol.

Num. Sol.

(b) Solution after 2 step

0 0.5 1

−1

0

1

X

Y

An. Sol.

Num. Sol.

(c) Solution after 3 step

0 0.5 1

−1

0

1

X

Y

An. Sol.

Num. Sol.

(d) Solution after 4 step

0 0.5 1

−1

0

1

X

Y

(e) Basis after 1 step

0 0.5 1

−1

0

1

X

Y

(f) Basis after 2 step

0 0.5 1

−1

0

1

X

Y

(g) Basis after 3 step

0 0.25 0.5 1

−1

0

1

X

Y

(h) Basis after 4 step

Figure 3.14: Shock Problem: basis and solution of first four standard hp-adaptive steps

3.4. Results 35

0 0.2 0.4 0.6 0.8 1

−1

0

1

X

Y

Analytical Solution

Numerical Solution

(a) Numerical Solution

0 0.2 0.4 0.6 0.8 1

5

9

7

6

8

4

2

3

X

ElementPolynomialDegree

(b) Polynomial degree distribution

Figure 3.15: Shock Problem (α = 120): numerical solution and polynomial degree distributionobtained after 40 multi-level hp-adaptive steps (CelementError = 0.7, Cdecay = −1)

0 50 100

10−4

10−3

10−2

10−1

100

101

102

103

−7.93 · 10−2

DOFs

Global

Error

inEnergy

Norm

stdFEM Analytical

stdFEM Estimated

mlFEM Analytical

mlFEM Estimated

e−7.93·10−2x+0.52


101 102

0

5

10

15

DOFs

Eff

ecti

vit

yIn

dex

stdFEM

mlFEM

Ideal Value: One


Figure 3.16: Shock Problem (α = 120): error estimates and effectivity index in 40 multi-levelhp-adaptive steps (CelementError = 0.7, Cdecay = −1), and comparison to adaptivity using standardmesh

3.4. Results 36

3.4.2 Results in 3D

The results of this section have been obtained with the object-oriented C++ implementa-tion of the error- and smoothness-estimator in AdhoC++ discussed in Section 3.3. For thePoisson’s examples, the values for the interior- and boundary- residual constants are simplychosen to be C2

1,T = C22,T = 1. Instead, a slightly more sophisticated choice has been made in

the context of linear elasticity. Indeed, the following quantities are chosen for the estimationconstants

C21,T =

0.4√λ+ 5ν

C22,T =

1.1√λ+ 5ν

,

where λ and ν are the Lame coefficients. These values correspond to the approximate upperbounds for C1,T , C2,T given in [CJ92].

In this section, the analytical absolute error (2.4) is computed as

‖e‖2E(Ω) = ‖u‖2E(Ω) − ‖uT ‖2E(Ω),

where u and uT are respectively the analytical and numerical solution of the considered prob-lem. The integral ‖uT ‖2E(Ω) is computed numerically, while ‖u‖2E(Ω) is computed analyticallywhere possible, otherwise numerically.

Linear Elastic Problem with Trigonometric Right Hand Side: Multi-Level h-Refinements

To test the error estimator in the context of linear elasticity, a simple problem has set up inorder to obtain the following solution with a trigonometric term

u(x, y, z) = xyz(x− 1)(y − 1)(z − 1)

sin(y)11

.

A problem that presents u as analytical solution can be formulated as

−∇ · σ(u) = f on Ω = [0, 1]3 (3.5)

u = 0 on ∂Ω,

where σ is the stress tensor defined in Section 2.6 and f is the right hand side obtainablefrom u such that f = −∇ · σ(u).

The obtained results confirm that what observed about Problem (3.2) in 1D, also hold for 3D.Indeed, Figure 3.17a shows that the error estimator indicates the same algebraic convergencebehaviour of the analytical error. Moreover, the corresponding efficiency index presents aconverging behaviour, as can be seen from Figure 3.17b. Finally, from the plots can be seenthe match of the estimated and analytical errors obtained by using multi-level hFEM orclassic hFEM match closely.

3.4. Results 37

102 103 104

10−2

10−1

DOFs

GlobalErrorin

EnergyNorm

stdFEM Analytical

stdFEM Estimated

mlFEM Analytical

mlFEM Estimated

(a) Error estimates compared to analytical error

102 103 104

2

4

6

DOFs

Eff

ecti

vit

yIn

dex

mlFEM

stdFEM

Ideal Value: One

(b) Effectivity index

Figure 3.17: Trigonometric linear elasticity: estimated error, analytical error and effectivity indexobtained by multi-level h-refinements

Linear Elastic Problem with Trigonometric Right Hand Side: p-Refinements ona Fixed Multi-Level Mesh

In this section, p-refinements on a fixed multi-level mesh are performed to test the error es-timator on a multi-level mesh with high-order elements. This is analogous to the benchmarkperformed on Problem (3.2).

Consider again Problem (3.5) and a mesh obtained by one multi-level h-refinement on thedomain Ω = [0, 1]3. Therefore, two leaf elements per spatial direction are attained. Finally,successive uniform p-refinements are performed.

Figure 3.18a compares analytical and estimated errors, while Figure 3.17b shows the efficiencyindex. Analogously to the 1D case, estimated and analytical errors obtained for the multi-level mesh match the ones corresponding to the standard mesh. Furthermore, the efficiencyindex show an oscillating behaviour, but not clearly diverging.

Singular Cube: Multi-Level hp-Adaptivity

The following problem has been considered

−∆u = λ(λ+ 1)rλ−2 on Ω = [0, 1]3

∇u · n = g on ∂Ω

u(0) = 0,

3.4. Results 38

5 10 15 20

10−9

10−7

10−5

10−3

10−1

DOFs1/3

GlobalError

inEnergyNorm

stdFEM Analytical

stdFEM Estimated

mlFEM Analytical

mlFEM Estimated


102 103 104

1

2

3

4

5

DOFs

Eff

ecti

vit

yIn

dex

mlFEM

stdFEM

Ideal Value: One


Figure 3.18: Trigonometric linear elasticity: estimated error, analytical error and effectivity indexobtained by p-refinements on a fixed multi-level mesh

where λ ∈ R, λ > 0, n is the outward normal vector to the boundary, r =√x2 + y2 + z2 and

g = λrλ−2

xyz

· n.

The analytical solution u = rλ has a gradient singular at x = y = z = 0 for λ < 2.

The problem has been solved by a multi-level hp-adaptive procedure automatically driven bythe error estimation and smoothness indicator. In this process the value λ = 2/3 has beenchosen. In Figure 3.19 the error level and polynomial ansatz order of the consecutive adaptivesteps from the second to the seventh is depicted. In this sequence it is possible to note thatin the initial steps the element containing the singularities contains the highest error. There-fore, a consecutive h-refinements toward the singularities are performed. During this process,the normalized error level of the coarser elements closer to the singularity grows gradually.In step number six, the geometric mesh reduces significantly the error obtained close to thesingular corner, and these elements turn to contain the maximum error. Therefore, they arerefined in the last step. Since, their spatial location is relatively far from the singularity, thesmoothness indicator has estimated these elements to contain a smooth solution. Hence, ap-refinement has been performed. The final numerical solution obtained after 25 multi-levelhp-adaptive steps is plotted in Figure 3.20 and its polynomial degree distribution is depictedin Figure 3.21. Overall, in the pictures it is observable a common pattern where the elementswith the same distance r to the singularity posses the same error level and ansatz polynomialorder.

In Figure 3.22a, the error estimates compared to the analytical error are plotted. As reportedin [Yos11], high-order approximations of the considered boundary value problem attain ana-

3.4. Results 39

Figure 3.19: Adaptivity on singular cube problem (λ = 2/3): normalized error level (left) and ansatzorder (right) of the adaptive steps from second (top) to seventh (bottom)

3.4. Results 40

(a) Numerical solution around the singularpoint

(b) Numerical solution far from the singularpoint

Figure 3.20: Adaptivity on singular cube Poisson problem (λ = 2/3): numerical solution after 25multi-level hp-adaptive steps (CelementError = 0.8, Cdecay = −1)

(a) Polynomial degree distribution around thesingular point

(b) Polynomial degree distribution far from thethe singular point

Figure 3.21: Adaptivity on singular cube problem (λ = 2/3): polynomial degree distribution ob-tained after 25 multi-level hp-adaptive steps (CelementError = 0.8, Cdecay = −1)

3.4. Results 41

lytical errors e satisfying the bound in the energy norm

‖e‖E(Ω) ≤ C exp( γNθ ) θ =1

4. (3.6)

Here, N is the number of DOFs, C, γ are two constants independent of N . Indeed, theconverge obtained by the automatic multi-level hp-adaptive procedure shows an exponentialbehaviour with respect to the fourth root of the DOFs. In the figure is depicted also theconvergence rate computed by a linear-regression fitting. In Figure 3.22b can be observedthat the effectivity index shows a converging behaviour.

2 4 6 8

10−5

10−4

10−3

10−2

10−1

100

−1.39

DOFs1/4

Global

Errorin

EnergyNorm

mlFEM Analytical

mlFEM Estimated


0 2,000 4,000

0

5

10

15

DOFs

Eff

ecti

vit

yIn

dex

mlFEM

Ideal Value: One


Figure 3.22: Adaptivity on singular cube problem (λ = 2/3): estimated error, analytical error andeffectivity index obtained on 25 multi-level hp-adaptive steps (CelementError = 0.8, Cdecay = −1)

Shock Problem: Multi-Level hp-Adaptivity

In this section, a 3D shock problem analogous to the 1D problem (3.4) is considered. Letα, r0 ∈ R and r =

√(x− x0)2 + (y − y0)2 + (z − z0)2. In order to define an analytical

solution u = arctan(α(r − r0))− arctan(α(0− r0)) that has a gradient depending on α andthat is zero at the origin, we consider the consider the problem

∆u =f on Ω = [0, 1]3

∇u · n = g on ∂Ω

u(0) = 0,

where

f =2α(1− α2r0(r − r0))

r(1 + α2(r − r0)2)2

g =α

r(1 + α2(r − r0)2)

x− x0

y − y0

z − z0

· n.

3.4. Results 42

As in Problem (3.4), the analytical solution u presents a gradient depending on α that canbe made really steep in a small region.

Figure 3.23a shows the numerical solution obtained after 35 multi-level hp-adaptive steps forα = 40, r0 =

√3, x0 = y0 = z0 = −1/4, while its gradient and the associated polynomial

degree distribution are shown in Figure 3.23b and Figure 3.23c, respectively.

(a) Numerical Solution (b) Gradient of Numerical Solu-tion

(c) Polynomial degree distribu-tion

Figure 3.23: Adaptivity on the 3D shock problem (α = 40, r0 =√

3, x0 = y0 = z0 = −1/4): nu-merical solution, numerical gradient and degree distribution obtained after 35 multi-level hp-adaptivesteps (CelementError = 0.8, Cdecay = −1)

The error estimates compared to the analytical error can be visualized in Figure 3.24a, whilethe obtained efficiency index is shown in Figure 3.24b. The convergence seems to have anexponential behaviour with respect to the third-root of the DOFs. This measure comparesto the work in [Yos11], where for this benchmark the bound (3.6) is given with θ = 1/3.Overall, the error estimation tracks correctly the analytical error, with a slowly improvingeffectivity index, as shown in Figure 3.24b.

3.4. Results 43

10 20 30 4010−2

10−1

100

101

−0.14

DOFs1/3

Global

Errorin

Energy

Norm mlFEM Analytical

mlFEM Estimated


0 2 4 6 8

·104

2

4

6

8

10

DOFs

Eff

ecti

vit

yIn

dex

mlFEM

Ideal Value: One


Figure 3.24: Adaptivity on the 3D shock problem (α = 40, r0 =√

3, x0 = y0 = z0 = −1/4):estimated error, analytical error and effectivity index obtained on 35 multi-level hp-adaptive steps(CelementError = 0.8, Cdecay = −1)

44

Chapter 4

Error Estimation for the Finite CellMethod

In this chapter, the Finite Cell Method (FCM) is motivated and introduced by means of thePoisson’s model problem (2.1). Then, error estimation is discussed furnishing a partial resulton an error upper-bound analogous to was done in Section 2.2.3, pointing out the encounteredproblematics and suggesting possible solutions. Finally, a small numerical investigation hasbeen conducted for one-dimensional Poisson’s problems.

4.1 General Idea

4.1.1 Motivation

One of the most time-consuming tasks in the finite element analysis pipeline is the mesh gen-eration. Indeed, this technique requires a mesh conforming to the geometry that, in general,might have very complex shapes. Quantitatively, according to [CHB09], about 20% of theoverall time is spent in meshing and 60% is spent in creating a geometry suitable for analysis,while only 20% of the time is spent to perform the actual analysis.

The Finite Cell’s Method (FCM) [PDR07] tackles this problem by performing analysis onunfitted structured meshes. Indeed, the key idea of the FCM is to embed the geometry ofinterest into a domain of simpler shape, called embedding domain. This way, uncomplicatedregular meshes can be used on this domain, eliminating the problem of error-prone and time-consuming mesh generation. Subsequently, the boundary of the real geometry are capturedin the integration by fully automatic techniques. The FCM is shown to achieve optimal rateof convergence theoretically [DDR14] and practically [DDR14, PDR07, DPYR08, SDR12].The method has been applied to many practical problems such as shells [RKSD11], transientelastodynamics [EZK+15], biomechanical applications [RTT+12], elastoplasticity [APDR14].Moreover, this method can be used in combination adaptive schemes [SDR12, ZBK+15].

4.1. General Idea 45

4.1.2 The Finite Cell Method

Consider the model problem introduced in Section 2.2.1 on a physical domain Ωphy

−∆u = f on Ωphy

∇u = g on ΓN (4.1)

u = 0 on ΓD.

Following the Finite Cell Method’s (FCM) approach, Ωphy is extended by a fictitious domainΩfict into an embedding domain Ω = Ωphy ∪ Ωfict , with Ωphy ∩ Ωfict = ∅. In this setting, theweak-formulation (2.2) can be equivalently rewritten as

Find u ∈H1D(Ω) such that

Bα(u, v) = Fα(v) ∀v ∈ H1D(Ω),

where

Bα(u, v) =

∫Ωα∇u · ∇v dΩ (4.2)

Fα(v) =

∫Ωαf · v dΩ +

∫ΓN

g · v dΓ

α(x) =

1 if x ∈ Ωphy

0 if x ∈ Ωfict .

Using the same notation as in Section 2.2.2, we define a mesh T (Ω) of the embedding do-main and a polynomial degree distribution p. Note that the meshing is constructed on theembedding domain Ω instead of on the physical domain Ωphy . This represents a powerfulstep of the FCM, as Ω can be chosen to have a very simple shape. The discrete finite elementproblem (2.3) makes then use of the newly defined Bα and Fα as follows

Find uT ∈ S (T , p) such that

Bα (uT , vT ) = Fα (vT ) ∀vT ∈ S (T , p). (4.3)

4.1.3 Adaptive Integration

The discrete FCM Problem (4.3) presents a discontinuity in the integrands of weak fromdue to the physical domain indicator function α. Therefore, the difficulties of generatingmeshes conforming to curved boundaries have been translated into computing efficiently andaccurately the integrals in (4.2). Indeed, the Gaussian quadrature adopted in standard FEMimplementation cannot be directly used, as it requires smoothness of the integrand. However,this difficulty can be overcome by using an adaptive integration technique. The underlyingbasic idea is to partition the integration domain adaptively into smaller integration cellsrefined toward the physical boundary. This way, the whole integral is computed by summingthe integral over each cell, and the discontinuity can be confined within an area of integrationthat can be made arbitrarily small. See Figure 4.1 (from [SR14]).

4.2. Error Upper Bound 46

Figure 4.1: Adaptive integration of curved boundaries for increasingly refinement level k (from[SR14])

Adaptive integration is not the only developed approach in the context of the FCM, but acomparison of other integration strategies can be found [APD+13].

4.2 Error Upper Bound

In this section we provide a starting point to prove an error bound derivation analogous toSection 2.2.3 in the context of the Finite Cell Method.

An upper bound to the error e = u− uT in energy norm can be derived starting from

Bα(e, v) = Bα(u, v)− Bα(uT , v)

= Fα(v)− Bα(uT , v)

=

∫Ωαf · v dΩ +

∫ΓN

g · v dΓ−∫

Ωα∇uT · ∇v dΩ

=∑

Ωe∈T (Ω)

∫Ωe

αf · v dΩ +

∫∂Ωe∩ΓN

g · v dΓ−∫

Ωe

α∇uT · ∇v dΩ

. (4.4)

As done in Chapter 2, we would like to apply the Gauss’ Theorem on the last integral inorder to obtain an expression based on the interior residual. For this purpose, we need acontinuously differentiable vector field as integrand. Hence, to eliminate the discontinuitygiven by α, we need to “break” the integral over Ω into the integral over the physical Ωphy


and fictitious domain Ωfict∫Ωe

α∇uT · ∇v dΩ =

∫Ωe∩Ωphy

α∇uT · ∇v dΩ +

∫Ωe∩Ωfict

α∇uT · ∇v dΩ

=

∫Ωe∩Ωphy

∇uT · ∇v dΩ

=

∫∂(Ωe∩Ωphy)

v · ∇uT · n dΓ−∫

Ωe∩Ωphy∆uT · v dΩ

=

∫∂(Ωe∩Ωphy)

v · ∇uT · n dΓ−∫

Ωe

α∆uT · v dΩ, (4.5)

where n(x) is the boundary outward normal at the point x. Plugging Equation (4.5) into(4.4) yields

Bα(e, v) =∑

Ωe∈T (Ω)

∫Ωe

αf · v dΩ +

∫∂Ωe∩ΓN

g · v dΓ−∫

∂(Ωe∩Ωphy)

v · ∇uT · n dΓ +

∫Ωe

α∆uT · v dΩ

=

∑Ωe∈T (Ω)

∫Ωe

α [∆uT + f ] · v dΩ +

∫∂Ωe∩ΓN

[g −∇uT · n] · v dΓ−∫

∂(Ωe∩Ωphy)\ΓN

v · ∇uT · n dΓ

=

∑Ωe∈T (Ω)

∫Ωe

α [∆uT + f ] · v dΩ +

∫∂Ωe∩ΓN

[g −∇uT · n] · v dΓ−∫

∂Ωe\ΓN

α v · ∇uT · n dΓ

.

To express the integral on ∂Ωe \ ΓN as inter-element boundary residual, we rearrange thesums in terms of the set of internal edges ∂T int. Since u = 0 on ΓD, we can write

Bα(e, v) =∑

Ωe∈T (Ω)

∫Ωe

α [∆uT + f ] · v dΩ +

∫∂Ωe∩ΓN

[g −∇uT · n] · v dΓ

+

+∑

γ∈∂T int

∫γα[∇uT ,Ωe1 −∇uT ,Ωe2

]· n dΓ.

As in Equation (2.8), Ωe1 and Ωe2 are the two elements sharing the internal interface γ, anduT ,Ωe1 , uT ,Ωe2 are associated local numerical solution, respectively. Therefore, we obtain

Bα(e, v) =∑

Ωe∈T (Ω)

∫Ωe

αRΩe(uT ) · v dΩ +

∫∂(Ωe∩Ωphy )

R∂Ωe(uT ) · v dΓ

, (4.6)

where RΩe(uT ) and R∂Ωe(uT ) are defined in Equations (2.10) and (2.11). Note that to makeuse of adaptive integration also along inter-element interfaces, Equation (4.6) can be rewritten


in the following form:

Bα(e, v) =∑

Ωe∈T (Ω)

∫Ωe

αRΩe(uT ) · v dΩ +

∫ΓN

R∂Ωe(uT ) · v dΓ +

∫∂Ωe\Γ

αR∂Ωe(uT ) · v dΓ

.

(4.7)

The classic derivation now would continue with the use of the Galerkin Orthogonality andthe estimates (2.12) and (2.13) from interpolation theory.

Bα(e, v) = Bα(e, v − ISv) (4.8)

=∑

Ωe∈T (Ω)

∫Ωe

αRΩe(uT ) · (v − ISv) dΩ +


R∂Ωe(uT ) · (v − ISv) dΓ

≤∑

Ωe∈T (Ω)

‖αRΩe(uT )‖L2(Ωe)‖v − ISv‖L2(Ωe)+

+ ‖R∂Ωe(uT )‖L2(∂(Ωe∩Ωphy ))

‖v − ISv‖L2(∂(Ωe∩Ωphy )).

Unfortunately, while Estimate (2.12) can be applied directly to ‖v−ISv‖L2(Ωe) (maybe withan over-estimation), this does not hold for Estimate (2.13) and ‖v − ISv‖L2(∂(Ωe∩Ωphy ))

. In-

deed, ∂(Ωe∩Ωphy) might include the Neumann boundary that, in general, can cut the interiorpart of the element Ωe by an arbitrary shape. Note that this estimate would be valid forelements that are not cut by the physical domain, simply because ∂(Ωe ∩ Ωphy) = ∂Ωe.The difficulties on estimating ‖v−ISv‖L2(∂(Ωe∩Ωphy ))

seem not to be easily avoided, because

cut-elements and imposition of weak boundary conditions are a fundamental concepts in theFCM, that cannot be taken out. To overcome this problems, a deeper investigation on inter-polation estimates for cut elements would have to be performed.

In this work, no further theoretical investigation is carried out. Therefore, we limit to adaptthe error upper-bound derivation given in Section 2.2.3 to the FCM up to this point. Here,the necessary precautions needed to treat the discontinuity in the integrands given by αwere pointed out. Moreover the points that still has to be clarified in order to completethe derivation were highlighted. However, two intuitive ideas on how to overcome the abovedifficulties are proposed in the next sections.

4.2.1 A Possible Strategy to Prove the Error Upper Bound

In this section, it is illustrated a simple idea on how to make use of the interpolation estimateson the curved boundary Γcut

Ωe“cutting” one element Ωe. Here, the key step is on consider

separately the boundaries ΓcutΩe

and ∂Ωe. Namely, let ΓcutΩe

= (ΓN ∩ Ωe) \ ∂Ωe. Then,∫∂(Ωe∩Ωphy )

R∂Ωe(uT ) · (v − ISv) dΓ =

∫∂(Ωe∩Ωphy )\Γcut

Ωe


+

∫Γcut

Ωe

R∂Ωe(uT ) · (v − ISv) dΓ .


Now, these two integrals can be estimated separately. Indeed,∫∂(Ωe∩Ωphy )\Γcut

Ωe


∫∂Ωe

αR∂Ωe(uT ) · (v − ISv) dΓ

≤ ‖αR∂Ωe(uT )‖L2(∂Ωe) ‖ (v − ISv) ‖L2(∂Ωe). (4.9)

Instead, regarding the integral over ΓcutΩe

, it is tempting to assert∫Γcut

Ωe

R∂Ωe(uT ) · (v − ISv) dΓ ≤ ‖R∂Ωe(uT )‖L2(ΓcutΩe

) ‖ (v − ISv) ‖L2(ΓcutΩe

)

≤ ‖R∂Ωe(uT )‖L2(ΓcutΩe

) ‖ (v − ISv) ‖L2(Ωe) . (4.10)

Indeed, this would allow to use the interpolation estimate over the volume Ωe. However,the statement ‖ (v − ISv) ‖L2(Γcut

Ωe) ≤ ‖ (v − ISv) ‖L2(Ωe) is delicate. In particular, this would

correspond to affirming that

‖ (v − ISv) ‖2L2(ΓcutΩe

) =

∫Γcut

Ωe

|v − ISv|22 dΓ

≤∫Ωe

|v − ISv|22 dΩ (4.11)

= ‖ (v − ISv) ‖2L2(Ωe).

At first sight, this step seems intuitive, as |v − ISv|22 ≥ 0 and ΓcutΩe⊂ Ωe. However, the

problem is that those two integrals are equipped with two different measures. In particular,Γcut

Ωeis a set of measure zero with respect to Ωe. Therefore, the validity of inequality (4.11)

should be further investigated.

Estimating a surface integral with a volume integral might not seem optimal. However, ingeneral Γcut

Ωecan assume really pathological shapes, occupying almost the whole volume Ωe.

Therefore, it might intuitively make sense to use the volume estimate for this term. Of course,assumptions on the shape of the boundary surface might allow to formulate better estimates.

Assuming inequality (4.11) to hold, the rest of the derivation could continue with standard


steps

Bα(e, v) = Bα(e, v − ISv)

=∑

Ωe∈T (Ω)

∫Ωe

αRΩe(uT ) · (v − ISv) dΩ +



≤∑

Ωe∈T (Ω)

‖αRΩe(uT )‖L2(Ωe)‖v − ISv‖L2(Ωe)+

+ ‖αR∂Ωe(uT )‖L2(∂Ωe) ‖ (v − ISv) ‖L2(∂Ωe) + ‖R∂Ωe(uT )‖L2(ΓcutΩe

) ‖ (v − ISv) ‖L2(Ωe)

≤∑

Ωe∈T (Ω)

C1,T he‖αRΩe(uT )‖L2(Ωe)‖v‖H1(Ωe)+

+ C2,T√he‖αR∂Ωe(uT )‖L2(∂Ωe)‖v‖H1(Ωe)

+ C1,T he‖R∂Ωe(uT )‖L2(ΓcutΩe

)‖v‖H1(Ωe)

≤ ∑

Ωe∈T (Ω)

C21,T h

2e‖αRΩe(uT )‖2L2(Ωe)

+

+ C22,T he‖αR∂Ωe(uT )‖2L2(∂Ωe)

+ C21,T h

2e‖R∂Ωe(uT )‖L2(Γcut

Ωe)

12

·

∑Ωe∈T (Ω)

‖v‖2H1(Ωe)

12

≤ K1‖v‖H1(Ω)

∑Ωe∈T (Ω)

C21,T h


+


+ C21,T h

2e‖R∂Ωe(uT )‖2L2(Γcut

Ωe)

12

,

for some constant K1 that depends just on T which takes into account that every elementis counted several times. Exactly as before, the last step made use of the shape-regularityassumption introduced in Section 2.2.2. Finally, proceeding as in the previous sections, weassume Bα(·, ·) to be coercive in H1(Ω), i.e.

∃K2 > 0 : ‖φ‖H1(Ω) ≤ K2‖φ‖E(Ω)∀φ ∈ H1(Ω).


We conclude

‖e‖2E(Ω) = Bα(e, e) (4.12)

≤ K1‖e‖H1(Ω)

∑Ωe∈T (Ω)

C21,T h


+


+ C21,T h


Ωe)

12

≤ K1K2‖e‖E(Ω)

∑Ωe∈T (Ω)

C21,T h


+


+ C21,T h


Ωe)

12

≤ C ∑

Ωe∈T (Ω)

C21,T h


+


+ C21,T h


Ωe)

,

where C = K21K

22 .

4.2.2 Another Possible Strategy to Prove the Error Upper Bound

Another possible approach on how to make use of the interpolation estimates (2.12) and(2.13) on the curved boundary “cutting” one element is to make use of the so-called surfacedelta function. This tool can be regarded as a generalization of the delta function to surfaces,i.e. to higher dimensions. Given domain Ω ⊂ Rd with a smooth boundary surface ∂Ω and itsindicator function 1Ω

1Ω =

1 if x ∈ Ω

0 if x 6∈ Ω,

the surface delta ∇1Ω of the surface ∂Ω can be defined as the normal derivative of theindicator function 1Ω, in the distribution sense

−∫Rd

g(x)∇1Ω · n dx =

∫∂Ω

g(x) dΓ,

where g is a function and n(x) is the vector outward normal to ∂Ω extended to be definedover the whole domain Ω, for example by taking the vector that is outward normal to thepoint in ∂Ω nearest to x.

Once again, the key step is to consider separately the boundaries ΓcutΩe

and ∂Ωe. Indeed, let


ΓcutΩe

= (ΓN ∩ Ωe) \ ∂Ωe, then∫∂(Ωe∩Ωphy )


∫∂(Ωe∩Ωphy )\Γcut

Ωe


+

∫Γcut

Ωe


with ∫∂(Ωe∩Ωphy )\Γcut

Ωe


∫∂Ωe

αR∂Ωe(uT ) · (v − ISv) dΓ

≤ ‖αR∂Ωe(uT )‖L2(∂Ωe) ‖ (v − ISv) ‖L2(∂Ωe) (4.13)

and ∫Γcut

Ωe


∫Ωe

−∇α · nR∂Ωe(uT ) · (v − ISv) dΩ

≤ ‖ −∇α · nR∂Ωe(uT )‖L2(Ωe) ‖ (v − ISv) ‖L2(Ωe)

= ‖R∂Ωe(uT )‖L2(ΓcutΩe

) ‖ (v − ISv) ‖L2(Ωe) . (4.14)

Here, the same inequalities (4.9) and (4.10) have been obtained. Therefore, assuming thesteps so far to be valid, the boundary interpolation estimate (2.13) could be applied directlyto the boundary ∂(Ωe ∩Ωphy) \ Γcut

Ωe. Namely, to the inter-element interfaces on the physical

domain and the part of physical boundary conforming to the element boundary. Moreover,the interior interpolation estimate (2.12) could be applied directly to the Neumann boundaryΓcut

Ωe“cutting” the element. The derivation of the error upper bound could then continue

exactly as in Section 4.2.1.

However, assumptions that enables the use of the surface delta function should be carefullyinspected. In particular, the validity of all the steps of the derivation should be checked.

4.2.3 Error Estimator

The derivation of an error upper-bound presented in Section 4.2 is furnishes just a partial re-sult, while the ideas illustrated in Sections 4.2.1 and 4.2.2 have not been formally investigateddeeply enough to assure their applicability. Therefore, the above section are not sufficientlydeveloped to justify the formulation of an error estimator for FCM. Nevertheless, in orderto perform a small numerical investigation, we define the error estimator as suggested by

4.3. Results 53

Equation (4.12).

‖e‖2E(Ω) ≈ η2 =∑

Ωe∈Tλ2

Ωe(4.15)

λ2Ωe

= η2Ωe

+ η2∂Ωe

η2Ωe

= C21,T h


η2∂Ωe

= C22,T he‖αR∂Ωe(uT )‖2L2(∂Ωe)

+ C21,T h


Ωe).

Furthermore, an high-order generalization analogous to the estimator (2.17) can be heuristi-cally formulated by following the form proposed in [Mel05]

‖e‖2E(Ω) ≈ η2 =∑

Ωe∈Tλ2

Ωe(4.16)

λ2Ωe

= η2Ωe

+ η2∂Ωe

η2Ωe

= C21,T

h2e

p2e

‖αRΩe(uT )‖2L2(Ωe)

η2∂Ωe

= C22,T

hepe‖αR∂Ωe(uT )‖2L2(∂Ωe)

+ C21,T

h2e

p2e

‖R∂Ωe(uT )‖2L2(ΓcutΩe

).

Note that this consists of a generalization of the Definition (4.16), as in the case Ωphy = Ω thesame estimator (4.16) is obtained. Indeed, α would satisfy α = 1 everywhere and Γcut

Ωe= ∅.

4.3 Results

A small investigation in 1D has been conducted to asset the performance of the estimator(4.16). The implementation is based on a modified version for FCM of the object-orientedMATLAB software discussed in Section 3.3.

The interior- and boundary- residual constants used are simply C21,T = C2

2,T = 1/24, asintroduced heuristically in [KGZ+83]. Moreover, to avoid an extreme ill-conditioning of thelinear system to solve, the domain indicator function α defined in Equation (4.2) has beenapproximated to

α(x) ≈

1 if x ∈ Ωphy

ε if x ∈ Ωfict ,

for a small value ε. The FCM is proven to be convergent for ε→ 0 [DDR14].

Exactly as it was done in Section 3.4.1 of the 1D results for the multi-level FEM, the analyticalabsolute error Equation (2.4) is obtained by means numerical integration. Furthermore, inthe plots vertical dotted lines indicate element boundaries.

4.3. Results 54

4.3.1 Results in 1D

Poisson Problem with Smooth Solution

The first problem considered is the analogous of Problem (3.2) in an embedding domaincontext

u′′(x) = − sin(8x) on Ωphy = [0, 1] ⊂ Ω = [0, 1 + δ] (4.17)

u(0) = 0

u′(1) = 0,

for some δ ∈ R, δ > 0, controlling the size of the fictitious domain Ωfict =]1, 1 + δ].

In Figure 4.2 the numerical solution obtained for different polynomial degrees is plotted. Thenumerical solution has been multiplied by α to restrict the focus just on the physical domainΩphy , as the numerical solution on the fictitious domain it is not relevant.

0 0.5 1

−3

−2

−1

0

·10−2

X

Y

An. Sol.

Num. Sol.

(a) p = 1

0 0.5 1

−3

−2

−1

0

·10−2

X

Y

An. Sol.

Num. Sol.

(b) p = 3

0 0.5 1

−3

−2

−1

0

·10−2

X

Y

An. Sol.

Num. Sol.

(c) p = 5

0 0.5 1

−3

−2

−1

0

·10−2

X

Y

An. Sol.

Num. Sol.

(d) p = 18

Figure 4.2: Smooth Problem (δ = 0.2, ε = 10−20): numerical solution (multiplied by α(x)) fordifferent polynomial degrees

Convergence behaviour and comparison to the error estimates are plotted in Figure 4.3a,while the associated effectivity index is shown in Figure 4.3b

Poisson Problem with Smooth Solution and Internal Hole

In this section, a problem similar to Problem (4.17) is considered. This time the physicaldomain is not extended by the fictitious domain just on one side, but it presents an internalhole. Let

Ωphy = Ωphyleft ∪ Ωphy

right ⊂ Ω = [0, 1]

Ωphyleft =

[0,

1

2− δ]

Ωphyright =

[1

2+ δ, 1

],

4.3. Results 55

5 10 15 2010−12

10−9

10−6

10−3

100

DOFs

GlobalError

inEnergyNorm

Analytical Error

Estimated Error


5 10 15 20

1

1.5

DOFs

Eff

ecti

vit

yIn

dex

mlFEM

Ideal Value: One


Figure 4.3: Smooth Problem (δ = 0.2, ε = 10−20): error estimates and effectivity index

for some hole radius δ ∈ R, 0 < δ < 1/2. To construct a smooth problem with trigonometric

solution on both Ωphyleft and Ωphy

right, we define

u′′(x) = f(x) on Ωphy (4.18)

u(0) = 0

u′(1

2− δ) = 0

u′(1

2+ δ) = 0

u(1) = k,

for some k ∈ R, controlling the vertical offset between solutions on Ωphyleft and Ωphy

right, and for

f =

1

(δ−1)2 sin(

8 x1−δ

)on

[0, 1

2 − δ]

1δ2 sin

(81+2δ−x

δ

)on

[12 + δ, 1

].

The analytical solution

u(x) =

x cos(8)8−8δ − 1

64 sin( 8x1−δ ) on

[0, 1

2 − δ]

k +2 cos(8)(4−4x+δ sin(8))−δ sin( 8+16δ−8x

δ)

64δ on[

12 + δ, 1

]is plotted in Figure 4.4 together with the numerical solution obtained for δ = 0.2, k = 0.1,ε = 10−12 a mesh T (Ω) = [0, 0.5, 1] and different uniform p-refinements.

Finally the error estimates and analytical error are plotted in Figure 4.5a, while the effectivityindex is shown in Figure 4.5b.

4.3. Results 56

0 0.5 1

0

5 · 10−2

0.1

X

Y

An. Sol.

Num. Sol.

(a) p = 1

0 0.5 1

0

5 · 10−2

0.1

X

Y

An. Sol.

Num. Sol.

(b) p = 3

0 0.5 1

0

5 · 10−2

0.1

X

Y

An. Sol.

Num. Sol.

(c) p = 5

0 0.5 1

0

5 · 10−2

0.1

X

Y

An. Sol.

Num. Sol.

(d) p = 16

Figure 4.4: Smooth Problem with Internal Hole (δ = 0.2, k = 0.1, ε = 10−12): numerical solutionobtained for a fixed mesh T (Ω) = [0, 0.5, 1] and degree distribution P = [p, p] for different values of p

0 10 20 3010−8

10−6

10−4

10−2

100

DOFs

Global

Error

inEnergy

Norm

Analytical Error

Estimated Error


0 10 20 30

0.8

1

1.2

1.4

DOFs

Eff

ecti

vit

yIn

dex

mlFEM

Ideal Value: One


Figure 4.5: Smooth Problem with Internal Hole (δ = 0.2, k = 0.1, ε = 10−12): error estimates andeffectivity index obtained for T (Ω) = [0, 0.5, 1] and uniform p-refinement

4.3. Results 57

Limitation of the Error Estimator for very tiny Internal Holes

Investigation of Problem (4.18) revealed that the problem numerically unstable for very smallinternal holes, i.e. as δ → 0. Consider for example Figure 4.6, where the numerical solutionis plotted for different values of δ. While the polynomial degree and the offset k are fixed, areduction of the hole radius yields an unexpected numerical solution. This, seems to be dueto the C0 continuity that is still enforced between elements of the mesh T (Ω) = [0, 0.5, 1].Indeed, a decrease in δ implies a reduction of the horizontal space in which the numericalsolution has to cover the vertical offset given by k, requiring steep gradients. It should benoted that the problem becomes harder to approximate, due to a constraint on a node that isin the fictitious domain, and not in the physical one. Of course, the problem could be solvedby removing the continuity constraints on the nodes outside the physical domain. However,in 2D and 3D, some additional problematics could be encountered, as edges and faces can bepartially contained in the physical- and partially in the fictitious-domain.

0 0.2 0.4 0.6 0.8 1

0

5 · 10−2

0.1

X

Y

Analytical Sol.

Numerical Sol.

(a) δ = 2 · 10−2

0 0.2 0.4 0.6 0.8 1

0

5 · 10−2

0.1

X

Y

Analytical Sol.

Numerical Sol.

(b) δ = 1 · 10−2

0 0.2 0.4 0.6 0.8 1

0

5 · 10−2

0.1

X

Y

Analytical Sol.

Numerical Sol.

(c) δ = 1 · 10−3

Figure 4.6: Numerical solution obtained for different values of the hole radius δ, polynomial degreep = 20, k = 0.1, ε = 10−12

However, it is of interest of this section to show that the error estimator does not fully detectthis problem. Indeed, it can be seen in Figure 4.6 that the estimated and the analyticalabsolute error present different behaviours as δ → 0. This can be also observed in theeffectivity index plotted in Figure 4.8a. Moreover, it also remarkable to observe that, whilesmall holes do not increase the condition number of the system to solve, this turns out tobe the case for big holes. This is confirmed in Figure 4.8b where, for a fixed the polynomialdegree, for example 15, the condition number is subject to a sudden increase as δ grows. Thecondition number has been computed with the MATLAB function cond.

4.3. Results 58

0

10

200

0.10.2

0

0.1

0.2

Polynomial Degree

δ

Absolute

AnalyticalError

(a) Absolute analytical error

0

10

20 00.1

0.2

0

0.1

0.2

0.3

0.4

Polynomial Degreeδ

Absolute

Estim

ated

Error

(b) Absolute estimated error

Figure 4.7: Smooth Problem with Internal Hole (k = 0.1, ε = 10−12): Analytical and estimatedabsolute error obtained as the polynomial degree p and the hole radius δ vary.

0

10

20 0

0.1

0.20

1

2

Polynomial Degree

δ

Eff

ecti

vit

yIn

dex

(a) Effectivity Index

0

10

20

0

0.1

0.20

0.5

1

·1012

Polynom

ial Degreeδ

Con

ditionNumber

(b) Condition Number

Figure 4.8: Smooth Problem with Internal Hole (k = 0.1, ε = 10−12): Effectivity index and conditionnumber error obtained as the polynomial degree p and the hole radius δ vary.

59

Chapter 5

Summary and Conclusions

5.1 Summary

The main goal of this thesis was to investigate the applicability of the residual-based errorestimation to the multi-level hp-Finite Element Method and to the Finite Cell Method.

To this end, an initial investigation on the concept of “element” for a multi-level basis wasgiven. Namely, it was suggested to consider the leaf element as classic elements of the multilevel-mesh. Using this definition, the derivation of the error upper bound resulted to be valid.Therefore, the classic residual-based error estimator could be adopted on multi-level meshes.

To confirm this, a residual-based error estimator was first implemented in a multi-level high-order Matlab finite element solver for one-dimensional Poisson’s problems. This allowed tosupport the above concepts by means of numerical experiments. After obtaining promisingresults, the same error estimator was implemented with a test-driven development for three-dimensional Poisson’s and linear elastic problems in AdhoC++, a complete finite elementsolver developed in C++ at the chair of Computation in Engineering. Once again, the sameideas where clearly confirmed numerically also in this context. Moreover, it was implementedin the software also a fully-automatic multi-level hp-adaptive refinement procedure driven bythe proposed error estimator. In this method, elements responsible for the highest estimatederror were marked for refinement by the error estimator, while a decision between h- or p-refinement was made by means of a smoothness indicator. The implemented procedure wasshown to achieve exponential convergence rate also for problems containing singularities orsteep gradients.

Finally, a small theoretical investigation was perform on the applicability of the error esti-mators to the Finite Cell Method. Such an analysis furnished just partial results, as it couldnot fit entirely on the time-scope of this work. Nevertheless, the encountered problematicswere pointed out and explained, furnishing a starting point for further investigation. Fur-thermore, possible approaches to overcome the faced difficulties were proposed. Despite thepartial theoretical results, a modest numerical investigation has been performed on an intu-itive formulation of the error estimator. These experiments have been conducted by means of

5.2. Conclusion 60

the Matlab one-dimensional Poisson’s solver. This way, sharp error estimates were obtainedfor simple scenarios, suggesting that an estimator of similar formulation could be successfullybe applied to the Finite Cell Method. However, also some limitations of the considered errorestimator were pointed out. Indeed, some estimates quite off the analytical error were ob-served to be produced in case of physical domains containing tiny holes.

5.2 Conclusion

In conclusion, the following results were achieved

• Confirmed that a residual-based error estimator can be applied to the multi-level finiteelements method. This was done by considering the leaf elements in the multi-levelbasis as classic elements in a standard mesh and applying the classic residual-basederror estimator for standard meshes to the leaf-elements.

• Confirmed that for each multi-level basis with high-order modes on the refined-mostelements and without hanging-nodes on two- and three-dimensional domains, there ex-ists a standard basis that spans the same finite element solution space. Such a basiswas proposed to be constructed by considering the leaf elements in the multi-level basisas classic elements in a standard mesh.

• Implemented and fully tested a residual-based error estimator for multi-level finite el-ement method in AdhoC++ that can be used for practical applications and furtherdevelopments.

• Implemented and fully tested an hp-adaptive procedure in AdhoC++ automaticallydriven by the error estimator and smoothness indicator that can be used for practicalapplications and further developments.

• Produced a partial theoretical investigation about the applicability of the residual-basederror estimator to the Finite Cell’s Method that can be used as inspiration or startingpoint for further studies. The difficulties encountered were highlighted and discussed.Moreover, a possible approach to overcome them was proposed.

• Produced a small numerical investigation about the applicability of the residual-basederror estimator to the Finite Cell’s Method, presenting scenarios in which good perfor-mance were achieved, along with some limitations of the considered error estimator.

5.2. Conclusion 61

LIST OF FIGURES 62

List of Figures

3.1 Example of multi level mesh with 4 levels . . . . . . . . . . . . . . . . . . . . 18

3.2 Example of Multi-Level basis and standard basis that spans the same space . 22

3.3 Example of multi-level element hierarchy . . . . . . . . . . . . . . . . . . . . 24

3.4 First three uniform multi-level h-refinements of a smooth Poisson problem . . 28

3.5 Smooth Poisson problem: error estimates and effectivity index obtained byuniform multi-level h-refinements . . . . . . . . . . . . . . . . . . . . . . . . 28

3.6 Smooth Poisson problem: first three uniform multi-level p-refinements on afixed mesh obtained by two uniform multi-level refinements . . . . . . . . . . 29

3.7 Smooth Poisson problem: error estimates and effectivity index obtained byp-refinements on a fixed mesh obtained by two uniform multi-level refinements 30

3.8 Non-smooth Poisson Problem (λ = 1.75): convergence and error estimation ofthe standard hFEM and pFEM . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.9 Non-smooth Poisson Problem (λ = 1.75): example of geometrically refinedmesh and numerical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.10 Non-smooth Poisson Problem (λ = 1.75): error estimates and effectivity indexobtained for p-refinements on a multi-level mesh geometrically refined towardx = 0 with optimal geometrical progression factor q = 1.7 and k = 8 andcomparison to p-refinements on equivalent standard mesh . . . . . . . . . . . 32

3.11 Adaptivity on non-smooth Poisson problem (λ = 0.75): numerical solutionand polynomial degree distribution obtained after 40 multi-level hp-adaptivesteps (CelementError = 0.7, Cdecay = −1) . . . . . . . . . . . . . . . . . . . . . . 32

3.12 Adaptivity on non-smooth Poisson problem (λ = 0.75): error estimates and ef-fectivity index in 40 multi-level hp-adaptive steps (CelementError = 0.7, Cdecay =−1) and comparison to adaptivity using standard mesh . . . . . . . . . . . . 33

3.13 Shock Problem: basis and solution of first four multi-level hp-adaptive steps . 34

3.14 Shock Problem: basis and solution of first four standard hp-adaptive steps . . 34

3.15 Shock Problem (α = 120): numerical solution and polynomial degree distribu-tion obtained after 40 multi-level hp-adaptive steps (CelementError = 0.7, Cdecay =−1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

LIST OF FIGURES 63

3.16 Shock Problem (α = 120): error estimates and effectivity index in 40 multi-level hp-adaptive steps (CelementError = 0.7, Cdecay = −1), and comparison toadaptivity using standard mesh . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.17 Trigonometric linear elasticity: estimated error, analytical error and effectivityindex obtained by multi-level h-refinements . . . . . . . . . . . . . . . . . . . 37

3.18 Trigonometric linear elasticity: estimated error, analytical error and effectivityindex obtained by p-refinements on a fixed multi-level mesh . . . . . . . . . . 38

3.19 Adaptivity on singular cube problem (λ = 2/3): normalized error level (left)and ansatz order (right) of the adaptive steps from second (top) to seventh(bottom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.20 Adaptivity on singular cube Poisson problem (λ = 2/3): numerical solutionafter 25 multi-level hp-adaptive steps (CelementError = 0.8, Cdecay = −1) . . . . 40

3.21 Adaptivity on singular cube problem (λ = 2/3): polynomial degree distribu-tion obtained after 25 multi-level hp-adaptive steps (CelementError = 0.8, Cdecay =−1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.22 Adaptivity on singular cube problem (λ = 2/3): estimated error, analyti-cal error and effectivity index obtained on 25 multi-level hp-adaptive steps(CelementError = 0.8, Cdecay = −1) . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.23 Adaptivity on the 3D shock problem (α = 40, r0 =√

3, x0 = y0 = z0 = −1/4):numerical solution, numerical gradient and degree distribution obtained after35 multi-level hp-adaptive steps (CelementError = 0.8, Cdecay = −1) . . . . . . . 42

3.24 Adaptivity on the 3D shock problem (α = 40, r0 =√

3, x0 = y0 = z0 = −1/4):estimated error, analytical error and effectivity index obtained on 35 multi-level hp-adaptive steps (CelementError = 0.8, Cdecay = −1) . . . . . . . . . . . . 43

4.1 Adaptive integration of curved boundaries for increasingly refinement level k(from [SR14]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2 Smooth Problem (δ = 0.2, ε = 10−20): numerical solution (multiplied by α(x))for different polynomial degrees . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.3 Smooth Problem (δ = 0.2, ε = 10−20): error estimates and effectivity index . . 55

4.4 Smooth Problem with Internal Hole (δ = 0.2, k = 0.1, ε = 10−12): numericalsolution obtained for a fixed mesh T (Ω) = [0, 0.5, 1] and degree distributionP = [p, p] for different values of p . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.5 Smooth Problem with Internal Hole (δ = 0.2, k = 0.1, ε = 10−12): error es-timates and effectivity index obtained for T (Ω) = [0, 0.5, 1] and uniform p-refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.6 Numerical solution obtained for different values of the hole radius δ, polynomialdegree p = 20, k = 0.1, ε = 10−12 . . . . . . . . . . . . . . . . . . . . . . . . . 57

LIST OF FIGURES 64

4.7 Smooth Problem with Internal Hole (k = 0.1, ε = 10−12): Analytical andestimated absolute error obtained as the polynomial degree p and the holeradius δ vary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.8 Smooth Problem with Internal Hole (k = 0.1, ε = 10−12): Effectivity indexand condition number error obtained as the polynomial degree p and the holeradius δ vary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

BIBLIOGRAPHY 65

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A Posteriori Error Estimator for the Multi-Level hp-Finite ...Angella.pdf · Namely, the multi-level Finite Element Method and the Finite Cell’s Method (FCM). The multi-level FEM