energy-corrected finite element methods · genwirkt, ist im fall von finite element methoden die...

Technische Universitat Munchen

Department of Mathematics

Master’s Thesis

Energy-Corrected Finite Element MethodsMarkus Huber

Supervisor: Prof. Dr. Barbara Wohlmuth

Advisor: Dr. Christian Waluga

Submission Date: 02. December 2013

I assure the single handed composition of this master’s thesis only supported by declaredresources.

Garching, 02. December 2013

........................................................

Knowledge is the small part of ignorance that we arrange and classify.

Ambrose Bierce.

Danksagung

Fur das Gelingen einer solchen Arbeit ist nicht nur der Verfasser verantwortlich, sondernauch diejenigen Personen, die ihm in dieser Zeit zur Seite standen.

So mochte ich mich bei Professor Dr. Barbara Wohlmuth und Dr. Christian Waluga furderen Forderung und Feedback, wahrend meiner Bearbeitungszeit bedanken. Daruberhinaus,dass sie mir mathematische Schwierigkeiten und Ideen naher gebracht haben, und fur jedeFrage ein offenes Ohr hatten. Danke an Christian fur die unzahligen Stunden, die er mitmir verbracht hat, um bei der Implementierung zu helfen und den Beweisen voranzukom-men, Unklarheiten in der Mathematik oder in der Kaffekuche zu beseitigen sowie fur diekonstruktive Durchsicht meiner Arbeit.

Danke an den kompletten Lehrstuhl M2 fur deren freundliche Aufnahme in ihre Mitte,die netten gemeinsamen Mittagsessen sowie Kaffee- und Kuchenpausen, und die inspiri-erenden Gesprache in den Denkpausen.

Danke an Dr. Helmut Schreck fur seine hilfreichen Ratschlage uber das vergangene Jahr.Seine wertvollen beruflichen als auch sozialen Tips haben mich sehr inspiriert und zumNachdenken angeregt.

Danke an meine Freunde fur die vielen tollen Momente und deren Wegbegleitung wahrendmeines Studiums.

Danke an meine langjahrige Freundin, Isabel, fur ihr Verstandnis wahrend des Studi-ums, meines Auslandssemesters, und meiner Masterarbeit sowie fur die wunderbare Zeit.

Ein Dank geht naturlich auch an meine Familie, an meine Eltern, Großeltern und meinenBruder fur deren Unterstutzung wahrend meines ganzen Studiums. Ohne sie ware dasnicht moglich gewesen. Danke.

Zusammenfassung

Bekanntlich ist die Regularitat fur elliptische Differentialoperatoren durch einspringendeEcken reduziert. Weiter noch konnen sich diese Effekte im Fall von springenden Koef-fizienten drastisch verschlimmern und haben somit negative Auswirkungen auf die Kon-vergenzrate von gitterbasierten Diskretisierungsverfahren. Ein Ansatz, der dem entge-genwirkt, ist im Fall von Finite Element Methoden die Energie-Korrektur, in der dieBilinearform im Bereich der Singularitat modifiziert und somit die volle Konvergenzratein gewichteten Sobolev Raumen wiedergewonnen wird. Im Folgenden soll zunachst derlineare und quadratische Ansatz dieser Methode in allgemeinen quasi-uniformen Gitternuntersucht werden, wobei fur den qudratischen Fall die Konvergenzanalyse erarbeitetwird. Weiter wird ein Newton-basiertes Verfahren entwickelt um die Korrekturparametermoglichst billig zu bestimmen. Ebenso wird im Falle von springenden Koeffizienten einea priori Aussage in gewichteten Sobolev Raumen bewiesen und eine Newton-Methodezur Bestimmung der Korrekturparameter angegeben. Die theoretischen Aussagen werdenan diversen numerischen Experimenten analysiert und bestatigt. Um Aussagen uber dieQualitat der Verfahren in Bezug auf schon existierende und verbreitete Strategien zu er-halten, vergleichen wir die Energie-Korrektur mit einem adaptiven Algorithmus auf Basiseines residualen Fehlerschatzers an ausgewahlten numerischen Beispielen.

Contents

1 Introduction........................................................................................... 12 Function Spaces ..................................................................................... 33 Finite Element Method ........................................................................... 5

I The Poisson Problem: Re-entrant Corners 11I.1 Singular Functions.................................................................................. 11I.2 Adaptivity ............................................................................................. 16I.3 Energy-Correction Method for Linear Finite Elements ............................... 28

I.3.1 Theory of the Energy-Correction Method........................................ 28I.3.2 Numerical Examples ..................................................................... 34

I.3.2.1 Energy-Correction in Symmetric Triangulations.................. 35I.3.2.2 Energy-Correction Method vs. Adaptivity.......................... 39I.3.2.3 Energy-Correction in Non-Symmetric Triangulations ........... 42I.3.2.4 Energy-Correction Method vs. Adaptivity.......................... 57

I.4 Energy-Correction Method for Second Order Finite Elements ..................... 61I.4.1 Theory of the Energy-Correction Method........................................ 61I.4.2 Numerical Examples ..................................................................... 76

I.4.2.1 Energy-Correction in Symmetric Triangulations.................. 76I.4.2.2 Energy-Correction in Non-Symmetric Triangulations ........... 84I.4.2.3 Energy-Correction Method vs. Adaptivity.......................... 86

II The Diffusion Problem: Jumping Coefficients 91II.1 Singular Functions.................................................................................. 91II.2 Adaptivity ............................................................................................. 98II.3 Energy-Correction Method for Linear Finite Elements ............................... 102

II.3.1 Theory of the Energy-Correction Method........................................ 102II.3.2 Numerical Examples ..................................................................... 112

II.3.2.1 Energy-Correction for One Singular Point .......................... 112II.3.2.2 Energy-Correction for Several Singular Points..................... 115II.3.2.3 Energy-Correction Method vs. Adaptivity.......................... 120

III Conclusion and Outline 123

References 125

1. Introduction

Finite element methods (FE-methods or FEMs) have become very popular over the past 50years in academical research and industrial applications. The applicability of the methodranges from linear partial differential equations (PDEs) to nonlinear time-dependent prob-lems in higher dimensions. For the simulation of PDEs, it is important to achieve anapproximation of real (physical) properties which models the equation as accurately aspossible. Even in simple cases there are limiting factors like the boundary, the differentialequation itself, the discretization (etc.) which may negatively harm the accuracy that canbe achieved by FE-solution.

For the Poisson equation, the presence of re-entrant corners with interior angle ω > π isa well-known problem [27, 28]. In general, the solution consists of singular componentsof the type rπ/ω even when the data is smooth [16,27,33]; here r denotes the distance tothe corner. This is the reason for reduced approximation accuracy and poor convergencerates when the standard finite element method on quasi-uniform meshes is used [8, 43].For the diffusion equation with jumping coefficients, the situation is even worse, since thesingular components are of the type rε with 0 < ε 1 [24–26].

The literature reveals many different approaches to improving the approximation andthe convergence rate. Commonly applied techniques to overcoming these problems aregraded meshes towards the singularity [3, 5]. Another strategy which is applied is theenrichment of the finite element space by proper singular functions [8,11,12,43]. The ideaof weighted least-squares formulations adds discrete versions of singular functions to thefinite element space and uses first-order system, least-squares approaches [7, 15, 31]. Forthe adaptive framework we refer for a general overview to Braess, Verfurth [10, 45]. Forthe interface problem, in this case special error estimators are developed which includethe jumps of the coefficients in their consideration [6, 13,38].

Most of the strategies mentioned above have in common an improved finite element ap-proximation near the singular point but need a strong modification of the finite elementcode, e.g., the extension of the finite element spaces. Nevertheless, in some applicationsthe quantity of interest can be computed by excluding or relaxing the influence of thesingularity, e.g., stress intensity factors, eigenvalues or the flux at some interface not in-cluding the singular points.

This is the motivation for using the energy-correction method. The initial idea was intro-duced for finite difference schemes [42, 46] and applied to finite elements [40]. A detailedanalysis for more general meshes was then provided [19] and proved that a local modifi-cation of the bilinear form yields optimal convergence rates in weighted Sobolev norms.This modification requires the exact knowledge of the correction parameters with compu-tations that were studied carefully [41]. Recently the method was applied to second-orderfinite elements, interface and eigenvalue problems [22].

As for the outline of the thesis, the introduction is followed by a presentation of therequired spaces and norms in the context of finite element methods, and a statment of an

2 Introduction

embedding theorem for weighted Sobolev spaces. In Section 3 we cover the basic idea ofthe finite element strategy and study the poor convergence in the presence of singularitiesthrough a first example. The work then splits into two main parts - the Poisson problemfor re-entrant corners and the diffusion equation with jumping coefficients.

In Section I.1 we develop the singular functions in the case of re-entrant corners andformulate an extension theorem. Then in Section I.2, we introduce the adaptive approachfor a residual-based error estimator and verify the optimal convergence results in nu-merical experiments. In Section I.3 we cover the ideas of the energy-correction methodfor linear finite elements in symmetric and non-symmetric triangulations, and formulatea nested-Newton method for the computation of the correction parameters. Further-more, numerical examples are provided, which show optimal convergence in symmetricand non-symmetric meshes by an optimal correction and parameters obtained by thenested-Newton method, as is a comparison with the adaptive method of Section I.2. Inthe following Section I.4, we study the energy-correction method for second order finiteelements, prove an optimal a priori result and verify our conclusions in numerical exper-iments. For a nested-Newton method similar numerical result are obtained. We draw acomparison with the adaptive method of Section I.2.In the second part we consider the existence of solutions for jumping coefficients and statean extension result (Section II.1). In Section II.2 we see an adaptive technique developedfor jumping coefficients, formulate an error estimator and apply it to numerical examples.In Section II.3 we analyze the energy-correction method for jumping coefficients, state anested Newton method, verify the theoretical results in numerical examples and finallycompare it with the adaptive method of Section II.2.

2. Function Spaces

In this section we recall the Sobolev function spaces and norms which are generally usedin finite element methods. To analyze the energy-correction approaches, we further defineweighted Sobolev spaces, and state an embedding theorem.

Let Ω ⊂ R2 be a bounded domain and D ⊂ Ω.

The space of continuous and m-times differentiable functions is defined by

Cm(D) = u is continuous in D and m-times differentiable in D. (2.1)

The space of square-integrable functions

L2(D) = u : ‖u‖0 <∞, (2.2)

is a Hilbert space with by the scalar product induced norm

(u, v) =

∫D

v · u dx, ‖u‖0 =√

(u, u). (2.3)

The space of essential bounded functions is defined by

L∞(D) = u : ‖u‖L∞(D) <∞ (2.4)

which is equipped with the norm

‖u‖L∞(D) = ess supx∈D|u(x)| . (2.5)

For m ≥ 0 we define the Sobolev-spaces

Hm(D) = u : Dβu ∈ L2(D) for 0 ≤ |β| ≤ m, (2.6)

where Dβ denotes the weak derivative with multi-index β. The spaces are Hilbert spaces,when we equip them with the usual Sobolev norms induced by

‖v‖m;D =

∑|β|≤m

‖Dβv‖20

1/2

. (2.7)

Let further r = r(x) denote the distance of the point x to a fixed but variable point xc,e.g., a singular point. For α ∈ R we define the weighted Sobolev spaces

Hmα (D) = u : rα+|β|−mDβu ∈ L2(D) for 0 ≤ |β| ≤ m. (2.8)

equipped with the norm

‖v‖m,α;D =

∑|β|≤m

‖rα+|β|−mDβv‖20

1/2

. (2.9)

4 Function Spaces

We omit the subscript D in the case D = Ω and denote by H1(D) the subspace of theSobolev space H1(D) with vanishing traces on ∂D. For further details on Lp - spaces,Sobolev spaces, weighted Sobolev spaces and weak derivatives see [1, 30]. Later, we fre-quently make use of the following embeddings which follow by [30, Cor. 6.7], classicalembeddings of Sobolev spaces [1, p. 97] and the definition (2.8) of the weighted Sobolevspaces.

Lemma 2.1. For any m ≥ 0, the following embeddings are continuous:

Hm+1α (Ω) → Hm

α−1(Ω), α ∈ R,Hm+2α (Ω) → Cm(Ω), α < 1.

3. Finite Element Method

In this section we describe the general setting in the finite element method and state an apriori result. Further, we consider numerical examples for which we once obtain optimaland the other time poor convergence rates.

We consider the Poisson equation with homogeneous boundary conditions

−∆u = f in Ω, u = 0 on ∂Ω (3.1)

on a bounded polygonal domain Ω ⊂ R2 and f ∈ L2(Ω). The variational formulation ofthe Poisson equation (3.1) is characterized by the following problem: Find u ∈ H1(Ω)such that

a(u, v) = (f, v), v ∈ H1(Ω), (3.2)

with bilinear form a(v, w) =∫

Ω∇v ·∇w dx for v, w ∈ H1(Ω). The existence and unique-

ness of a weak solution is given by the Lax-Milligram lemma [21, p. 215, Lemma 6], sincethe bilinear form is continuous and coercive.

In the context of finite elements the aim is to derive the solution of the variationalformulation (3.2) of the Poisson equation. The problem (3.2) is formulated in infinitedimensional Sobolev spaces, namely, the ansatz space and test space H1(Ω). The finiteelement approach discretizes the problem (3.2) in finite dimensional spaces. Thus, oneneeds a discretization of Ω and the finite spaces associated with the ansatz and the testspace in our case H1(Ω).

We start with the domain Ω. For that, let T1 denote a regular triangulation of the domainΩ in the sense of Ciarlet [14] and let Thh≥1 be a sequence of uniform refinements of thecoarse mesh T1. The refinements are obtained by subsequently dividing each triangularin four concurrent ones, cf. Figure 3.1.

Figure 3.1: Refinement of a triangle into four concurrent ones.

By definition each mesh of Thh is quasi-uniform, i.e., it holds

hT ≤ h . ρT for all T ∈ Th (3.3)

with a constant depending on the refinement level. The parameters ρT and hT in (3.3)denote the diameter of the largest ball which can be inscribed in T , and the smallestball that contains T , respectively. Further, h = maxhT : T ∈ Th is referred to as themeshsize, cf. Figure 3.2.

6 Finite Element Method

Remark 3.1. In the finite element approaches it is often assumed that Ω is polygonal.This is due to the triangulation of the domain and no additional occurring error in theapproximation of the domain.

Here and throughout this paper we write a . b for a ≤ Cb with some constant C whichdepend on parameters like the domain that are not relevant for our analysis.

hT

ρT

Figure 3.2: Triangle diameter; Left: Smallest ball containing the triangle; Right: Largestball that can be inscribed in the triangle.

We choose as finite dimensional ansatz and test space a subspace of H1(Ω) which consistsof continuous functions which are piecewise polynomials of degree k

V kh = v ∈ H1(Ω) : v|T ∈ Pk(T ) for all T ∈ Th. (3.4)

The finite element space V kh is the standard conforming space associated with the mesh

Th. Hence, we can formulate the finite element method: Find uh ∈ V kh such that

a(uh, v) = (f, v), v ∈ V kh . (3.5)

The existence and uniqueness of the finite element solution uh is again guaranteed by theLax-Milgram lemma, since ansatz and test spaces are subspaces of H1(Ω).Our next propose is to derive an approximation result for the finite element method whichgive us an a priori estimate of the error u− uh in proper norms like L2, H1. We assumethat the domain Ω is convex, then, e.g. by Strang, Fix [43] or Braess [10] the followingholds:

Theorem 3.1. Let V k−1h be the finite element space and u ∈ Hk(Ω). Then, the finite

element approximation uh differs from the true solution u by

‖u− uh‖0 . hk‖u‖k (3.6)

in the L2-norm and in the H1-norm

‖u− uh‖1 . hk−1‖u‖k. (3.7)

Energy-Corrected Finite Element Methods 7

Remark 3.2. Here, we analyze the Poisson equation with homogeneous boundary condi-tion (3.1), by a simple modification of the bilinear form, the ansatz and test space, thisis also valid for any elliptic partial differential equation and boundary condition. Theapproximation result further holds under certain additional conditions, cf. [43].

We now consider two numerical examples: One verifying the result of the optimal con-vergence rates and the other showing that the requirements on the domain are necessary.First, let us introduce the Poisson equation with Dirichlet boundary conditions

−∆u = 0 in Ω, u = r sin(ϕ) + r2 sin(2ϕ) on ∂Ω, (3.8)

where (r, ϕ) denotes the polar coordinates centered at the origin and Ω is a polygonalapproximation of the semicircle, cf. Figure 3.3.

Figure 3.3: Left: Polygonal approximation of the semi-circle; Right: Initial mesh T1 forthe convergence analysis.

We want to apply the result of this section to the equation (3.8) and need to transform theproblem into the form of the Poisson equation with homogeneous boundary conditions(3.1). Therefore, let u0 ∈ H1(Ω) with u0|∂Ω = r sin(ϕ) + r2 sin(2ϕ), then the problem(3.8) can be rewritten: Find w = u− u0 ∈ H1(Ω) such that

a(w, v) = (f, v), v ∈ H1(Ω). (3.9)

Hence, we write (3.8) in the form of (3.9) and then apply our results.

For our convergence analysis we refine the initial mesh T1 presented in Figure 3.3 sixtimes and study the convergence of the linear finite elements approximation to the exactsolution of (3.8) which is given by its Dirichlet boundary condition. The results are shownin Table 3.1.

l 100× ‖u− uh‖0 rate 10× ‖u− uh‖1 rate1 3.80930 — 4.73580 —2 0.95369 2.00 2.36390 1.003 0.23854 2.00 1.18080 1.004 0.05964 2.00 0.59035 1.005 0.01491 2.00 0.29517 1.006 0.00931 2.00 0.14758 1.00

Table 3.1: Convergence analysis of the Poisson equation for linear finite elements in thesemi-circle.

8 Finite Element Method

In the next Table 3.2 the analysis for quadratic finite element is presented. Here, we addan additional term to the boundary condition of (3.8) such that

u = r sin(ϕ) + r2 sin(2ϕ) + r3 sin(3ϕ) on ∂Ω.


Table 3.2: Convergence analysis of the Poisson equation with second order finite elementsin the semicircle.

In Table 3.1 and Table 3.2 we exactly observe the results of the Theorem 3.1, for thelinear finite elements a h2 and h convergence and for the second order finite elements a h3

and h2 convergence in the L2, H1 - norm, respectively. All requirements of the Theorem3.1 are fulfilled, namely, the domain Ω is a polygonal semi-circle, hence, convex, and theexact solutions r sin(ϕ) + r2 sin(2ϕ) and r sin(ϕ) + r2 sin(2ϕ) + r3 sin(3ϕ) are smooth.

In the next experiment we study the Poisson equation (3.8) with Dirichlet condition

u = r2/3 sin(3

2ϕ) + r4/3 sin(

4

3ϕ) on ∂Ω

and in the (circular) L-shape domain dipicted in Figure 3.4.

Figure 3.4: Left: Polygonal approximation of the (circular) L-shape domain; Right: Initialmesh T1 for the convergence analysis.

Again, we apply the standard finite element method of this section. The convergenceresults are presented in Table 3.3.



Table 3.3: Convergence analysis of the Poisson equation with the linear finite element inthe (circular) L-shape domain.

In contrast to the example before we observe a reduced convergence rate in the L2- andH1-norm. This is due to the L-shape domain, which is obviously not convex. Since thesolutions given by the boundary condition are not in H2(Ω) and H3(Ω), the conditionsfor the optimal convergence rates of 2 and 3, resp., of Theorem 3.1 are violated.

Part I.The Poisson Problem: Re-entrantCorners

I.1. Singular Functions

In this section we consider an expansion of the solution of the Poisson equation. We willsee that the solution in general consists of a singular part and a regular part. The singu-larities occur if the boundary is not smooth, i.e., if the boundary has re-entrant corners(interior angle > π).First, we derive by analytic tools an expansion of the Poisson equation and state an ex-istence theorem with a regularity estimate in weighted Sobolev spaces. Further, we willfocus on some properties of the singular functions and verify them in numerical examples.

Again, we consider the Poisson equation with homogeneous boundary conditions

−∆u = f in Ω, u = 0 on ∂Ω (I.1)

on a bounded polygonal domain Ω ⊂ R2. We further assume that Ω has a single re-entrant corner with interior angle π < ω < 2π which is located at the origin and one edgeof the boundary touching the singularity lies in the positive x-axis, cf Figure I.1. Sinceour arguments are locally, the one re-entrant corner case can easily be extended to severalcorners.

Ω1

r = r0

Ω

ϕ = 0ϕ = ω

Figure I.1: Domain with a single re-entrant corner at the origin and one edge of theboundary touching the singularity lies in the positive x-axis.

For the extension of the solution of the Poisson problem we follow some ideas of Strandand Fix [43, p. 258]. Let us assume that the data f is analytic and the boundary issmooth expect of the corner at the origin, cf. Figure I.1. Then, by Weyl’s lemma [21, p.199] u is analytic in Ω except of the origin. To describe the solution u local near theorigin, we define

Ω1 = (r, ϕ) : 0 < r < r0, 0 < ϕ < ω ⊂ Ω (I.2)

12 Singular Functions

with polar coordinates (r, ϕ) centered at the origin; cf. Figure I.1. Since u is analytic inΩ1, we are able to expand the solution u in a Fourier series in Ω1. Let us assume we canwrite the solution u in Ω1 by a separation of variables as

u(r, ϕ) = R(r)Φ(ϕ). (I.3)

We write the Poisson equation (I.1) in polar coordinates

−urr −1

ruϕ −

1

r2uϕϕ = f in Ω1,

u(r, 0) = u(r, ω) = 0 for 0 < r < r0.(I.4)

To deal with inhomogeneity in (I.4) we use a variation of the constant, i.e.,

u(r, ϕ) = U(r, ϕ) + V (r, ϕ) (I.5)

with functions U and V satisfying the homogeneous and the inhomogeneous variant of(I.4), both composed as in (I.3) and the same notation. Let us first consider the homoge-neous solution U . We plug it into (I.4) and derive

−R′′Φ− 1

rR′Φ− 1

r2RΦ′′ = 0. (I.6)

By some calculus we determine the following equality which is constant and set this toλ ∈ R, since the left hand side only depends on the radius and the right hand side onlyon the angle.

−R′′ − 1/rR′

1/r2R=Φ′′

Φ≡ const. = λ. (I.7)

The problem (I.4) reduces to two ordinary differential equations (ODEs) in r and ϕ

r2R′′ + rR′ − λR = 0, (I.8)

Φ′′ + λΦ = 0. (I.9)

Due to (I.9) and the induced boundary condition is λ > 0. Then, the solution of the ODE(I.9) in the angle ϕ is obtained by including the boundary conditions U(r, 0) = U(r, ω) = 0

Φ(ϕ) = sin(kπ

ωϕ) for k ∈ Z (I.10)

and for (I.8) we have

R(r) = rkπω for k ∈ Z. (I.11)

Note, that the negative exponents can be dropped, since we have an analytic solution inΩ1. Thus, we can combine these two solutions and by superposition we get the solutionof the homogeneous problem

U(r, ϕ) =∞∑k=1

αkrkπω sin(

kπ

ωϕ), (I.12)

where αk is the Fourier coefficient of u(r0, ϕ). This concludes the derivation of the solutionof the homogeneous problem. For the inhomogeneous problem, which can be obtained,


e.g., by variation of the constant, we refer to Strang, Fix [43]. The full representationnear the corner is provided by Lehman [32], but for us this is not of main interest, sincethe leading r -term has an exponent of 2, hence, is smooth enough.

Before, we can state the lemma which summarizes the derived expansion above we intro-duce some notation. Let us define the singular functions for k ∈ N

sk(r, ϕ) = rkπω sin(

kπ

ωϕ) (I.13)

andsk(r, ϕ) = η(r)sk(r, ϕ) (I.14)

in polar coordinates (r, ϕ). The smooth cut-off function η(r) is identically one in a neigh-borhood Ω′ ⊂ Ω of the singularity; for details see [8]. Then, due to the expansion we sawin (I.12) and the notes about the inhomogeneous solution we can expanded the solutionof the Poisson problem into singular functions (I.14) and some regular component. Werefer for details to the Russian works of Kondratiev [27] and Maz´ja, Plamenevskii [33],on the French work of Grisvard [20] and to Blum, Dobrowolski [8].

Lemma I.1. Let f ∈ Hmβ (Ω) for β < 1 + m and 1 + m − β 6= nπ

ωfor all n ∈ Z. Then,

the solution u of the Poisson problem (I.1) admits the expansion

u =∑n∈I∩N

knsn +W, (I.15)

with I = (0, ωπ

(1 +m− β)) and W ∈ Hm+2α (Ω). Moreover, there holds∑

n∈I∩N

|kn|+ ‖W‖2,β . ‖f‖0,β. (I.16)

Let us state some remarks regarding this lemma.

Remark I.1. The coefficients kn in the representation (I.15) are called stress intensityfactors. Their exact formulation can be found in [8].

Remark I.2. In the case of m = 0, in Lemma I.1, for β > 1 − nπω

it is sufficient toconsider (n − 1) singular components and in the case 1 − π

ω< β < 1 the solution is

smooth, i.e. in H2β(Ω).

Next, let us study the regularity of the singular functions (I.14).

Lemma I.2. For any α > 1 − nπω

there holds sn ∈ H2−α(Ω) and sn ∈ H2α(Ω), but

sn /∈ H1+nπ/ω(Ω) and sn /∈ H21−nπ/ω(Ω).

Proof. We only proof the results for the weighted Sobolev spaces, for the usual Sobolevspaces we refer to Grisvard [20]. To check the result in the weighted spaces it is sufficientto verify

‖∇2sn‖0,α <∞. (I.17)


We further assume that sn ≡ 0 in Ω\Ω1, i.e., we set in the definition of the singular func-tions (I.14) Ω1 = Ω′. In the calculation we use the transformation into polar coordinatesand that the cut-off function η(r) and the term Φ(ϕ) are automatically bounded.

‖∇2sn‖20,α .

∫ r0

r=0

∫ ω

ϕ=0

r2αr2nπω−4Φ(ϕ)r dϕ dr .

∫ r0

r=0

r2α+2nπω−4+1dr <∞, (I.18)

which holds for 2α + 2nπω− 4 + 1 > −1⇔ α > 1− nπ

ω.

The first singular function s1 is not in Hk(Ω) for k ≥ 2 by Lemma I.2, thus, also thesolution of the Poisson equation (I.1) not. Recall that a solution needs to be at least inH2(Ω) for optimal convergence in the L2 - norm (Theorem 3.1), which certainly cannot besatisfied in the presence of re-entrant corners. Also, if we measure the error in a weightedSobolev norm, which reduces the influence of the neighborhood of the singularity, weobserve sub-optimal convergence rates for the solution on the whole domain, an effectwhich is commonly referred to as ”pollution” [5, 8, 9, 43].

Lemma I.3. (Pollution effect). Let u be the solution of (I.1) with f ∈ H0−α(Ω) for

some α > −1. If k1 6= 0 in (I.15), then

‖u− uh‖0,α & ‖∇(u− uh)‖20 & h

2πω . (I.19)

Proof. We proof both inequalities separately. For the first inequality, we use that f isin H0

−α(Ω)

‖∇(u− uh)‖20 = |a(u, u)− a(uh, uh)|

= |f(u)− f(uh)| ≤ ‖f‖0,−α‖u− uh‖0,α.(I.20)

For the second inequality we use the expansion of (I.1) and the regulartiy of the firstsingular function. By applying the Galerkin orthogonality, twice, as well as the boundnessof Φ(ϕ) and the cut-off function η(r), we obtain

‖∇(u− uh)‖20 & ‖∇(s1 − s1,h)‖2

0 = a(s1 − s1,h, s1 − s1,h) = ‖∇s1‖20

&∫ h

r=0

∫ ω

ϕ=0

r2πω−2Φ(ϕ)r dϕ dr &

∫ h

r=0

r2πω−1dr = h2π/ω,

(I.21)

where we used that the singular function s1 is non-zero in a neighborhood of the singu-larity.

Finally, we consider a numerical example which illustrates the poor convergence rate inthe weighted norms independent of the order of the finite elements. We analyze thePoisson equation

−∆u = 0 in Ω, u = s1 + s2 on ∂Ω (I.22)


with Ω as the circular L-shape domain like in Section 3 in Figure 3.4. In comparison tothe Section 3 we analyze linear and quadratic finite elements in weighted L2-norms. Theconvergence results are depicted in Table I.1.

l 100× ‖u− uh‖0,0.39 rate 100× ‖u− uh‖0,1.39 rate1 2.70080 — 0.14776 —2 0.88115 1.62 0.04991 1.573 0.30947 1.51 0.01948 1.364 0.11438 1.44 0.00772 1.345 0.04361 1.39 0.00306 1.336 0.01692 1.37 0.00121 1.33

Table I.1: Convergence analysis of the Poisson equation for linear and quadratic finiteelement in the (circular) L-shape domain in weighted L2-norms.

In the convergence analysis of Table I.1 we observe the reduced rate stated in TheoremI.3, for the L-shape domain, this is 1.33.

I.2. Adaptivity

In this section we consider the adaptive method to regain optimal convergence rates for fi-nite element methods in the presence of corner singularities. This approach is well-knownand studied for many kinds of partial differential equations [4,10,17,23,39,45]. Differentadaptive approaches were made but all of them have a fine mesh in regions of the domainwith singularities and a coarse mesh where one suppose smooth solutions, in common.The application of adaptivity poses three issue that have to be solved: a refinement strat-egy, error indication and error estimation. In the following we use the bisection for therefinement, the Dorfer strategy for the error indication which is the most efficient wayto reduce the error and a residual-based error estimator. Finally, we apply this residualbased adaptive method to corner singularities in numerical examples and obtain optimalconvergence rates for linear and second order finite elements.

Let us consider the setting in a convergence analysis. In the purpose of the SectionI.1 and Section 3 we refined the elements of the mesh in each step in an uniform way,whereas the adaptive approach only refines the elements which have a large error fraction.In the case of the Poisson equation with homogeneous boundary condition with a singlere-entrant corner like in Section I.1, we know that the finite element approximation hasa large error near the corner. Thus, a refinement in this corner needs to be done. Ina general framework we do not know where singularities of the solution are, thus, it isessential to detect the regions, where the error is dominant using error estimators [2, 10].

Definition I.1. Let ηex = ‖u − uh‖? denotes the exact error in a suitable norm ‖ · ‖?.Then, we call ηes an error estimator, if 0 < c,C <∞ exist such that

cηes ≤ ηex ≤ Cηes. (I.23)

Many kinds of error estimators have been developed, e.g., residual-based estimators byBabuska, Reinboldt [4], hierarchical error estimators by Deuflhard, Leinen, Yserentant [17]and dual error estimators by Johansen, Rannacher, Vexler [23, 39]. In the following, weconcentrate on the residual-based estimators.

For an element T ∈ Th we defined the residual-based estimator as

η2T,Res = h2

T‖∆uh + f‖20,T +

1

2

∑e∈∂T

he‖[∇uh · ne]‖20,e, (I.24)

where e denotes an edge of the boundary of the triangle T and [∇uh · ne] = ∇uh|T1 ·ne − ∇uh|T2 · ne with ne the outer unit normal of T1 and e = ∂T1 ∩ ∂T2. We refer toBraess [10, p. 155] for the proof of the following Lemma I.4 of the efficiency of the errorestimator (I.24).

Lemma I.4. The error estimator defined in (I.24) satisfies condition (I.23) with ‖ · ‖? =‖ · ‖0.


By (I.24) we can estimate the error in each element. The next purpose is to definea strategy which ensures an error reduction in each step of the convergence analysis.Strategies are for instance the Maximum - strategy or the Mean - values strategy, butthe most common used one is the Dorfler strategy which enables at least a reduction inthe estimated error, cf. [18]. For the summarized description of the algorithm compareAlgorithm 1.Let η2

T,Res be the local error of each element T ∈ Th which was computed, here, by theerror estimator (I.24). We sum up all local errors and define the global estimated error

η2DS =

∑T∈Th

η2R,Res

and sort the local errors by their magnitude, i.e.

ηT1,Res ≥ ηT2,Res ≥ ... ≥ ηTnT ,Res

with nT = |Th|. Then, we define M ∈ N as the minimum such that

M∑i=1

ηTi,Res ≥ (1− σ)2η2DS

for a σ ∈ [0, 1]. We mark all elements Ti with i ≤M .

Algorithm 1 Dorfler strategy

Choose σ ∈ [0, 1] and set η2DS =

∑T∈Th

η2R,Res and nT = |Th|.

Sort ηT,Res such thatηT1,Res ≥ ηT2,Res ≥ ... ≥ ηTnT ,Res.

Define M ∈ N as the minimum such thatM∑i=1

ηTi,Res ≥ (1− σ)2η2DS.

Mark all elements Ti with i ≤M .

Remark I.3. In the Dorfler Algorithm 1 one needs to choose the parameter σ in a properway. The choice σ = 0 means an uniform refinement, since all elements are marked,whereas σ = 1 no marking and refinement takes place. Usually the choice of σ is between0.4 to 0.6.

Remark I.4. Due to the ordering of the local error the adaptive algorithm distributes theglobal error over all elements uniformly.

Trough the Dorfler Algorithm 1 it is ensured that the elements with the biggest errorfraction are marked. Then, these marked elements are refined by a refining strategy. Apossible choice is the bisection. The algorithm of the bisection is presented in Algorithm2 for the refinement of one triangle. Note, Algorithm 2 ensures that the triangulationsatisfies the condition of Ciarlet [14] in each step.

18 Adaptivity

Algorithm 2 Bisection

Set S = T ∈ Th : T is marked.while S 6= ∅ do

Choose T ∈ S and connection the longest edge of T with the opposite vertex.Remove the mark of T .Mark all elements with hanging vertex in the midpoint of a edge and add theseelements to S.

end while

We want to come back to analyze re-entrant corners in numerical experiments. Let usconsider the Poisson equation

−∆u = 0 in Ω, u = s1 + s2 on ∂Ω (I.25)

in a domain Ω ⊂ R2 with a single re-entrant corner as described in Section I.1. We study(I.25) in the polygonal section with interior angle ω ∈ 5

4π, 3

2π, 7

4π, 2π, where we also call

the domain with angle 32π L-shape domain and 2π slit-domain. The domains are depicted

in Figure I.2.

Figure I.2: Domains 5/4π-section, L-shape domain, 7/4π-section and slit-domain.


First, let us present the initial symmetric triangulation of the domains in Figure I.3 andstudy the convergence when applying the adaptive method for linear finite elements.

Figure I.3: Symmetric initial mesh T1 for domains 5/4π-section, L-shape domain, 7/4π-section and slit domain.

In the convergence plots of Figure I.5 for the 5/4π-section, Figure I.6 for the L-shapedomain, Figure I.7 for the 7/4π-section and Figure I.8 for the slit-domain we observean optimal convergence rate for the FE-solution obtained by the adaptive method (blue)of this section and a reduce rate for the standard FE-solution (green). This poor ratecorresponds to the one of Lemma I.3. We also depict the by the bisection Algorithm 2refined meshes for the four geometries, cf. Figure I.4. As argued in the beginning of thissection the mesh is refined closely at the re-entrant corner, since there the solution needsto be resolved exactly.

Figure I.4: Refined meshes obtained by a bisection method with 633 DOFs (5/4π-section),631 DOFs (L-shape domain), 630 DOFs (7/4π-section) and 525 DOFs (slitdomain); Initial mesh: symmetric (linear FE).

20 Adaptivity

101 102 103 104 105 106

DOFs

10−6

10−5

10−4

10−3

10−2

10−1

100

Figure I.5: Error ‖u − uh‖0,α with α ≈ 0.25 in the 5/4π-section; Dashed red: h2 refer-ence line; Blue: Error of the FE-solution by adaptivity; Green: Error of thestandard FE-solution.

101 102 103 104 105 106

DOFs

10−6

10−5

10−4

10−3

10−2

10−1

100

Figure I.6: Error ‖u − uh‖0,α with α ≈ 0.38 in the L-shape domain; Dashed red: h2

reference line; Blue: Error of the FE-solution by adaptivity; Green: Error ofthe standard FE-solution.


101 102 103 104 105 106

DOFs

10−6

10−5

10−4

10−3

10−2

10−1

100

Figure I.7: Error ‖u − uh‖0,α with α ≈ 0.48 in the 7/4π-section; Dashed red: h2 refer-ence line; Blue: Error of the FE-solution by adaptivity; Green: Error of thestandard FE-solution.

101 102 103 104 105 106

DOFs

10−6

10−5

10−4

10−3

10−2

10−1

100

Figure I.8: Error ‖u − uh‖0,α with α ≈ 0.55 in the slit domain; Dashed red: h2 refer-ence line; Blue: Error of the FE-solution by adaptivity; Green: Error of thestandard FE-solution.

22 Adaptivity

Next, we study the convergence in the case of a non-symmetric triangulation in the fourtest scenarios of Figure I.2. We start with the triangulation depicted in Figure I.9.

Figure I.9: Non-symmetric initial mesh T1 for domains 5/4π-section, L-shape domain,7/4π-section and slit domain.

The convergence analysis of the four examples is depicted in Figure I.11, Figure I.12, Fig-ure I.13 and Figure I.14, respectively. Again, optimal rates are obtained by the adaptivemethod, whereas the standard approach yields to a reduced rate similar as in the case ofthe symmetric triangulation. The adaptive meshes are depicted in Figure I.10 with a finetriangulation near the corner.

Figure I.10: Refined meshes obtained by the bisection method with 539 DOFs (5/4π-section), 575 DOFs (L-shape domain), 640 DOFs (7/4π-section) and 507DOFs (slit domain); Initial mesh: Non-symmetric (linear FE).


101 102 103 104 105 106

DOFs

10−6

10−5

10−4

10−3

10−2

10−1

100

Figure I.11: Error ‖u − uh‖0,α with α ≈ 0.25 in the 5/4π-section with a non symmetrictriangulation; Dashed red: h2 reference line; Blue: Error of the FE-solutionby adaptivity; Green: Error of the standard FE-solution.

101 102 103 104 105 106

DOFs

10−6

10−5

10−4

10−3

10−2

10−1

100

Figure I.12: Error ‖u−uh‖0,α with α ≈ 0.38 in the L-shape domain with a non symmetrictriangulation; Dashed red: h2 reference line; Blue: Error of the FE-solutionby adaptivity; Green: Error of the standard FE-solution.

24 Adaptivity

101 102 103 104 105 106

DOFs

10−6

10−5

10−4

10−3

10−2

10−1

100

Figure I.13: Error ‖u − uh‖0,α with α ≈ 0.48 in the 7/4π-section with a non symmetrictriangulation; Dashed red: h2 reference line; Blue: Error of the FE-solutionby adaptivity; Green: Error of the standard FE-solution.

101 102 103 104 105 106

DOFs

10−6

10−5

10−4

10−3

10−2

10−1

100

Figure I.14: Error ‖u − uh‖0,α with α ≈ 0.55 in the slit domain with a non symmetrictriangulation; Dashed red: h2 reference line; Blue: Error of the FE-solutionby adaptivity; Green: Error of the standard FE-solution.


Finally, we apply the adaptive method also to second order finite elements. We considerthe symmetric triangulations of Figure I.3 for the four examples. The results of theconvergence analysis are presented in Figure I.16, Figure I.17, Figure I.18 and Figure I.19.The optimal decay of the error h3 is observed in the weighted L2 norms. The standardsecond order approach yields to the same poor rate as for linear FE. The adaptive meshesare presented in Figure I.15, the singularity in the corner is resolved by these meshes.

Figure I.15: Refined meshes obtained by a bisection method with 591 DOFs (5/4π-section), 662 DOFs (L-shape domain), 556 DOFs (7/4π-section) and 531DOFs (slit domain); Initial mesh: Symmetric (second order FE).

101 102 103 104 105

DOFs

10−7

10−6

10−5

10−4

10−3

10−2

10−1

Figure I.16: Error ‖u − uh‖0,α with α ≈ 1.25 in the 5/4π section with a symmetric tri-angulation; Dashed red: h3 reference line; Blue: Error of the second orderFE-solution by adaptivity; Green: Error of the second order FE-solution.

26 Adaptivity

101 102 103 104 105

DOFs

10−7

10−6

10−5

10−4

10−3

10−2

10−1

Figure I.17: Error ‖u − uh‖0,α with α ≈ 1.38 in the L-shape domain with a symmetrictriangulation; Dashed red: h3 reference line; Blue: Error of the second orderFE-solution by adaptivity; Green: Error of the second order FE-solution.

101 102 103 104 105 106

DOFs

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

Figure I.18: Error ‖u − uh‖0,α with α ≈ 1.48 in 7/4π section with a symmetric trian-gulation; Dashed red: h3 reference line; Blue: Error of the second orderFE-solution by adaptivity; Green: Error of the second order FE-solution.


101 102 103 104 105 106

DOFs

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

Figure I.19: Error ‖u − uh‖0,α with α ≈ 1.55 in the slit domain with a symmetric tri-angulation; Dashed red: h3 reference line; Blue: Error of the second orderFE-solution by adaptivity; Green: Error of the second order FE-solution.

In this section we saw a residual-based adaptive method applied to the Poisson equation.We applied this approach to the four scenarios with re-entrant corner, the 5/4π-section,the L-shape domain, the 7/4π-section and the slit domain. The mesh is refined by theadaptive method, fine near and coarse away from the corner due to the singularity in there-entrant corner. We observed that the adaptive approach enables us to regain optimalconvergence rates for symmetric and non-symmetric triangulations in the case of linearFE and for symmetric meshes in the case of second order FE.

I.3. Energy-Correction Method for Linear Finite Elements

In this section we consider an alternative way to the adaptive method in the last SectionI.2 to regain optimal convergence rates - the energy-correction method. In the adaptivityapproach we refined the mesh in such a way that the singularities are resolved properly,whereas in the energy-correction context we modify the bilinear form in the neighborhoodof the singularity. This modification relaxes the influence of the singularity but in manycases the quantity of interest allows this proceeding and hence, avoids the pollution effectof the corner.The basic idea was originally introduced in the context of finite difference methods onstructured meshes in [42, 46] and extended to linear finite elements in [19, 40]. Further,in [41] algorithms were presented which enable to find a proper correction parameter inan efficient way. Recently, the method was extended to second order finite elements, in-terface and eigenvalue problems in [22].

In the first part of this section we want to concentrate on the theory of the energy-correction methods for linear finite elements. We extend the work of [19] such thatsymmetric as well as non-symmetric triangulations can be considered. Therefore, we willenhance the modification of the bilinear form. We will see that the new introduced modi-fication is a generalized formulation of the one in [19]. In the numerical examples we studyour four test scenarios of Section I.2 and observe that different correction domains canbe chosen to obtain h2 convergence. Further, we will develop a nested Newton method inorder to determine the correction parameters in an efficient way similar to [41]. Finally,we will compare the energy-correction method for linear FE with the adaptive method ofthe Section I.2 by concentrating on the accuracy of the methods in weighted norms. Ausage of the energy-correction approach can then be recommended in the weighted normframework.

I.3.1. Theory of the Energy-Correction Method

Let us introduce the energy-correction method for linear finite elements for the Poissonequation (I.1) on bounded domains Ω with one single re-entrant corner as described inSection I.1. In the energy-correction method we modify the bilinear form near the re-entrant corner, thus, we rewrite the standard formulation (3.5) for the Poisson problem(I.1) by: Find umh ∈ V 1

h such that

ah(umh , v) = (f, v), v ∈ V 1

h , (I.26)

where ah(·, ·) is a mesh-depended bilinear form defined by

ah(w, v) = a(w, v)− ch(w, v). (I.27)

In (I.27) a(·, ·) is the origin bilinear form of the standard Galerkin method of Section 3.By setting ch(·, ·) = 0 the standard Galerkin approximation is recovered. In the followingwe restrict our modification ch(·, ·) on some basic assumptions. The modification ch(·, ·)is only supported locally in the neighborhood

Ωh =⋃

T for T ∈ Th with dist(T, 0) ≤ κh (I.28)


of the singularity, for some κ > 0. Further, we postulate the same properties of the mod-ification ch(·, ·) as hold for the bilinear form a(·, ·) - coercivity, continuity and symmetry:

(C1) a(u, u)− ch(u, u) ≥ ‖u‖21 for all u ∈ H1(Ω),

(C2) ch(u, v) ≤ ‖u‖1,Ωh‖v‖1,Ωh for all u, v ∈ H1(Ω),

(C3) ch(u, v) = ch(v, u) for all u, v ∈ H1(Ω).

A part of our results requires an additional symmetry property of mesh at the singularity.

(S) The set of T ⊂ TH for which 0 ∈ T forms a symmetric partition, i.e., the coarsemesh T is locally symmetric around the singular point, cf. also Figure I.20.

Ωh

Figure I.20: Local symmetry assumption on the mesh (S) .

First, we show basic properties of the modified method: existence and uniqueness of themodified solution, Galerkin orthogonality and minimization property.

Lemma I.5. Let (C1)and (C2) be satisfied. Then, the modified FE-solution umh isuniquely defined and for the modified Galerkin orthogonality holds

a(u− umh , v) + ch(umh , v) = 0 for all v ∈ V k

h , (I.29)

for k ∈ N. If in addition the symmetry condition (C3) is guaranteed, then for all v ∈ V 1h

the minimization property

a(u− umh , u− umh )− ch(umh , umh ) ≤ a(u− v, u− v)− ch(v, v) (I.30)

is valid. The conditions (I.26), (I.29) and (I.30) are equivalent.

This result is not proven in [19], hence, this is carried out here.

30 Energy-Correction Method for Linear Finite Elements

Proof. We start to proof that umh is well-defined and unique. We use the Lax-Milgramlemma [21, p. 215], hence, the bilinear form has to be continuous and coercive. For thecontinuity we use the triangle inequality, the continuity of a(·, ·), property (C1) and normestimates.

|a(v, w)− ch(v, w)| ≤ |a(v, w)|+ |ch(v, w)|. ‖v‖1‖w‖1 + ‖∇v‖0,Ωh‖∇w‖0,Ωh . ‖v‖1‖w‖1

for all v, w ∈ H1(Ω). Thus, also continuous on the subspace V kh of H1(Ω). The coercivity

is simply given by (C2) on H1(Ω) and, hence, on V kh , too. Then, by the Lax-Milgram

lemma the existence and uniqueness of umh is given.

For the modified Galerkin orthogonality (I.29) we use the modified (I.26) and the standardvariational formulation (3.2):

0 = (f, v)− (f, v) = a(u, v)− a(umh , v) = a(u− umh , v) + ch(umh , v)

for all v ∈ V kh .

Finally, we proof the minimization property of the modified FE-solution.

a(u− v,u− v)− ch(v, v)

= a(u− v + umh − umh , u− v + umh − umh )

− ch(v + umh − umh , v + umh − umh )

= a(u− umh , u− umh ) + a(u− umh , umh − v)

+ a(umh − v, u− umh ) + a(umh − v, umh − v)

−(ch(v − umh , v − umh ) + ch(v − umh , umh )

+ ch(umh , v − umh ) + ch(u

mh , u

mh ))

= a(u− umh , u− umh )− ch(umh , umh )

+ 2 · (a(u− umh , umh − v) + ch(umh , u

mh − v))

+ a(umh − v, umh − v) + ch(umh − v, umh − v)

≥ a(u− umh , u− umh )− ch(umh , umh )

for all v ∈ V kh . Above we used the modified Galerkin orthogonality (I.30) and condition

(C3).


The following theorem shows that the full convergence rate can be recovered in weightedSobolev norms

Theorem I.1. Let f ∈ H0−α(Ω) for some 1− π

ω< α < 1. Further, assume that conditions

(C1) - (C3) are valid and that the modification satisfies

a(s1 − sm1,h, s1 − sm1,h)− ch(sm1,h, sm1,h) = O(h2) (I.31)

anda(s1 − sm1,h, s2 − sm2,h)− ch(sm1,h, sm2,h) = O(h2). (I.32)

If π < ω ≤ 23π or if 2

3π < ω < 2π and (S) hold, (I.32) can be dropped. Then, the

convergence rates are of optimal order, i.e.

‖u− umh ‖0,α . h2‖f‖0,−α and ‖∇(u− umh )‖0,α . h‖f‖0,−α. (I.33)

Remark I.5. Condition (I.31) and (I.32) of Theorem I.1 yield the estimate

a(u, u)− a(umh , umh ) = O(h2),

which is the energy defect of the solution u and the modified FE-solution umh . This con-dition is necessary (cf. [19]) and sufficient (cf. Theorem I.1) to avoid the pollution effectand gives the method its name: Energy-correction method.

A direct consequence of Theorem I.1 is a L2- and H1-estimate when we omit a vicinity ofthe re-entrant corner.

Corrolary I.1. Assume that (I.31) and (I.32) hold. Then for any domain Ω′′ ⊂ Ω withpositive distance to the singularity point we have

‖u− umh ‖0,Ω\Ω′′ = O(h2) and ‖∇(u− umh )‖1,Ω\Ω′′ = O(h). (I.34)

For the application of the energy-correction method it is necessary to define a modificationch(·, ·). Let us define two mesh-dependent bilinear forms

d1h(u, v) =

∫ω1h

∇u · ∇v dx (I.35)

and

d2h(u, v) =

∫ω2h

∇u · ∇v dx (I.36)

with domains ω1h ∪ ω2

h = Ωh and ω1h ∩ ω2

h 6= ∅ depending on the parameter κ > 0 in thedefinition of Ωh in (I.28).


In the following we define two concrete types of modification ch(·, ·):

One parameter correction for γ ∈ [0, 1):

ch(u, v) = γd1h(u, v) with ω1

h = Ωh (I.37)

Two parameter correction for (γ1, γ2) ∈ [−1, 1)2:

ch(u, v) = γ1d1h(u, v) + γ2d

2h(u, v). (I.38)

Remark I.6. The one parameter correction (I.37) is the modification which was intro-duced in [19] and is a special case of the two parameter correction (I.38), by settingω1h = Ωh, ω2

h = ∅ and restrict γ1 on the interval [0, 1).

Remark I.7. Recalling, that the Poisson equation models a membrane, the parameters(γ1, γ2) ∈ [−1, 1)2 can be regarded as softening (γi > 0) or stiffening (γi < 0) of thematerial in ωih, respectively.

The next Lemma verifies that a linear combination of the two parameter correction sat-isfies the conditions (C1) - (C3).

Lemma I.6. The two parameter correction (I.38) fulfills the conditions (C1) - (C3).

Proof. We start with the coercivity. Let u ∈ H10 (Ω).

a(u, u)− γ1d1h(u, u)− γ2d

2h(u, u)

= ‖∇u‖20 − γ1‖∇u‖2

0,ω1h− γ2‖∇u‖2

0,ω2h

& ‖u‖21 −maxγ1‖∇u‖2

0,ω1h, γ2‖∇u‖2

0,ω1h

& ‖u‖21,

where we used Poincare-Friedrich inequality for a(·, ·) and γ1, γ2 < 1.

Second, the continuity (C2). By the Cauchy-Schwarz inequality we obtain

γ1d1h(u, v) + γ2d

2h(u, v)

≤ ‖∇u‖0,ω1h‖∇v‖0,ω1

h+ ‖∇u‖0,ω2

h‖∇v‖0,ω2

h

. ‖∇u‖0,Ωh‖∇v‖0,Ωh

for all v, w ∈ H1(Ω).

The symmetry follows by the definition of the bilinear forms d1h(·, ·) and d2

h(·, ·).

In Egger, Rude, Wohlmuth [19] the one parameter correction (I.37) was considered, sinceit was restricted to the problem where only the condition (I.31) needs to be satisfied.


It was shown that a γ can be garantueed such that the modification (I.37) satisfies thecondition of Theorem I.1. For our further considerations the one parameter correction(I.37), two parameter correction (I.38) and their respective FE-solutions play importantroles, hence, we introduce the following notation. We denote the FE-solution of the oneparameter correction and two parameter correction by

uh(γ) and uh(γ1, γ2) (I.39)

in order to specify the dependence on the correction parameters. Thus, by notation(I.39) the standard Galerkin approximation is uh(0), uh(0, 0), respectively. According toTheorem I.1 the size of

a(s1 − s1,h(γ), s1 − s1,h(γ))− γd1h(s1,h(γ), s1,h(γ))

is a quantity of interest. Since the singular function s1 is only supported at the singularity,this can be reduced to a local statement depending on the angle, the shape and the numberof elements attached to singular point. Hence, we can formulate an auxiliary problem toobtain the correction parameter: Find the root γ ∈ [0, 1) of

g(γ) = a(s1 − s1,h(γ), s1 − s1,h(γ))− γd1h(s1,h(γ), s1,h(γ)), (I.40)

where g(·) is the energy defect induced by the modified FE-approximation of

−∆u = 0 in Ω, u = s1 on ∂Ω. (I.41)

Remark I.8. In the formulation of the auxiliary problem (I.41) we can use the wholedomain Ω, since we have only one re-entrant corner, and, thus, one singularity. When thedomain consists of more than one singularity, we need to define sufficient subproblems foreach singular point which preserve the angle, the shape and number of elements attachedto the singular point and has no interaction with other singularities. A possibility to dothat is to take the elements attached to the singularity, refine them uniformly and considerthe subproblem (I.41) and determine the energy defect on the new created submesh.

By Theorem I.1 a good approximation of the correction parameter of (I.40) already fulfillsthe assertion for optimal convergence. In Rude, Waluga, Wohlmuth [41] different meth-ods are considered to determine such a approximated γ, e.g., nested Newton methodsor a nonlinear fit, to reduce the computational costs. A good approximation of such aparameter is already determined by a one-step Newton method.Let us consider the algorithm for the energy-correction function (I.40) for the one param-eter correction. For an initial guess γ0 = 0 ∈ [0, 1) we set for l = 1, 2...

γl+1 = γl − g(γl)

g′(γl)=a(s1, s1)− a(s1,h(γ

l), s1,h((γl))

d1h(s1,h(γl), s1,h(γl))

, (I.42)

where γl is the correction parameter used for the one-level coarser mesh. The analysis ofthis method is done in [41].

The one-step Newton method for the one parameter correction motivates to proceed in a


similar way for the two parameter correction (I.38). First, we define the energy-correctionfunction by adding the additional condition (I.32)

g(γ1h, γ

2h) =

(a(s1 − s1,h(γ1, γ2), s1 − s1,h(γ1, γ2))a(s1 − s1,h(γ1, γ2), s2 − s2,h(γ1, γ2))

−γ1d1h(s1,h(γ1, γ2), s1,h(γ1, γ2))− γ2d

2h(s1,h(γ1, γ2), s1,h(γ1, γ2))

−γ1d1h(s1,h(γ1, γ2), s2,h(γ1, γ2))− γ2d

2h(s1,h(γ1, γ2), s2,h(γ1, γ2))

). (I.43)

Note, that in this case we need to solve an additional auxiliary problem (I.41) with bound-ary condition s2.

Similar, to the one parameter case we derive a one-step Newton method in the twodimensional case. We compute the Jacobi matrix of (I.43)

∇g(γ1, γ2) =

(d1h(s1,h(γ1, γ2), s1,h(γ1, γ2)) d2

h(s1,h(γ1, γ2), s1,h(γ1, γ2))d1h(s1,h(γ1, γ2), s2,h(γ1, γ2)) d2

h(s1,h(γ1, γ2), s2,h(γ1, γ2))

). (I.44)

The exact calculation of the Jacobi matrix is similarly to the 1d case in [41]. Thus, weformulate the one-step Newton method for (I.43): For an initial guess (γ0

1 , γ02) = (0, 0) ∈

[−1, 1)2 we set for l = 1, 2, ...(γ1

γ2

)l+1

=

(γ1

γ2

)l−∇g(γl1, γ

l2)−1g(γl1, γ

l2)

=∇g(γl1, γl2)−1

(−a(s1, s1) + a(s1,h(γ

l1, γ

l2), s1,h(γ

l1, γ

l2))

−a(s1, s2) + a(s1,h(γl1, γ

l2), s2,h(γ

l1, γ

l2))

), (I.45)

where (γ1, γ2)l are the parameters used form the one-level coarser mesh. The analysis ofthis method is not the main issue of this thesis and will be done elsewhere. A study fora two dimensional one-step Newton method is done in [22].

I.3.2. Numerical Examples

In this subsection we consider numerical experiments which illustrate the theoretical re-sults of Section I.3.1. We consider the energy-correction method for symmetric and non-symmetric triangulations. Further, we compare the energy-correction method with theadaptive approach of Section I.2.

We consider the Poisson problem

−∆u = 0 in Ω, u = s1 + s2 on ∂Ω, (I.46)

in the four different domains, the 5/4π-section, the L-shape domain, the 7/4π-section andthe slit domain, cf. Figure I.2. The Dirichlet boundary conditions u = s1 + s2 are chosensuch that it is the exact solution. Note, that the second singularity function s2 does notreduce the convergence rates of the uncorrected approach due to sufficient regularity.


I.3.2.1. Energy-Correction in Symmetric Triangulations

In this subsection we consider symmetric triangulations of the four geometries like pre-sented in Figure I.3. For symmetric triangulation we use the one parameter correction(I.37) and choose the elements attached to singularity as correction domain, that is,

Ωh = T ∈ Th : (0, 0) ∈ ∂T. (I.47)

The correction domains (I.47) for the triangulations of the four geometries are depictedin Figure I.21.

Figure I.21: Marked Elements in the neighborhood of the singularity for meshes T1 andT2 of the 5/4π-section, L-shape domain, 7/4π-section and slit-domain.


We conduct our analysis on a series of uniform refined meshes, i.e., we refine the initialmesh T1 seven times and study the convergence to the exact solution u = s1 + s2 inweighted Sobolev norms. First, we present the correction parameters, the roots of theenergy-correction function (I.40) obtained by a full Newton-method (γopt) and the resultsobtained by the one-step Newton method (γoN) in Table I.2.

l 100× γopt 100× γoN1 3.39949 4.342382 3.76150 4.429653 4.06103 4.456614 4.28960 4.671715 4.46273 4.751956 4.59378 4.812767 4.69300 4.85884

Table I.2: Optimal and one-step Newton correction parameters for symmetric triangula-tion of the 5/4π-section.

We study the problem (I.46) in the 5/4π-section for different correction parameters,namely, parameters of Table I.2, the standard Galerkin method by setting γ = 0 andtwo static choices γ = 0.05, 0.1. The convergence analysis is depicted in Table I.3.We observe a non-optimal convergence of the standard Galerkin method. Also, the choiceof the correction parameter 0.1 yields a reduced convergence rate. The parameters of thefull and one-step Newton method presented in Table I.2 as well as the static choice of0.05 converge with a rate of 2.0. The one-step Newton parameters and the static param-eter 0.05 are good approximations for the correction parameters ,and therefore, have asimilar accuracy as the optimal choice. Whereas the standard FE-solution and the staticparameter 0.1 yield worse approximations.

l \ γ 0.00000 rate γopt rate γoN rate 0.05000 rate 0.10000 rate1 2.58530 — 2.60950 — 2.62440 — 2.63690 — 2.78850 —2 0.66998 1.95 0.63384 2.04 0.63140 2.06 0.63034 2.06 0.66354 2.073 0.18016 1.89 0.15503 2.03 0.15345 2.04 0.15243 2.05 0.16411 2.024 0.05005 1.85 0.03831 2.02 0.03780 2.02 0.03746 2.02 0.04353 1.915 0.01435 1.80 0.00953 2.01 0.00939 2.01 0.00929 2.01 0.01233 1.826 0.00425 1.76 0.00238 2.00 0.00234 2.00 0.00232 2.00 0.00369 1.747 0.00129 1.72 0.00059 2.00 0.00059 2.00 0.00058 2.00 0.00115 1.68

Table I.3: Errors 100 × ‖u − uh(γ)‖0,α with α ≈ 0.25; Data of the convergence study ofthe PDE (I.46) in a symmetric triangulation of the 5/4π-section.

Similar results are obtained for the other geometries, the L-shape domain, the 7/4π-section and the slit-domain. When we compare the optimal parameters with the onesobtained by the one-step Newton method, we observe that they are good approximations.Hence, both yield similar errors in the convergence analysis. The values are presented inTable I.4 for the L-shape domain, in Table I.6 for the 7/4π-section and in Table I.8 for


slit-domain. The convergence analysis yields similar results for the four domains. Theenergy-correction method with optimal and by the one-step Newton method obtainedparameters has optimal rates in weighted L2-norms. For the standard Galerkin method,we observe a poor convergence and a reduced accuracy. The static parameters only yieldoptimal rates when they are a good approximation of the roots of the energy-correctionfunction. The convergence study for the L-shape domain is presented in Table I.5, for the7/4π-section in Table I.7 and for the slit-domain in Table I.8.

l γopt γoN1 0.10214 0.120092 0.10827 0.116943 0.11229 0.117734 0.11482 0.118245 0.11641 0.118566 0.11741 0.118777 0.11804 0.11889

Table I.4: Optimal and one-step Newton correction parameters for a symmetric triangu-lation of the L-shape domain

l \ γ 0.00000 rate γopt rate γoN rate 0.10000 rate 0.20000 rate

1 2.70080 — 2.51370 — 2.53220 — 2.51260 — 2.81430 —2 0.88115 1.62 0.62394 2.01 0.61971 2.03 0.63102 1.99 0.75065 1.913 0.30947 1.51 0.14916 2.06 0.14667 2.08 0.15806 2.00 0.22092 1.764 0.11438 1.44 0.03575 2.06 0.03495 2.07 0.04174 1.92 0.07788 1.505 0.04361 1.39 0.00867 2.04 0.00845 2.05 0.01198 1.80 0.03016 1.376 0.01691 1.37 0.00213 2.03 0.00207 2.03 0.00379 1.66 0.01200 1.337 0.00663 1.35 0.00053 2.02 0.00051 2.02 0.00131 1.54 0.00479 1.32

Table I.5: Errors 100 × ‖u − uh(γ)‖0,α with α ≈ 0.39; Data of the convergence study ofthe PDE (I.46) in a symmetric triangulation of the L-shape domain.

l γopt γoN1 0.18017 0.208822 0.18692 0.196193 0.19056 0.195614 0.19253 0.195315 0.19361 0.195156 0.19420 0.195057 0.19453 0.19500

Table I.6: Optimal and one-step Newton correction parameters for a symmetric triangu-lation of the 7/4π-section.


l \ γ 0.00000 rate γopt rate γoN rate 0.30000 rate 0.40000 rate1 3.39390 — 2.82770 — 2.88050 — 2.85940 — 3.33940 —2 1.33000 1.35 0.75890 1.90 0.76137 1.92 0.76372 1.90 1.07220 1.643 0.55947 1.25 0.18702 2.02 0.18613 2.03 0.18632 2.04 0.37116 1.534 0.24457 1.19 0.04471 2.06 0.04426 2.07 0.04420 2.08 0.15199 1.295 0.10890 1.17 0.01062 2.07 0.01048 2.08 0.04420 2.08 0.15199 1.296 0.04891 1.15 0.00254 2.07 0.00250 2.07 0.00269 1.97 0.03059 1.147 0.02206 1.15 0.00061 2.06 0.00060 2.06 0.00081 1.74 0.01389 1.14

Table I.7: Errors 100 × ‖u − uh(γ)‖0,α with α ≈ 0.48; Data of the convergence study ofthe PDE (I.46) in symmetric triangulations of the 7/4π-section.

l γopt γoN1 0.25828 0.300652 0.26496 0.274483 0.26789 0.272474 0.26928 0.271575 0.26997 0.271116 0.27031 0.270887 0.27048 0.27076

Table I.8: Optimal and one-step Newton correction parameters for a symmetric triangu-lation of the slit-domain.

l \ γ 0.00000 rate γopt rate γoN rate1 4.24550 — 3.27920 — 3.41920 — 0.30000 rate 0.40000 rate2 1.87480 1.18 0.97714 1.75 0.99053 1.79 3.41630 — 4.11090 —3 0.88907 1.08 0.26461 1.88 0.26632 1.90 0.29847 1.82 0.60627 1.364 0.43471 1.03 0.06847 1.95 0.06861 1.96 0.08965 1.74 0.27031 1.175 0.21521 1.01 0.01738 1.98 0.01737 1.98 0.03257 1.46 0.13070 1.056 0.10711 1.01 0.00437 1.99 0.00436 1.99 0.01437 1.18 0.06489 1.017 0.05344 1.00 0.00109 2.00 0.00109 2.00 0.00694 1.05 0.03242 1.00

Table I.9: Errors 100 × ‖u − uh(γ)‖0,α with α ≈ 0.55; Data of the convergence study ofthe PDE (I.46) in a symmetric triangulation of the slit domain.

In the case of a symmetric triangulation we confirmed the theoretical result that an opti-mal convergence rate of 2.0 can be obtained. We also observed that it is not necessary touse the root of the energy-correction function but a sufficient close approximation. Goodresults are obtained by the one-step Newton method, which can be used instead of thefull-iterative Newton approach for the exact root. The convergence rates and the approxi-mation errors in weighted L2-norms are improved in comparison to the standard Galerkinmethod. Thus, the energy-correction method is an alternative to regain better approxi-mation results, when we are in possession or easily able to compute correction parameters.


I.3.2.2. Energy-Correction Method vs. Adaptivity in Symmetric Triangulations

As presented in the above Section I.3.2.1 and Section I.3.1 the energy-correction method isan approach to regain optimal convergence rates. In Section I.2 we saw that the adaptivemethod also achieves this performance. In this section we will study whether a methodcan be preferred in the setting of re-entrant corners. We neglect the computational timeand costs, and only concentrate on the accuracy of the approximated solution by a givennumber of degrees of freedom (DOFs).We use in this section the residual-based adaptive method described in Section I.2 andcompare it with the energy-correction method of this section with optimal parameters.We study again the Poisson problem (I.46) in the four geometries, the 5/4π-sections, theL-shape domain, the 7/4π-section and the slit-domain for symmetric triangulations.

The results of the 4 geometries are presented for the 5/4π-section in Figure I.22, for theL-shape domain in Figure I.23, for the 7/4π-section in Figure I.24 and for the slit-domainin Figure I.25. The energy-correction method and the adaptive algorithm convergenceboth with order h2, but the energy-correction has in all four test cases a higher accuracy.

Note, that we measure the error in another - a weaker - norm as in the above stud-ies. This is reasonable, when we are not interest in an approximation at the re-entrantcorner which is the aim of the energy-correction function and relax the singularity in astronger way. Our theoretical results allow use of these weaker norms.In this context one sees that the energy-correction method has advantages compared tothe adaptive method, when an accurate approximation of the singularity in the re-entrantcorner is not of interest. One reason for this observation is that the error is measured inweighted norms, but the adaptive method preferentially refines the mesh in this neigh-borhood which is relaxed and accurately resolves the singularity.

In our above analysis we neglected the computational cost and time, but when we againconsider the convergence plots in Figure I.22, I.23, I.24 and I.25 each blue dot represents acomputation of the FE-solution. Thus, this has the consequence that many resolving stepsare needed to obtain good approximations by adaptivity. Whereas in the energy-correctionthe resolving costs are limited, but one needs to derive the correction parameters beforethe analysis which can also be expansive and time intensive.


101 102 103 104 105 106

DOFs

10−6

10−5

10−4

10−3

10−2

10−1

100

Figure I.22: Convergence analysis of the energy-correction method with optimal parame-ter (green) and adaptive method (blue) of (I.46) in a symmetric triangulationof the 5/4π-section. Dashed red: h2 reference line; Error measured in theweighted L2-norm with weight α ≈ 1.25.

101 102 103 104 105 106

DOFs

10−6

10−5

10−4

10−3

10−2

10−1

100

Figure I.23: Convergence analysis of the energy-correction method with optimal param-eter (green) and adaptive method (blue) of (I.46) in the L-shape domain.Dashed red: h2 reference line; Error measured in the weighted L2-norm withweight α ≈ 1.39.


101 102 103 104 105 106

DOFs

10−6

10−5

10−4

10−3

10−2

10−1

100

Figure I.24: Convergence analysis of the energy-correction method with optimal parame-ter (green) and adaptive method (blue) of (I.46) in the 7/4π-section. Dashedred: h2 reference line; Error measured in the weighted L2-norm with weightα ≈ 1.47.

101 102 103 104 105 106

DOFs

10−6

10−5

10−4

10−3

10−2

10−1

100

Figure I.25: Convergence analysis of the energy-correction method with optimal parame-ter (green) and adaptive method (blue) of (I.46) in the slit-domain. Dashedred: h2 reference line; Error measured in the weighted L2-norm with weightα ≈ 1.55.


I.3.2.3. Energy-Correction in Non-Symmetric Triangulations

In this subsection we again consider the Poisson problem (I.46) on the four geometries, the5/4π-section, the L-shape domain, the 7/4π-section and the slit-domain. In comparisonto the above sections we triangulate the domains in a non-symmetric way. The meshesof the initial triangulation T1 are depicted in Figure I.9. In Section I.3.1 we saw that itdepends on the interior angle of the re-entrant which correction, i.e., how many correctionparameters - the one parameter (I.37) or the two parameter correction (I.38) - are neededfor optimal convergence results. In general, we can also use the two parameter correctionin the case of an interior angle ≤ 3

2π. Then, the two parameter correction simply reduces

to a one parameter correction. In the two parameter correction (I.38) it remains to choiceproper correction domains for Ωh = ω1

h ∪ ω2h. We will define two sufficient ways - the

two-element-layer method and the one-element-layer method.

a) Method I: Two-Element-Layer

As correction domain for symmetric meshes we used the elements attached at the singu-larity. A natural extension is to include a further element layer to the correction domain,i.e.,

ω1h = T ∈ Th : (0, 0) ∈ ∂T and ω2

h = T ∈ Th : ∂T ∩ ω1h 6= ∅.

For our meshes the correction domains are marked in Figure I.26 for T2, the pink elementsdefine ω1

h and the blue elements ω2h. Note, that we start with the mesh T2, since for T1

the correction domain is the whole domain Ω.

Figure I.26: Marked Elements in the neighborhood of the singularity for meshes T2 ofthe 5/4π-section, L-shape domain, 7/4π-section and slit-domain; ω1

h: Pinkelements; ω2

h: Blue elements.


−0.5 −0.4 −0.3 −0.2 −0.1 0.0 0.1γ1

−0.2

−0.1

0.0

0.1

0.2

0.3

0.4

γ2

-0.120

-0.090

-0.060

-0.030

0.000

0.030

0.060

-0.024

-0.016

-0.008

0.000

0.008

Figure I.27: Contour lines of the energy-correction function (I.43) in the 5/4π-section;First component: Red; Second component: Green

−0.1 0.0 0.1 0.2 0.3 0.4 0.5γ1

−0.5

−0.4

−0.3

−0.2

−0.1

0.0

0.1

γ2

-0.080-0.040

0.000

0.040

0.080

0.120

-0.016

-0.008

0.000

0.008

0.016

0.024

Figure I.28: Contour lines of the energy-correction function (I.43) in the L-shape domain;First component: Red; Second component: Green


In Figure I.27 and Figure I.28 the contour plots of the energy-correction function (I.43)are depicted. The red lines are the contour of the first component of (I.43) and the greenlines of the second one. The x-axis shows the range of γ1 and the y-axis of γ2.

The optimal correction parameters (γ1,opt, γ2,opt), parameters obtained by the one-stepNewton method (I.45) for two parameters (γ1,oN , γ2,oN) and by the one-step Newtonmethod (I.42) for one parameter γoN are shown in Table I.10 for the 5/4π-section and inTable I.11 for the L-shape domain.

l γopt,1 γopt,2 γoN,1 γoN,2 γoN2 0.06315 -0.00185 0.08469 -0.00633 0.071863 0.05635 0.00280 0.07036 -0.00050 0.069304 0.04635 0.00858 0.05704 0.00621 0.068645 0.03309 0.01568 0.04136 0.01384 0.067996 0.01524 0.02479 0.02173 0.02343 0.067467 -0.00914 0.03671 -0.09088 0.03571 0.06705

Table I.10: Optimal energy-correction parameters, correction parameters obtained by theone-step Newton method for two parameter and for one parameter for the5/4π-section for non-symmetric triangulations (two-element-layer).

l γopt,1 γopt,2 γoN,1 γoN,2 γoN2 0.26802 -0.10337 0.34486 -0.11490 0.172343 0.27871 -0.11448 0.29239 -0.11990 0.155144 0.29001 -0.12673 0.29713 -0.12978 0.151045 0.30240 -0.14009 0.30686 -0.14199 0.148306 0.31529 -0.15398 0.31810 -0.15517 0.146527 0.32833 -0.16812 0.33013 -0.16886 0.14539

Table I.11: Optimal energy-correction parameters, correction parameters obtained by theone-step Newton method for two parameter and for one parameter for theL-shape domain in non-symmetric triangulations (two-element-layer).

In the convergence analysis of the 5/4π-section and the L-shaped domain for a non-symmetric triangulation we observe optimal convergence rates for the energy-correctionwith one parameter and two parameters, cf. Table I.12, Table I.13 respectively. Theaccuracy of all energy-correction methods is similar. In our analysis we also study theconvergence results, when setting a static parameter γ2 = −0.5, 0.2 and then obtaining aparameter γ1,oN by the one-step Newton method. Here, the results are also optimal andyield comparable accuracy.


l \ γ (0.0000, 0.0000) rate γoN rate (γopt,1, γopt,2) rate2 1.11840 — 1.05230 — 1.05740 —3 0.28860 1.95 0.26021 2.02 0.26307 2.014 0.07564 1.93 0.06481 2.01 0.06573 2.005 0.02018 1.91 0.01618 2.00 0.01645 2.006 0.00551 1.87 0.00405 2.00 0.00412 2.007 0.00154 1.83 0.00101 2.00 0.00104 1.99

l \ γ (γoN,1, γoN,2) rate (γoN,1,−0.5000) rate (γoN,1, 0.2000) rate2 1.04240 — 2.65560 — 1.66760 —3 0.26006 2.00 0.42638 2.64 0.39433 2.084 0.06503 2.00 0.09875 2.11 0.09088 2.125 0.01628 2.00 0.02377 2.05 0.02153 2.086 0.00408 2.00 0.00576 2.05 0.00520 2.057 0.00102 1.99 0.00140 2.04 0.00127 2.03

Table I.12: Errors 100×‖u− uh(γ1, γ2)‖0,α with α ≈ 0.38; Data of the convergence studyof the PDE (I.46) in the 5/4π-section for non-symmetric triangulations (two-element-layer).



Table I.13: Errors 100×‖u− uh(γ1, γ2)‖0,α with α ≈ 0.33; Data of the convergence studyof the PDE (I.46) in the L-shape domain for non-symmetric triangulations(two-element-layer).


In contrast to the 5/4π-section and the L-shape domain the theory tells us that for the7/4π-section and the slit-domain a one parameter correction does not yield optimal con-vergence rates. This is already observed in the contour lines of the energy-correctionfunction (I.43) in Figure I.29 for the 7/4π-section and in Figure I.30 for the slit domain.

We compute the roots of the energy-correction function (γ1,opt, γ2,opt), the one, two pa-rameters by the one-step Newton method, resp., cf. Table I.14 for the 7/4π-section andTable I.15 for the slit-domain.


Table I.14: Optimal energy-correction parameters (γ1, γ2) for the 7/4π-section for non-symmetric triangulations.


Table I.15: Optimal energy-correction parameters (γ1, γ2) for the slit-domain for non-symmetric triangulations.

The convergence study for the 7/4π-section and for the slit-domain is presented in TableI.16 and in Table I.17, respectively. Optimal rates are only obtained for optimal param-eters and the two parameters obtained by the one-step Newton method shown in TableI.14 and Table I.15, respectively. A correction with one parameter yields a reduced con-vergence rate but an improvement in comparison to the standard Galerkin method. Thestatic choices of γ2 = −0.5, 0.2 only yields optimal rates when γ1 can be derived by theone-step Newton method such that both parameters are close to a root of the energy-correction function (I.43). The accuracy of the methods with optimal rates are similarand by a factor more than 100 more accurate than the standard FE-solution.


γ1

γ2

−0.1 0.0 0.1 0.2 0.3 0.4 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0.0

0.1 -0.050

-0.025

0.0000.025

0.050

0.075

-0.006-0.003

0.000

0.003

0.006

Figure I.29: Contour lines of the energy-correction function (I.43) in the 7/4π-section;First component: Red; Second component: Green

0.1 0.2 0.3 0.4 0.5 0.6 0.7−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

-0.090

-0.060

-0.030

0.000

0.030

0.060

0.090

-0.008-0.004

0.000

0.004

0.008

Figure I.30: Contour lines of the energy-correction function (I.43) in the slit-domain; Firstcomponent: Red; Second component: Green




Table I.16: Errors 100×‖u− uh(γ1, γ2)‖0,α with α ≈ 0.48; Data of the convergence studyof the PDE (I.46) in the 7/4π-section for non-symmetric triangulations (two-element-layer).


l \ γ (γoN,1, γoN,2) rate (γ1,oN ,−0.5000) rate (γ1,oN , 0.2000) rate2 3.50850 — 2.36800 — 1.53020 —3 0.44361 2.98 0.37293 2.67 0.50473 1.604 0.07073 2.65 0.09666 1.95 0.15085 1.745 0.01655 2.10 0.02564 1.91 0.04439 1.766 0.00408 2.02 0.00688 1.90 0.01328 1.747 0.00102 2.01 0.00189 1.86 0.00490 1.70

Table I.17: Errors 100×‖u− uh(γ1, γ2)‖0,α with α ≈ 0.55; Data of the convergence studyof the PDE (I.46) in the slit-domain for non-symmetric triangulations (two-element-layer).


b) Method II: One-element-layer

Next, we consider a different approach to choose the modification domain Ωh. Whenan even number of elements are attached to the singularity, we allocate the one half ofthe elements to ω1

h and the other half to ω2h, i.e.,

ω1h = T ∈ Th : (0, 0) ∈ ∂T and α(x) ≤ ωoL for all x ∈ T,ω2h = T ∈ Th : (0, 0) ∈ ∂T and α(x) ≥ ωoL for all x ∈ T,

where α(x) denotes the angle coordinate of the polar coordinates of x, and the ωoL is theangle such that there is the same number of elements on each side of the angle, cf. FigureI.31.

ω1h

ω2h

ωoL

Figure I.31: Definition of ω1h and ω2

h in the one-element-layer case.

For our four geometries, the 5/4π-section, the L-shape domain, the 7/4π-section and theslit-domain the correction domains are marked in Figure I.32 for the triangulations T1

and T2. The prink elements define ω1h and the blue elements ω2

h. An obvious advantage ofthis correction domain is that in comparison to the two-element-layer correction domainit is compact at the singularity.

First, let us consider the contour lines of the energy-correction function (I.43) (red: firstcomponent, green: second component) of the 5/4π-section in Figure I.33 and of the L-shape domain in Figure I.34. The x-axis shows the range of γ1 and the y-axis of γ2.


Figure I.32: Marked Elements in the neighborhood of the singularity for meshes T1 and T2

of the domains 5/4π-section, L-shape domain, 7/4π-section and slit-domain;ω1h: Pink elements; ω2

h: Blue elements.


−0.2 −0.1 0.0 0.1 0.2 0.3 0.4γ1

−0.3

−0.2

−0.1

0.0

0.1

0.2

0.3

γ2

-0.120

-0.090

-0.060

-0.030

0.0000.030

0.060

-0.045

-0.030

-0.015

0.000

0.015

0.030

Figure I.33: Contour lines of the energy-correction function (I.43) in the 5/4π-section;First component: Red; Second component: Green (one-element-layer).

−0.2 −0.1 0.0 0.1 0.2 0.3 0.4γ1

−0.3

−0.2

−0.1

0.0

0.1

0.2

0.3

γ2

-0.120

-0.080

-0.040

0.000

0.0400.080

0.120

-0.080

-0.060

-0.040

-0.0200.0000.020

0.040

Figure I.34: Contour lines of the energy-correction function (I.43) in the L-shape domain;First component: Red; Second component: Green (one-element-layer).


We derive the parameters for the new chosen correction domain. The optimal valuesand the values obtained by the one-step Newton method are depicted in Table I.18 forthe 5/4π-section and in Table I.19 for the L-shape domain. Note, by setting γ1 = γ2

we achieve a one parameter correction, thus, the values for the one parameter correctionobtained by the one-step Newton method are the same as for the two-element-layer.

l γopt,1 γopt,2 γoN,1 γoN,2 γoN1 0.05335 0.05889 0.06264 0.08676 0.070262 0.05792 0.06328 0.06400 0.05241 0.069653 0.06425 0.05620 0.06885 0.07032 0.069304 0.07063 0.04593 0.07410 0.05671 0.068645 0.07777 0.03243 0.08038 0.04073 0.067996 0.08656 0.01441 0.08972 0.02086 0.067467 0.09783 -0.00993 0.09927 -0.00483 0.06705

Table I.18: Optimal energy-correction parameters (γ1, γ2) for the 5/4π-section in non-symmetric triangulations.

l γopt,1 γopt,2 γoN,1 γoN,2 γoN1 0.10652 0.21656 0.12335 0.27429 0.173622 0.10108 0.23885 0.10984 0.25612 0.159993 0.09557 0.24818 0.10113 0.25900 0.154994 0.08877 0.25749 0.09232 0.26423 0.151045 0.08168 0.26771 0.08394 0.27191 0.148306 0.07469 0.27851 0.07613 0.28114 0.146527 0.06783 0.28961 0.06875 0.29127 0.14539

Table I.19: Optimal energy-correction parameters (γ1, γ2) for the L-shape domain in non-symmetric triangulations.

The convergence analysis is presented in Table I.20 for the 5/4π-section and in Table I.21for the L-shape domain. The energy-correction method for one and two parameter yieldsoptimal rates and similar high accuracy of the solution. Also when we choose a staticparameter γ2 = −0.5, 0.2 the one-step Newton method adjusts γ1 such that the optimalcorrection can be observed. We only observe poor convergence for the standard Galerkinmethod and a less accurate solution.


l \ γ (0.0000, 0.0000) rate γoN rate (γopt,1, γopt,2) rate1 4.42770 — 4.34010 — 4.35010 —2 1.11840 1.99 1.05237 2.04 1.05970 2.043 0.28860 1.95 0.26021 2.02 0.26256 2.014 0.07564 1.93 0.06480 2.01 0.06551 2.005 0.02018 1.91 0.01618 2.00 0.01638 2.006 0.00551 1.87 0.00405 2.00 0.00409 2.007 0.00154 1.83 0.00101 2.00 0.00103 2.00

l \ γ (γoN,1, γoN,2) rate (γoN,1,−0.5000) rate (γoN,1, 0.2000) rate1 4.32660 — 5.08550 — 4.24790 —2 1.05100 2.04 1.24510 2.03 1.03010 2.043 0.26017 2.01 0.29941 2.06 0.25638 2.014 0.06490 2.00 0.07286 2.04 0.06416 2.005 0.01622 2.00 0.01790 2.02 0.01606 2.006 0.00406 2.00 0.00442 2.02 0.00402 2.007 0.00102 2.00 0.00110 2.01 0.00100 2.00

Table I.20: Errors 100×‖u− uh(γ1, γ2)‖0,α with α ≈ 0.55; Data of the convergence studyof the PDE (I.46) in the 5/4π-section for non-symmetric triangulations (one-element-layer).


l \ γ (γoN,1, γoN,2) rate (γ1,oN ,−0.5000) rate (γ1,oN , 0.2000) rate1 3.24830 — 5.06600 — 3.30860 —2 0.77333 2.07 1.35610 1.90 0.79557 2.063 0.18718 2.05 0.35089 1.97 0.19323 2.044 0.04599 2.03 0.08973 1.97 0.04754 2.025 0.01140 2.01 0.02296 1.97 0.01181 2.016 0.00284 2.00 0.00590 1.96 0.00295 2.007 0.00071 2.00 0.00151 1.96 0.00074 1.99

Table I.21: Errors 100×‖u− uh(γ1, γ2)‖0,α with α ≈ 0.33; Data of the convergence studyof the PDE (I.46) in the L-shape domain for non-symmetric triangulations(one-element-layer).


Again, for 7/4π-section and slit-domain we expect a different property as for the first twogeometries. First, we consider the contour lines of both domains. The red lines representthe first component and the green lines the second component of the energy-correctionfunction (I.43) in Figure I.35 for the 7/4π-section and in Figure I.36 for the slit-domain.

We provide the optimal correction parameters and the values obtained by the one-stepNewton method for one and two parameters in Table I.22 for the 7/4π-section and inTable I.23 for the slit-domain.


Table I.22: Optimal energy-correction parameters (γ1, γ2) for the 7/4π-section in non-symmetric triangulations.


Table I.23: Optimal energy-correction parameters (γ1, γ2) for the slit-domain in non-symmetric triangulations.

The convergence analysis, cf. Table I.24 for the 7/4π-section and Table I.25 for the slit-domain, confirms the statements of the theory. The optimal convergence rates can beachieved, when we correct with two parameters - the optimal ones or the ones obtainedby the one-step Newton method. Thus, the parameters of the one-step Newton approachare good approximations of the exact roots of the energy-correction function (I.43). Theone parameter correction and two parameter correction with static γ2 (= −0.5, 0.2) and γ1

obtained by the one-step Newton method improve the convergence rate but is not optimal.High accuracy of the FE-solution is only derived for the two parameter correction and bya factor of more than 100 more accurate than the standard Galerkin solution.


−0.1 0.0 0.1 0.2 0.3 0.4γ1

0.0

0.1

0.2

0.3

0.4

0.5

γ2

-0.150

-0.100-0.050

0.000

0.050

0.100

0.150-0.075

-0.050

-0.025

0.000

0.025

0.050

Figure I.35: Contour lines of the energy-correction function (I.43) in the 7/4π-section;First component: Red; Second component: Green.

−0.1 0.0 0.1 0.2 0.3 0.4γ1

0.0

0.1

0.2

0.3

0.4

0.5

γ2

-0.120

-0.0600.000

0.060

0.120

0.180

0.240-0.120

-0.080

-0.040

0.000

0.040

0.080

Figure I.36: Contour lines of the energy-correction function (I.43) in the slit-domain; Firstcomponent: Red; Second component: Green.



l \ γ (γoN,1, γoN,2) rate (γoN,1,−0.5000) rate (γoN,1, 0.2000) rate1 2.88660 — 6.56940 — 3.19560 —2 0.81987 1.82 1.93690 1.76 0.88595 1.853 0.21992 1.90 0.57183 1.76 0.23425 1.924 0.05683 1.95 0.16807 1.77 0.06140 1.935 0.01451 1.97 0.04955 1.76 0.01634 1.916 0.00369 1.98 0.01473 1.75 0.00445 1.887 0.00093 1.98 0.00441 1.74 0.00124 1.84

Table I.24: Errors 100×‖u− uh(γ1, γ2)‖0,α with α ≈ 0.48; Data of the convergence studyof the PDE (I.46) in the 7/4π-section for non-symmetric triangulations.


l \ γ (γoN,1, γoN,2) rate (γ1,oN ,−0.5000) rate (γ1,oN , 0.2000) rate1 2.68830 — 9.13530 — 3.96700 —2 0.75040 1.84 2.84850 1.68 1.27130 1.643 0.19987 1.91 0.92405 1.62 0.39165 1.704 0.05171 1.95 0.30853 1.58 0.11948 1.705 0.01322 1.97 0.10394 1.57 0.03725 1.716 0.00337 1.97 0.03543 1.55 0.01196 1.647 0.00086 1.98 0.01220 1.54 0.00394 1.60

Table I.25: Errors 100×‖u− uh(γ1, γ2)‖0,α with α ≈ 0.55; Data of the convergence studyof the PDE (I.46) in the slit-domain for a non-symmetric triangulation


Summarized, the stated theory in Subsection I.3.1 for non-symmetric meshes for linearFE methods is confirmed in the numerical examples. For an interior angle ≤ 3

2π the

one parameter correction is sufficient but a two parameter correction does not harm theapproximation. Moreover, in this case an arbitrary second parameter can be chosen, sincethe first parameter is adjusted in such a way that optimal rates are obtained. For domainswith an interior angle > 3

2π a two parameter correction is necessary to regain optimal

convergence rates and a high accuracy of the FE-solution. Further, we identified thatthe values obtained by a one-step Newton method instead of the calculation of the exactroot of the energy-correction function are adequate. The results, which we obtained forthe two different correction domains - the one-element-layer and the two-element-layer,yield the same convergence property. An advantage of the one-element-layer is that thecorrection domain is compactly placed at the singularity.

I.3.2.4. Energy-Correction Method vs. Adaptivity in Non-SymmetricTriangulations

In this section we compare the approximation error of the energy-correction method ofSection I.3.2.1 and the adaptive method of Section I.2 in non-symmetric triangulations.The adaptive method can be applied to symmetric and non-symmetric meshes withoutany difference in the algorithm of Section I.2. Whereas in the energy-correction methodthe symmetry, non-symmetry of the mesh causes certain problems, respectively.In the following we consider the Poisson equation (I.46) in the four geometries, the5/4π-section, the L-shape domain, the 7/4π-section and the slit-domain, with the non-symmetric triangulation of the above section. We compare the approximation error inthese domains in weighted L2-norms but neglect the computational time and cost of thetwo methods. As adaptive algorithm we choose the residual-based one of Section I.2,for the energy-correction method with interior angle ≤ 3

2π the energy correction method

with one parameter obtained by the one-step Newton method and for greater angles theenergy-correction method with two parameter, also obtained by the one-step Newtonmethod (one-element-layer).


In Figure I.37 we see the energy-correction method (green), the adaptive method (blue)and dashed red reference line for the 5/4π-section. For both methods the convergence isoptimal and the accuracy almost identical.

101 102 103 104 105 106

DOFs

10−6

10−5

10−4

10−3

10−2

10−1

100

Figure I.37: Approximation errors of the PDE (I.46) derived by the energy-correctionmethods and adaptivity in the 5/4π-section in the weighted L2-norm withα ≈ 2.25; Green: Energy-correction; Blue: Adaptivity.

In the other three cases we observe different properties for the L-shape domain in FigureI.38, the 7/4π-section in Figure I.39 and the slit-domain in Figure I.40. The energy-correction method produces a more accurate solution than the adaptive approach for afixed number of DOFs. Nevertheless, both methods converge of order h2.

101 102 103 104 105 106

DOFs

10−6

10−5

10−4

10−3

10−2

10−1

100

Figure I.38: Approximation errors of the PDE (I.46) derived by the energy-correctionmethods and adaptivity in the L-shape domain in the weighted L2-normwith α ≈ 2.38; Green: Energy-correction; Blue: Adaptivity.


101 102 103 104 105 106

DOFs

10−6

10−5

10−4

10−3

10−2

10−1

100


101 102 103 104 105 106

DOFs

10−6

10−5

10−4

10−3

10−2

10−1

100

Figure I.40: Approximation errors of the PDE (I.46) derived by the energy-correctionmethods and adaptivity in the slit domain in the weighted L2-norm withα ≈ 1.55; Green: Energy-correction; Blue: Adaptivity.

We saw that the energy-correction method yields FE-solution in domains with re-entrantcorners which are more accurate than (or as good as) the solutions obtained by the adap-tive method for a fixed number of DOFs when neglecting computational cost and time.One reason for this is that the adaptive approach refines the mesh at the re-entrant cornerwhich we neglect by the weighted norm. Note, that we measured the error in a weakernorm than before, compare Section I.3.2.2.

When we reconsider Figures I.37, I.38, I.39 and I.40, we see that the adaptive methodneeds to resolve the FE-solution many times and thus, a lot of steps are necessary to geta good approximation. Whereas the energy-correction needs for the calculation of the


FE-solution correction parameters which are obtained in a pre-processing.

In this section we obtained optimal convergence rates by the energy-correction method forsymmetric and non-symmetric triangulations in case of re-entrant corners. For symmetricmeshes a one parameter correction is sufficient, whereas for the non-symmetric triangu-lations with an interior angle ≥ 3

2π we need a two parameter correction. In both cases

the parameters can be sufficiently calculated by a one-step Newton method. Further,the comparison with an adaptive residual-based method yields a recommendation of theenergy-correction method, when we measure the error in weighted norms.

I.4. Energy-Correction Method for Second Order FiniteElements

In the last Section I.3 we considered the linear finite element method in the case ofre-entrant corners and saw that optimal convergence rates can be achieve by the energy-correction method. In this section we will study the energy correction method for secondorder finite element methods. The standard idea is similar to one for the linear finiteelements worked out in Section I.3 and [19]. Again, we modify the bilinear form in theneighborhood of the re-entrant corner such that the pollution effect is avoided and optimalconvergence rates of order h3 in weighted L2-norms are obtained.In the first part of this section we consider the theory of the energy-correction method forsecond order FE and obtain optimal convergence results for weighted L2 and H1-norms.In the proofs we apply several times the Aubin-Nitsche trick for the deviation of the apriori results. Further, we verify our theoretical results by numerical examples in our fourtest scenarios for re-entrant corners for symmetric triangulations. Finally, we compare theenergy-correction method with the adaptive method of the Section I.2 by concentratingon the accuracy of the methods in weighted norms. A usage of the energy-correctionapproach can be recommended in the weighted norm framework.

I.4.1. Theory of the Energy-Correction Method

In this section we develop the theory of the energy-correction for an a priori approxima-tion result for second order FEM. We proceed similar to the linear case in Egger, Rude,Wohlmuth [19] and carry the analysis for linear case to the quadratic setting.

Again, we consider the Poisson problem with homogeneous Dirichlet boundary condi-tions (I.1) on a bounded domain with one re-entrant corner located at the origin as inSection I.3. Similar to Lemma I.3, the standard Galerkin ansatz for quadratic FE yieldsa reduced convergence rate. Analogously, to the linear finite element case of Section I.3we want to correct the bilinear form in the neighborhood of the re-entrant corner. There-fore, we consider the modification ah(·, ·) defined in (I.27) with the properties (C1), (C2),(C3) as in Section I.3 and the modified weak formulation (I.26) but with the quadraticansatz and test space V 2

h . Additionally, we restrict ourselves for a part of our results tosymmetric meshes at the singularity (S) .

First, we prove some approximation and interpolation results to be able to derive ourmain result. The next lemma considers an interpolation result in weighted-norms. There-fore, we introduce the standard nodal interpolation operator for the linear and quadraticfinite element spaces V 1

h and V 2h

I ih : C(Ω)→ V ih (I.48)

for i = 1, 2, cf. [10, Rem. 5.6].

62 Energy-Correction Method for Second Order Finite Elements

Lemma I.7. (Interpolation in weighted-norms). Let I ih : C(Ω) → V ih be the inter-

polation operator of (I.48) for i = 1, 2. Then, for u ∈ H1+iα (Ω) with α < i it holds

‖u− Ihu‖0,β . h1+i+β−α‖u‖1+i,α, α− (i+ 1) ≤ β ≤ α,

‖∇(u− Ihu)‖0,β . hi+β−α‖u‖1+i,α, α− i ≤ β ≤ α.(I.49)

Proof. The case of i = 1 has been worked out in [19, Lem. A.1 ]. The proof is analogousto the one below for i = 2.

First, we show that the interpolation operator is well-defined. By the continuous em-bedding properties of Lemma 2.1, we have

H3α(Ω) → H2

α−1(Ω) → C(Ω) (I.50)

for α < 2.

Let us consider i = 2. For (I.49) we proceed in an element-by-element fashion: Fortriangles T not attached to the singular vertex we have r ≈ cT on T for some constantcT > 0 and for k ∈ 0, 1

‖∇k(u− Ihu)‖0,β;T . cβT‖∇k(u− Ihu)‖L2(T ) . cβTh

3−k‖∇3u‖L2(T )

. cβ−αT h3−k‖∇3u‖0,α;T . h3−k+β−α‖∇3u‖0,α;T ,(I.51)

where we used the definition of the weighted-norm, standard interpolation property andcT & h for α ≥ β.If the singularity is a vertex of T , we get

‖∇k(u− Ihu)‖0,β;T . hβ−k‖rβ∇k(u− Ihu)‖L2(T )

. hβ−k‖rα∇3u‖L2(T ) . h3−k+β−α‖rα∇3u‖L2(T )

. h3−k+β−α‖u‖3,α;T ,

(I.52)

where we used the transformation to the reference element T , embedding properties ofLemma 2.1, i.e., H3

α(Ω) → Hkβ(Ω) for α − (i + 1 − k) ≤ β, the Braemble-Hilbert lemma

and transformation backwards to T . By summing over all elements we derive the claims.

Next, let us consider sharp interpolation error error estimates for singular functions.

Lemma I.8. The following interpolation bounds hold for i = 1, 2

‖sn − I ihsn‖0 = O(h1+nπ/ω) and ‖sn − I ihsn‖1 = O(hnπ/ω) (I.53)

Proof. Let us again proceed in an element-wise fashion: For elements attached to thesingularity we have

‖sn − I ihsn‖0,T . h1+i−nπ/ω−ε‖s1‖1+i,nπ/ω−ε,T . h1+nπ/ω


by Lemma I.7 and explicit integration similar to Lemma I.2. For elements not attachedto the singularity we simply have

‖sn − I ihsn‖0,T . hi+1‖sn‖i+1,T ,

since sn is smooth. By summing other all elements we obtain the stated L2 result. Theprocedure for H1-bound is the same.

In some of our results we need an inverse estimate in weighted-norms, which is presentedin [19, Lem. A.3.].

Lemma I.9. For any v ∈ V kh with k ∈ N and α > −1 there holds

‖vh‖1,α . h−1‖vh‖0,α (I.54)

Further, we need an auxiliary result for an estimation of the standard Galerkin approx-imation in weighted-norms. We obtain the next Lemma by extending [9, Lem. 2.1] tosecond order finite elements.

Lemma I.10. Define ρ = (r2 + θh2)1/2 for some sufficiently large but fixed θ > 0. Thenfor any exponent 1− π/ω < α′ < 1 there holds

‖ρ−α′∇(u− uh)‖0 . ‖ρ−α′∇(u− I2

hu)‖0 + h−1‖ρ−α′(u− I2

hu)‖0 (I.55)

and

‖ρ−α′(u− uh)‖0 . h‖ρ−α′∇(u− I2

hu)‖0 + ‖ρ−α′(u− I2

hu)‖0. (I.56)

Proof. The proof is similar to Lemma [9, Lem. 2.1] by setting γ = 1 − α′, η = 1 andreplacing the linear Ritz projection and linear interpolation operator by their quadraticcounterparts.

For the proof we will use interpolation error estimates in the ρ-norm which are iden-tical to the bounds of Lemma I.7 just using ρ instead of weight r.For simplifying our notation set e ≡ u− uh and let us start with the L2-bound. Considerthe solution v ∈ H1

0 (Ω) of the dual problem

−∆v = ρ−2α′I2he in Ω, v = 0 on ∂Ω.

Note, that the right hand side of the dual problem is in L2(Ω) due to the special form ofρ. Thus, by a regularity estimate [9, Lem. 2.2] it holds

2∑k=1

‖rk+α′∇kv‖0 . ‖ρ−α′I2he‖0. (I.57)


We start to evolve the L2-bound by the definition of the dual problem, Green’s formula,uh ∈ V 2

h , Galerkin orthogonality and Cauchy Schwarz inequality

‖ρ−α′e‖2

0 = (ρ−α′e, ρ−α

′(u− I2

hu)) + (ρ−α′e, ρ−α

′(I2hu− uh))

= (ρ−α′e, ρ−α

′(u− I2

hu)) + (e, ρ−2α′I2he)

= (ρ−α′e, ρ−α

′(u− I2

hu)) + (e,−∆v)

= (ρ−α′e, ρ−α

′(u− I2

hu)) + (∇e,∇(v − I1hv))

≤ ‖ρ−α′e‖0‖ρ−α(u− I2

hu)‖0 + ‖ρ−α′∇e‖0‖ρα′(v − I1

hv)‖0

(I.58)

By a similar consideration as given in Lemma I.7 we obtain

‖ρα′(v − I1

hv)‖0 . h‖rα′∇2v‖0

for the interpolation error of the dual problem and, thus, with (I.57) we derive

‖ρα′(v − I1

hv)‖0 . h‖ρ−α′I2he‖0. (I.59)

Finally, we conclude with a provisional L2-estimate by using (I.58), (I.59), ‖ρ−α′I2he‖0 .

‖ρ−α′e‖0 and dividing by ‖ρ−α′

e‖0

‖ρ−α′e‖0 . ‖ρ−α

′(u− I2

hu)‖0 + h‖ρ−α′∇e‖0 (I.60)

Now, we continue with the H1-estimate. We use the definition of e, I2he, Galerkin orthog-

onality for I2h

(ρ−2α′

I2he)∈ V 2

h and Cauchy-Schwarz inequality

‖ρ−α′∇e‖20 = (ρ−α

′∇e, ρ−α′∇(e− I2he)) + (∇e, ρ−2α′∇(I2

he))

= (ρ−α′∇e, ρ−α′∇(e− I2

he)) + (∇e,∇(ρ−2α′I2he))− (∇e,∇(ρ−2α′

)I2he)

= (ρ−α′∇e, ρ−α′∇(e− I2

he))

+ (∇e,∇(ρ−2α′

I2he− I2

h

(ρ−2α′

I2he))

)− (∇e,∇(ρ−2α′)I2he)

Consequently, we have by Cauchy-Schwarz inequality and dividing by ‖ρ−α′∇e‖0

‖ρ−α′∇e‖0 . ‖ρ−α′∇(u−I2

hu)‖0 +‖ρα′∇(ρ−2α′

I2he−I2

h

(ρ−2α′

e))‖0 +‖ρ−α′−1I2

he‖0. (I.61)

The interpolation bound for the ρ-norm and the product rule yield

‖ρα′∇(ρ−2α′I2he−I2

h(ρ−2α′e))‖0 . h2‖ρα′∇3(ρ−2α′

I2he)‖0

. h2(‖ρ−α′−3I2he‖0 + ‖ρ−α′−2∇(I2

he)‖0 + ‖ρ−α′−1∇2(I2he)‖0)

(I.62)

Then, by combining (I.61) and (I.62), using h . ρ, the inverse inequality (I.9) (the statedLemma also holds for the ρ-norm), ‖ρ−α′

I2he‖0 . ‖ρ−α′

e‖0 and the provisional L2-bound


(I.60), we obtain the H1-bound

‖ρ−α′∇e‖0 . ‖ρ−α′∇(u− I2

hu)‖0

+ h2(‖ρ−α′−3I2he‖0 + ‖ρ−α′−2∇(I2

he)‖0 + ‖ρ−α′−1∇2(I2he)‖0)

+ ‖ρ−α′−1I2he‖0

. ‖ρ−α′∇(u− I2hu)‖0 + ‖ρ−α′−1I2

he‖0

. ‖ρ−α′∇(u− I2hu)‖0 + h−1‖ρ−α′

e‖0

. ‖ρ−α′∇(u− I2hu)‖0 + h−1(‖ρ−α′

(u− I2hu)‖0 + h‖ρ−α′∇e‖0)

. ‖ρ−α′∇(u− I2hu)‖0 + h−1‖ρ−α′

(u− I2hu)‖0

Substituting in the H1 estimate in (I.60) we derive the stated L2 bound.

Let us next prove a generalized a priori estimate for the modified Galerkin approximation.

Lemma I.11. Let (C1)- (C3) hold, and assume that u ∈ H3α(Ω) ∩ H1(Ω) for some

0 ≤ α < 2. Then for all 1− π/ω < β < 1

‖∇(u− umh )‖0 . h2−α‖u‖3,α and ‖u− umh ‖0 . h2−α−β‖u‖3,α. (I.63)

Proof. The modified bilinear form is continuous and H1(Ω) - coercive by (C1) and (C2),thus, we employ the result of Berger, Scott and Strang [10, Lem. III.1.2] and derive

‖u− umh ‖1 . infv∈V 2

h

‖u− v‖1 + supw∈V 2

h

|ah(u,w)− f(w)|‖w‖1

. ‖u− I2hu‖1 + ‖∇u‖0,Ωh ,

(I.64)

where we used the definition of ah(·, ·) and (C2) for the second estimate. Then, the firstterm is treated by the interpolation result of Lemma I.7 and the second term ‖∇u‖0,Ωh isexpanded by Lemma I.1. Further, we use to bound ‖∇sn‖0 explicit integration.

For the L2 estimate let us consider the dual problem

−∆z = (u− umh ) in Ω and z = 0 on ∂Ω.

We use the definition of the dual problem, Green’s formula, Galerkin orthogonality ofLemma I.5 and obtain

‖u− umh ‖20 = (u− umh , u− umh ) = (u− umh ,−∆z) = a(u− umh , z)

= a(u− umh , z − I1hz)− ch(umh , I1

hz)

= ‖∇(u− umh )‖0‖∇(z − Ihz)‖0 + ‖∇umh ‖0‖∇I1hz‖0

(I.65)


We consider both terms separately. By the interpolation results of Lemma I.7 we derive

‖∇(u− umh )‖0‖∇(z − Ihz)‖0 . h2−α‖u‖3,αh1−β‖z‖2,β

. h3−α−β‖u‖3,α‖u− umh ‖0.(I.66)

for the first term. For the second term we have by triangle inequality and a priori boundof Lemma I.11

‖∇umh ‖0,Ωh . hα(‖∇(umh − u)‖0,−α,Ωh + ‖∇u‖0,−α,Ωh)) . h2−α‖u‖3,−α (I.67)

and‖∇I1

hz‖0,Ωh . ‖∇(I1hz − z)‖0,Ωh + ‖∇z‖0,Ωh

. h1−β‖z‖2,β . h1−β‖u− umh ‖0.(I.68)

Combining the results of (I.66), (I.67) and (I.68) yield the L2 result.

Remark I.9. In Lemma I.11 we derived a priori estimates for the L2 and H1-norms.These bounds also hold in the case of weighted-norms for 2 − π/ω < α < 2 in the sensethat the full rates are obtained, i.e. h2 in the weighted L2-norm and h3 for the weightedH1-norm.

In the following Lemma we derive an optimal estimate for regular Galerkin approximationin weighted-norms.

Lemma I.12. Let U ∈ H3−α(Ω) ∩ H1(Ω) for α = α′ + 1 some 1− π/ω < α′ < 1. Then

‖∇(U − Uh)‖0,−α′ . h2‖U‖3,−α and ‖U − Uh‖0,−α . h3‖U‖3,−α′ . (I.69)

Proof. Let us start with the H1 estimate. An essential part of this proof is Lemma I.10.We will estimate in such a way that we can employ it. Let us proceed in an element-by-element based way. First, consider the elements T not attached to the singularity, wherewe have

‖∇(U − Uh)‖0,−α′,T . ‖ρ−α′∇(U − Uh)‖0

. ‖ρ−α′∇(U − I2hU)‖0 + h−1‖ρ−α′

(U − Uh)‖0

. ‖r−α′∇(U − I2hU)‖0 + h−1‖r−α′

(U − Uh)‖0

. h2‖U‖3,−α′ . h2‖U‖3,−α.

by r−α′ ≈ ρ−α

′for these kinds of elements, Lemma I.10 and interpolation results Lemma

I.7. Now, let T have a vertex that is the singular point. By applying the triangle inequalitywe get

‖∇(U − Uh)‖0,−α′,T . ‖∇(U − I2hU)‖0,−α′,T + ‖∇(I2

hU − Uh)‖0,−α′,T (I.70)


For the second term we use Holder’s inequality, exact integration of ‖r−α′‖0,T (α′ < 1),Markov’s inequality [29, Thm. 1], ρ . h and derive

‖∇(I2hU − Uh)‖0,−α′,T = ‖r−α′∇(I2

hU − Uh)‖0,T

. ‖∇(I2hU − Uh)‖L∞(T )‖r−α

′‖0,T . h1−α′‖∇(I2hU − Uh)‖L∞(T )

. h−1h1−α′‖∇(I2hU − Uh)‖0,T = ‖h−α′∇(I2

hU − Uh)‖0,T

. ‖ρ−α′(I2hU − Uh)‖0,T

Then, by triangle inequality we have

‖ρ−α′∇(I2hU − Uh)‖0,T . ‖ρ−α′

(I2hU − U)‖0,T + ‖ρ−α′∇(U − Uh)‖0,T . (I.71)

By using the results of (I.70), (I.71), Lemma I.10, r . ρ and interpolation results I.7, weobtain

‖∇(U − Uh)‖0,−α′,T . ‖∇(U − I2hU)‖0,−α′,T + ‖ρ−α′∇(U − Uh)‖0,T

. ‖∇(U − I2hU)‖0,−α′,T + ‖ρ−α′∇(U − I2

hU)‖0,T + h−1‖ρ−α′(U − Uh)‖0,T

. ‖∇(U − I2hU)‖0,−α′,T + ‖r−α′∇(U − I2

hU)‖0,T + h−1‖r−α′(U − Uh)‖0,T

. h2‖U‖3,−α′,T . h2‖U‖3,−α,T .

By summing over all elements T , we obtain the H1 result.

Next, we consider the L2-bound. We define the dual problem

−∆z = r−2α′(U − Uh) in Ω and z = 0 on ∂Ω.

The right hand side of the dual problem r−2α′(U −Uh) is in H0

α′(Ω), since U ∈ H0−α′(Ω) ⊂

H3−α′(Ω). Thus, we have by Lemma I.1

‖z‖2,α′ . ‖r−2α′(U − Uh)‖0,α′ = ‖U − Uh‖0,−α′ .

This yields

‖U − Uh‖20,−α′ = (U − Uh, r−2α′

(U − Uh)) = (U − Uh,−∆z)

= a(U − Uh, z) = a(U − Uh, z − I1hz)

≤ ‖∇(U − Uh)‖0,−α′‖∇(z − I1hz)‖0,α′

. h2‖U‖3,−α′h‖z‖2,α′ . h3‖U‖3,−α‖U − Uh‖0,−α′

where we applied the definition of the dual problem, Green’s formula, standard Galerkinorthogonality, the H1 result and interpolation Lemma I.7. By dividing by ‖U − Uh‖0,−α′

we determine the L2 estimate.

The next result considers the smooth remainder of the expansion of the Poisson equation(I.1) in negative weighted norms and yield optimal convergence. This is the analogousresult for the modified Galerkin approximation as we have already seen in Lemma I.12for the standard one.


Lemma I.13. Let (C1)- (C3) hold and assume that U ∈ H3−α(Ω)∩ H1(Ω) for α = α′+1

for some 1− π/ω < α′ < 1. Then

‖U − Umh ‖0,α′ . h3‖U‖3,−α and ‖∇(U − Um

h )‖0,−α′ . h2‖U‖3,−α. (I.72)

Proof. Let us first proof the L2 result. We consider the solution of the dual problem

−∆z = r−2α′(U − Um

h ) in Ω and z = 0 on ∂Ω, (I.73)

and denote by zmh ∈ V 1h its modified Galerkin approximation defined by

ah(zmh , v) = (r−2α′

(U − Umh ), v)0 = (−∆z, v)0 = a(z, v) for all v ∈ V 1

h ,

where we used Green’s formula. Note, that the right hand side of the dual problemr−α

′(U − Um

h ) is in H0α′(Ω), since U ∈ H0

α′(Ω) ⊂ H3−α(Ω). Thus, the solution of the dual

problem admits the regularity estimate of Lemma I.1

‖z‖2,α′ . ‖r−2α′(U − Um

h )‖0,α′ = ‖U − Um

h ‖0,−α′ .

Using Green’s formula, the (modified) Galerkin orthogonality for zmh , umh , Uh, triangle andCauchy-Schwarz inequality, we derive

‖U − Umh ‖2

0,−α′ = (U − Umh , r

−2α′(U − Um

h ))0

= (U − Umh ,−∆z)0 = a(U − Um

h , z)

= a(U − Umh , z − zmh )− ch(Um

h , zmh )

= a(U − Uh, z − zmh )− ch(Uh, zmh )

= a(U − Uh, z)− ch(Uh, zmh )

. ‖U − Uh‖0,−α′‖∆z‖0,α′ + ‖∇Uh‖0,Ωh‖∇zmh ‖0,Ωh .

Now, we treat each term separately. The first term can be estimated in the sense ofLemma I.12 such that

‖U − Uh‖0,−α′ . h3‖U‖3,−α

and by the definition of the dual problem we have

‖∆z‖0,α′ = ‖U − Umh ‖0,−α′ .

For the third term we use that r . h in Ωh, the triangle inequality, and Lemma I.12

‖∇Uh‖0,Ωh ≤ ‖Uh − U‖1,Ωh + ‖U‖1,Ω

≤ hα′(‖Uh − U‖1,−α′,Ωh + ‖U‖1,−α′,Ωh)

. h2+α′‖U‖3,−α.

We estimate the fourth term by the triangle inequality, a priori result in weighted norms[19, Lem. 3.2], which is similar to Lemma I.11, and the estimate of the dual problem

‖∇zmh ‖0,Ωh ≤ ‖z − zmh ‖1,Ωh + ‖∇z‖0,Ωh

. h1−α′‖z‖2,α′ . h1−α′‖U − Umh ‖0,−α′ .


Taking all estimates given above together and by dividing by ‖U − Umh ‖0,−α′ we obtain

the L2 result.

It remains to show the H1 estimate:

‖∇(U − Umh )‖0,−α′ ≤ ‖∇(U − I2

hU‖0,−α′ + ‖∇(I2hU − Um

h )‖0,−α′

. h2‖U‖3,−α + h−1‖I2hU − Um

h ‖0,−α′

. h2‖U‖3,−α + h−1(‖I2hU − U‖0,−α′ + ‖U − Um

h ‖0,−α′)

. h2‖U‖3,−α.

We used the interpolation estimates of Lemma I.7, the inverse inequality of Lemma I.9and the L2 estimate.

The following lemma contains our main result of this section with slightly stronger as-sumption as necessary for the ease of the proof. Later we will weaken these assumption.

Lemma I.14. Assume that ch(·, ·) satisfies (C1)-(C3) and, in addition, let

a(si − smi,h, sj − smj,h)− ch(smi,h, smj,h) = O(h3) (I.74)

hold for 1 ≤ i ≤ 3, 1 ≤ j ≤ 2. Then, for any f ∈ H1−α(Ω) with α = α′ + 1 for some

1− π/ω < α′ < 1, we have

‖u− umh ‖0,α . h3‖f‖1,−α and ‖∇(u− umh )‖0,α . h3‖f‖1,−α.

Proof. Let us begin with the proof of the L2 estimate. We consider the dual problem

−∆v = r2α(u− umh ) in Ω and v = 0 on ∂Ω.

The right hand side r2α(u−umh ) is in H0−α(Ω), since by Lemma (I.1) u ∈ H3

α(Ω) ⊂ H0−α(Ω).

Thus, by Lemma I.1 we can expand the solution of the original Poisson problem u andthe solution v of the dual problem such that

u =3∑i=1

λisi + U and v =2∑j=1

µj sj + V (I.75)

with U ∈ H3−α(Ω), V ∈ H2

−α(Ω). Further, the following estimates hold

3∑i=1

|λi|+ ‖U‖3,−α . ‖f‖1,−α,

2∑j=1

|µj|+ ‖V ‖2,−α . ‖u− umh ‖0,α.

(I.76)


Then, we treat the L2 error by Green’s formula, modified Galerkin orthogonality, and theexpansions of u, v, thus,

‖u− umh ‖20,α = (u− umh , r2α(u− umh ))0 = (u− umh ,−∆v)0

= a(u− umh , v) = a(u− umh , v − vmh )− ch(umh , vmh )

=3∑i=1

2∑j=1

λiµj(a(si − smi,h, sj − smj,h)− ch(smi,h, smj,h)

)+

2∑j=1

µj(a(U − Um

h , sj − smj,h)− ch(Umh , s

mj,h))

+3∑j=1

λi(a(si − smi,h, V − V m

h )− ch(smi,h, V mh ))

+ a(U − Umh , V − V m

h )− ch(Umh , V

mh ).

Now, we consider each term individually. We can estimate the first term by the assumptionand by the estimates of the expansions, i.e.

λiµj(a(si − smi,h, sj − smj,h)− ch(smi,h, smj,h)

). ‖f‖1,−α‖u− umh ‖0,αh

3.

For the second term we apply the modified Galerkin orthogonality Lemma I.5, Green’sformula, Cauchy Schwarz inequality, the fact that ‖∆sj‖0,α′ is bounded by a domainconstant and the estimates of Lemma I.13 as well as the regularity estimates

µi(a(U − Um

h , sj − smj,h)− ch(Umh , s

mj,h))

= µja(U − Umh , sj) = µj(U − Um

h ,−∆sj)

≤ |µi| ‖U − Umh ‖0,−α′‖∆sj‖0,α′

. ‖u− umh ‖0,αh3‖U‖3,−α

. ‖u− umh ‖0,αh3‖f‖−1,α

We treat the third term by the modified Galerkin orthogonality Lemma I.5, Green’sformula, Cauchy-Schwarz inequality, the a priori result of Lemma I.11 and Remark I.9and the estimates of the expansion

λi(a(si − smi,h, V − V m

h )− ch(smi,h, V mh ))

= λia(si − smi,h, V ) = λi(si − smi,h,−∆V )0

≤ |λi| ‖si − smi,h‖0,α‖∆V ‖0,−α

. |λi| h3‖si‖3,α‖V ‖2,−α

. ‖f‖1,−α h3‖u− umh ‖0,α

For the fourth term we use the Galerkin orthogonality, Green’s formula, Cauchy-Schwarzinequality, the a priori result of Lemma I.11 and Remark I.9 and the estimates of the


expansion

a(U − Umh , V − V m

h )− ch(Umh , V

mh )

= a(U − Umh , V ) = (U − Um

h ,−∆V )0

≤ ‖U − Umh ‖0,α′‖∆V ‖0,−α′

≤ h3‖U‖3,α‖V ‖2,−α

. h3‖f‖1,−α‖u− umh ‖0,α

Finally, we divide by ‖u− umh ‖0,−α and obtain the L2 statement.

Next, we consider the H1 estimate:

‖∇(u− umh ‖0,α ≤ ‖∇(u− I2hu)‖0,α + ‖∇(I2

hu− umh )‖0,α

. h2‖u‖3,α + h−1‖I2hu− umh ‖0,α

≤ h2‖u‖3,αh−1(‖Ihu− u‖0,α + ‖u− umh ‖0,α)

. h2‖u‖3,α . h2‖f‖1,−α

The H1 bound is given by the interpolation estimate of Lemma I.7, the inverse inequalityof Lemma I.9, the above L2 estimate and the bound of the expansion. This concludes theproof.

Our next goal is to derive some relaxations of Lemma I.14. First, we weaken the conditionon the combination of the second singularity functions in the case of small interior anglesand apply a similar argumentation as in [19, Lem. 4.2.].

Lemma I.15. Let (C1)- (C3) hold and let the interior angle be ω ≤ 43π. Then

a(s2 − sm2,h, s2 − sm2,h)− ch(sm2,h, sm2,h) = O(h3).

Proof. We apply the modified Galerkin orthogonality, the Cauchy-Schwarz inequalityand an estimate used in the a priori estimate in Lemma I.11.

|a(s2 − sm2,h, s2 − sm2,h)− ch(sm2,h, sm2,h)| ≤ ‖∇(s2 − sm2,h)‖2 + ‖∇s2‖20,Ωh

. ‖s2 − I1hs2‖2

1 + ‖∇s2‖20,Ωh

. h4πω . h3

for ω ≤ 43π. In the last bound we use

‖∇s2‖20,Ωh

.∫ h

0

r4π/ω−2 · r dr . h4π/ω

and by Lemma I.8‖s2 − I1

hs2‖1 . O(h4π/ω).

to derive to the stated result.


Next, we will employ that the singular functions coincide with symmetric and asymmetricfunctions in the neighborhood of the origin and, thus, are orthogonal. This will allow usto relax the condition in Lemma I.14 for symmetric triangulations in the neighborhoodof the singularity for i+ j = 3 and i+ j = 5. The result is similar to [19, Lem. 4.2]

Lemma I.16. Let (C1)- (C3)and (S) hold. Then,

a(si − smi,h, sj − smj,h)− ch(smi,h, smj,h) = O(h3). (I.77)

for (i, j) = (1, 2), (2, 1), (3, 2).

Proof. Without loss of generality we can restrict ourselves to (i, j) = (1, 2), (3, 2), since(i, j) = (2, 1) is satisfied by symmetry of a(·, ·) and ch(·, ·).

Let us define a cut-off function

χ(r) =

1 on Ω\Ω1,0 on Ω2 ⊂⊂ Ω1.

Further, without a restriction we can assume that the support of the modification satisfiesΩh ⊂⊂ Ω2. For the definition of Ω1, Ω2, Ω3 see Figure I.41.

x?

Ω1

x Ω3

Ω2

Ω1

Figure I.41: Defintion of support of Ω1, Ω2 and Ω3.

We start with our bounds by using the fact that χ vanishes on Ωh and the modifiedGalerkin orthogonality

a(si − smi,h, s2 − sm2,h)− ch(smi,h, sm2,h) = a(si − smi,h, s2) (I.78)

= a(si − smi,h, χs2 − I2h(χs2)) + a(si − smi,h, (1− χ)s2) = (T1) + (T2).


We consider the last two terms (T1) and (T2) of (I.78) separately. We apply the Cauchy-Schwarz inequality and that χs2 − Ih(χs2) is equal to zero on Ω2. Thus,

a(si − smi,h, χs2 − I2h(χs2)) . ‖si − smi,h‖1,Ω\Ω2‖χs2 − I2

h(χs2)‖1,Ω\Ω2 . (I.79)

We see that‖χs2 − I2

h(χs2)‖1,Ω\Ω2 = O(h2),

since χ and the singular function are smooth on Ω\Ω2. By the localization technique ofNitsche and Schatz [37], we obtain

‖si − smi,h‖1,Ω\Ω2 . ‖si − I2hsi‖1,Ω\Ω3 + ‖si − smi,h‖0,Ω = O(h),

since the singular functions are smooth away from the singularity and Lemma I.11. Alltogether the first term is

(T1) = O(h3).

For the second term (T2) in (I.78) we have

a(si, (1− χ)s2) = 0,

since the singular functions s1 and s3 are even functions and s2 odd. In general si,h willnot have this symmetry property. Let us define a s?i,h ∈ V 2

h in Ω1 by

s?i,h(x) = si,h(x?) on ∂Ω1

a(s?i,h, v) = a(si,h, v) in v ∈ V 2h ∩ H1(Ω1)

(I.80)

Here, x? denotes the mirror point to x, cf. Figure I.41. By definition si,h+s?i,h is symmetricon the boundary ∂Ω1 and we have

ah(s?i,h + si,h, v) = 2ah(si,h, v) = 2a(si,h, v) for all v ∈ V 2

h ∩ H1(Ω1)

where we used the definition of s?i,h and of the modified bilinear form. We employ thatch(·, ·) is symmetric, the symmetry of the mesh and si, and obtain that s?i,h + si,h is

symmetric, since the values coincide on the triangle T and the mirror triangle T . Now,we turn back to the second term of (I.78)

(T2) = a(−smi,h,(1− χ)s2) +1

2a(s?i,h + si,h, (1− χ)s2) =

1

2a(s?i,h − smi,h, (1− χ)s2)

=1

2a(s?i,h − smi,h, (1− χ)s2 − Ih((1− χ)s2))

. ‖s?i,h − smi,h‖1,Ω1‖(1− χ)s2 − Ih((1− χ)s2)‖1,Ω1 .

At the singularity we have

‖(1− χ)s2 − Ih((1− χ)s2)‖1,Ω1 = O(h).

By definition, s?i,h− smi,h is discrete harmonic on Ω1 and thus we derive by using an Lemmaof [44, Lemma 4.10]

‖s?i,h − smi,h‖1,Ω1 . ‖s?i,h − smi,h‖1/2,∂Ω1

≤ ‖s?i,h − si‖ 12,∂Ω + ‖si − smi,h‖ 1

2,∂Ω1

. ‖s1 − sm1,h‖ 12,∂Ω1


For functions in H1(Ω), the H12 (∂Ω1) –norm is equivalent to the H

1200(∂Ω1\∂Ω) –norm

with the trace theorem, cf. [20, Theorem 1.5.2.1], yields to

‖si − smi,h‖1/2,∂Ω1 . ‖si − smi,h‖1,Ω\Ω1 = O(h2)

since the singularity is smooth away from the origin. This concludes the proof.

Lemma I.17. For π ≤ ω ≤ 53π we have


Proof. For interior angles ω ≤ 53π we can neglect the symmetric condition on the mesh

for i = 3 and j = 2.

a(s3 − sm3,h, s2 − sm2,h)− ch(sm3,h, sm2,h)

= a(s3 − sm3,h, s2) . ‖s3 − sm3,h‖1‖∇s2‖0

. (‖s3 − I1hs3‖1 + ‖∇s3‖0,Ωh)‖∇s2‖0

. h2π/ωh3π/ω . h3

where we used modified Galerkin orthogonality, Cauchy-Schwarz inequality, similar ar-guments as in Lemma I.11 and explicit integration of ‖∇s2‖0,Ωh , ‖∇s3‖0,Ωh and ‖∇s2‖0.Note, that s2 is supported in a small neighborhood of the corner.

Lemma I.18. For π ≤ ω ≤ 43π we have

a(s3 − sm3,h, s1 − sm1,h)− ch(sm3,h, sm1,h) = O(h3). (I.81)

Proof. The proof is analogous to the proof of Lemma I.17.


Now, we can formulate our main result for the energy-correction of second order finiteelements by summarizing the Lemma I.14 and the relaxations in Lemma I.15 - I.18.

Theorem I.2. Let f ∈ H1α(Ω) for some 2 − π

ω< α < 2, and assume (C1)-(C3). If the

modification satisfies either of the following sets of assumptions

• for π ≤ ω ≤ 43π

a(s1 − sm1,h, s1 − sm1,h)− ch(sm1,h, sm1,h) = O(h3)

and (S) is valid, otherwise, in addition,

a(si − smi,h, sj − smj,h)− ch(smi,h, smj,h) = O(h3),

for (i, j) = (1, 2), (2, 1).

• for 43π < ω ≤ 5

3π

a(si − smi,h, si − smi,h)− ch(smi,h, smi,h) = O(h3) for i = 1, 2,

a(s3 − sm3,h, s1 − sm1,h)− ch(sm3,h, sm1,h) = O(h3),

and (S) is valid, otherwise, in addition

a(si − smi,h, sj − smj,h)− ch(smi,h, smj,h) = O(h3)

for (i, j) = (1, 2), (2, 1).

• for 53π < ω ≤ 2π

a(si − smi,h, si − smi,h)− ch(smi,h, smi,h) = O(h3) for i = 1, 2,

a(s3 − sm3,h, s1 − sm1,h)− ch(sm3,h, sm1,h) = O(h3),

and (S) are valid, otherwise, in addition,

a(si − smi,h, si − smi,h)− ch(smi,h, smi,h) = O(h3)

for (i, j) = (1, 2), (2, 1), (3, 2).

Then, convergence rates of optimal order holds, i.e.

‖u− umh ‖0,α . h3‖f‖1,−α and ‖∇(u− umh )‖0,α . h2‖f‖1,−α.

Remark I.10. In the above Theorem we summarized the statements of this section un-der which conditions we obtain optimal convergence rate for second order finite elementmethods by using energy-correction. The conditions yield to the estimate

a(u, u)− ah(umh , umh ) = O(h3)

of the energy defect. Further, this condition is necessary (by Theorem I.2) and sufficient(similar to Theorem 2.2 in [19]) to avoid the pollution effect.


I.4.2. Numerical Examples

In this section we present numerical examples, which verify the theoretical results forthe energy-correction method for second order FE. We will formulate a modified bilinearform that satisfies the conditions of Theorem I.2 and a sufficient correction domain Ωh.Different approaches are made to obtain optimal correction like a full-Newton methodor a one-step Newton method. Finally, we compare results from the adaptive approachand the energy-correction method by neglecting the computational time and cost andconcentrating on the accuracy of FE solution.

We consider the Poisson problem

−∆u = 0 in Ω, u = s1 + s2 + s3 on ∂Ω, (I.82)

in four different domains, the 5/4π-section, the L-shape domain, the 7/4π-section and theslit-domain, cf. Figure I.2. The Dirichlet boundary conditions u = s1 + s2 + s3 are chosensuch that it is the exact solution. Note, that the third singularity function s3 does notreduce the convergence rates of the uncorrected approach due to sufficient regularity.

I.4.2.1. Energy-Correction in Symmetric Triangulations

In this subsection we consider symmetric triangulations of the four geometries, cf. FigureI.2. By Theorem I.2 we see that by the theory at most three conditions have to be satisfiedby the modified bilinear form. Nevertheless, numerical experiments show that for all fourtriangulations of Figure I.42 we can drop the condition


Thus, it remains that our modification satisfies two conditions similar to the case of linearfinite elements in non-symmetric meshes. Hence, by following the idea of non-symmetrictriangulations of linear FE in Section I.3 a good choice is the two parameter correction

ch(u, v) = γ1d1h(u, v) + γ2d

2h(u, v) = γ1

∫ω1h

∇u · ∇vdx+ γ2

∫ω2h

∇u · ∇vdx (I.83)

for correction parameters (γ1, γ2) ∈ [−1, 1)2 and disjoint correction domains ω1h, ω

2h ⊂ Ωh.

We will show that a good choice for the correction domains is the two-element-layer, cf.Section I.3.2.3. The correction domains for the four geometries are depicted in FigureI.42. The pink elements define ω1

h and the blue elements ω2h.


Figure I.42: Marked elements in the neighborhood of the singularity of the 5/4π-section,L-shape domain, 7/4π-section and slit domain; ω1

h: Pink elements; ω2h: Blue

elements.

As seen in the linear FE in Section I.2 the correction parameters can be obtained byderiving the root of the energy-correction function, namely,

g(γ1, γ2) =

(a(s1 − s1,h(γ1, γ2), s1 − s1,h(γ1, γ2))a(s2 − s2,h(γ1, γ2), s2 − s2,h(γ1, γ2))

−γ1d1h(s1,h(γ1, γ2), s1,h(γ1, γ2))

−γ1d1h(s2,h(γ1, γ2), s2,h(γ1, γ2))

−γ2d2h(s1,h(γ1, γ2), s1,h(γ1, γ2))

−γ2d2h(s2,h(γ1, γ2), s2,h(γ1, γ2))

).

(I.84)Since we consider domains with one re-entrant corner, we can use the singular functionss1 and s2 instead of s1 ,s2, compare Remark I.8. Then, it is obvious that the root of theenergy-correction function satisfies the condition of Theorem I.2 in the case of symmetricmeshes for each size of the interior angle.In the linear case we observed that the correction parameters obtained by the one dimen-sional as well as the two dimensional one-step Newton method, cf. Section I.3 and [41],are sufficient choices. Similar, we apply this approach to the second order FE case. Wealso refer to [22], where the method for second order FE is presented and by numerical re-sults showing that the gammas calculated by this one-step Newton method stay bounded.

The one-step Newton method in this case is given by: Given the initial guess γ0 =(0, 0) ∈ (−1, 1)2 on the coarse mesh T0, we set for l = 0, 1, ...

(γl+11 , γl+1

2 ) = [∇gh(γl1, γl2)]−1

(a(s1,h(γ

l1, γ

l2), s1,h(γ

l1, γ

l2))− a(s1, s1)

a(s2,h(γl1, γ

l2), s2,h(γ

l1, γ

l2))− a(s2, s2)

), (I.85)


with

∇gh(γ) =

(a1,h(s1,h(γ1, γ2), s1,h(γ1, γ2)) a2,h(s1,h(γ1, γ2), s1,h(γ1, γ2))a1,h(s2,h(γ1, γ2), s2,h(γ1, γ2)) a2,h(s2,h(γ1, γ2), s2,h(γ1, γ2))

).

Note that the index l denotes the refinement level, and on each level typically only one-step Newton method is carried out. The start value for the Newton on level l + 1 is thevalue computed on level l.

In the following convergence analysis we compare the results obtained by the correc-tion parameters which are the roots of the energy-correction function (I.84), calculatedby a one-step Newton parameter for two parameters like in (I.85) for one parameter likein Section I.3 in (I.42), respectively, and two static choices.

Let us start with the 5/4π-section. In Figure I.43 the contour lines are plotted for theinitial mesh. The red lines denote the first component of the energy-correction functionand the green ones the second component. The bullet denotes the root of the energy-correction function. The optimal correction parameters and the values obtained by theone-step Newton method for one and two parameters are depicted in Table I.26.

γ2

γ1

−0.10 −0.05 0.00 0.05 0.10 0.15 0.20−0.10

−0.05

0.00

0.05

0.10

0.15

0.20

-0.075

-0.050

-0.025

0.000

0.025

-0.060

-0.045

-0.030

-0.015

0.000

0.015

Figure I.43: Contour lines of the energy-correction function (I.84) in the 5/4π-section;First component: Red; Second component: Green; The root is denoted bythe black bullet.


l γopt,1 γopt,2 γoN,1 γoN,2 γoN2 0.00626 -0.00051 0.00637 -0.00053 0.005283 0.00623 -0.00050 0.00625 -0.00050 0.005234 0.00621 -0.00049 0.00622 -0.00049 0.005235 0.00620 -0.00048 0.00621 -0.00049 0.00523

Table I.26: Optimal energy-correction parameters, correction parameter obtained by theone-step Newton method for two parameter and for one parameter for the5/4π-section for symmetric triangulation (Second order FEM).

The Table I.27 contains the convergence analysis for the 5/4π-section of the energy-correction method for second order FEM in symmetric triangulations. The energy-correction for two optimal parameters and the values obtained by the one-step Newtonmethod for one, two parameters, respectively, yield optimal convergence rates and simi-lar accuracy. Thus, a one parameter correction can be used instead of a two parametercorrection. The static choices of parameters (γ1, γ2) = (0.0060,−0.0005), (0.005, 0.0000)yield optimal rates and a higher accuracy when they are close to the optimal values. Apoor convergence and lower error reduction can be observed for the standard Galerkinmethod.

l \ γ (0.0000, 0.0000) rate γoN rate (γopt,1, γopt,2) rate2 1.20100 — 1.53640 — 0.81492 —3 0.32364 1.89 0.18924 2.02 0.10234 2.994 0.10314 1.65 0.02365 3.00 0.01298 2.985 0.03386 1.61 0.00299 2.98 0.00168 2.95

l \ γ (γoN,1, γoN,2) rate (0.0060,−0.0005) rate (0.005, 0.0000) rate2 1.53770 — 1.53460 — 1.53480 —3 0.18944 3.02 0.18957 3.02 0.18943 3.024 0.02367 3.00 0.02407 2.98 0.02402 2.985 0.00299 2.98 0.00336 2.84 0.00332 2.85

Table I.27: Errors 104×‖u− uh(γ1, γ2)‖0,α with α ≈ 0.38; Data of the convergence studyof the PDE (I.82) in the 5/4π-section with a symmetric triangulation (Secondorder FEM).


γ1

γ2

−0.10 −0.05 0.00 0.05 0.10 0.15 0.20−0.10

−0.05

0.00

0.05

0.10

0.15

0.20-0.100

-0.075

-0.050

-0.025

0.000

0.025

0.050

-0.080

-0.060

-0.040

-0.020

0.000

0.020

Figure I.44: Contour lines of the energy-correction function (I.84) in the L-shape domain;First component: Red; Second component: Green; The root is denoted bythe black bullet.

γ2

γ1

−0.10 −0.05 0.00 0.05 0.10 0.15 0.20−0.10

−0.05

0.00

0.05

0.10

0.15

0.20

-0.090

-0.060

-0.030

0.000

0.030

0.060

-0.100

-0.075

-0.050

-0.025

0.000

0.025

Figure I.45: Contour lines of the energy-correction function (I.84) in the 7/4π-section;First component: Red; Second component: Green; The root is denoted bythe black bullet.


γ1

γ2

−0.10 −0.05 0.00 0.05 0.10 0.15 0.20−0.10

−0.05

0.00

0.05

0.10

0.15

0.20

-0.090

-0.060

-0.030

0.0000.030

0.060

0.090

-0.120

-0.090

-0.060

-0.030

0.000

0.030

Figure I.46: Contour lines of the energy-correction function (I.84) in the slit domain; Firstcomponent: Red; Second component: Green; The root is denoted by the blackbullet.

The contour lines of the energy-correction function (I.84) are presented in Figure I.44 forthe L-shape domain, in Figure I.45 for the 7/4π-section and the slit domain in the FigureI.46.

Table I.28 for the L-shape domain, Table I.29 for the 7/4π-section and Table I.30 forthe slit domain present the parameters for the energy-correction method obtained by afull-Newton method and by a one-step Newton method for one and two parameters.


Table I.28: Optimal energy-correction parameters, correction parameter obtained by theone-step Newton method for two parameter and for one parameter for theL-shape domain for symmetric triangulation (Second order FEM).



Table I.29: Optimal energy-correction parameters, correction parameter obtained by theone-step Newton method for two parameter and for one parameter for the7/4π-section for symmetric triangulation (Second order FEM).


Table I.30: Optimal energy-correction parameters, correction parameter obtained by theone-step Newton method for two parameter and for one parameter for the slitdomain for symmetric triangulation (Second order FEM).

The convergence study for the L-shape domain in Table I.31 is presented, for the 7/4π-section in Table I.32 and for the slit domain in Table I.33. The optimal parameters andthe ones obtained by the one-step Newton method (I.85) for two parameters yield optimalrates and similar accuracy. An improved rate and accuracy is derived by a one parametercorrection. For the static choices we always get optimal rates when the parameters areclose to the optimal values. A poor rate is observed for the standard Galerkin method.


l \ γ (γoN,1, γoN,2) rate (0.0310,−0.0050) rate (0.0200,−0.0060) rate2 1.86970 — 1.78930 — 2.88860 —3 0.20312 3.20 0.20174 3.10 1.04330 1.474 0.02402 3.08 0.02567 2.94 0.40996 1.355 0.00299 3.00 0.00466 2.46 0.16247 1.34

Table I.31: Errors 104×‖u− uh(γ1, γ2)‖0,α with α ≈ 0.38; Data of the convergence studyof the PDE (I.82) in the L-shape domain for symmetric triangulations (Secondorder FEM).




Table I.32: Errors 104×‖u− uh(γ1, γ2)‖0,α with α ≈ 0.38; Data of the convergence studyof the PDE (I.82) in the 7/4π-section for symmetric triangulations (Secondorder FEM).



Table I.33: Errors 103×‖u− uh(γ1, γ2)‖0,α with α ≈ 0.38; Data of the convergence studyof the PDE (I.82) in the slit domain for symmetric triangulations (Secondorder FEM).

In this section we verified the theoretical results of Theorem I.2 for second order FEenergy-correction method for symmetric triangulations. For interior angles ≤ 4

3π a cor-

rection with one parameter is sufficient for optimal rates and for greater angles a twoparameter correction. We also studied different approach to derive the correction param-eters, the full-Newton method for the exact roots of the energy-correction function (I.84)on each mesh level, the one-step Newton method and static choices for the parameters.The one-step Newton method yields parameters close to the optimal ones. We also ob-served that the manual adjusted static parameters which are close to the optimal valuesgive solutions of much better quality. By fine-tuning of the parameters the convergencerate and quantitative error can be improved even if the parameters are unknown. A re-duced rate and error decay can be observed by the standard Galerkin method and thestatic parameters which are not close to the optimal ones.


I.4.2.2. Energy-Correction in Non-Symmetric Triangulations

In this section we consider the Poisson equation (I.82) with non-symmetric triangulationsand verify the results of Theorem I.2. We apply the energy-correction method with a twoparameter correction.

In Theorem I.2 we saw that for a non-symmetric triangulation several conditions needto be satisfied. Thus, we restrict ourselves to the case of interior angle ≤ 4

3π, where

by the theory only two condition have to be valid. Further, we also study the case ofa non-symmetric triangulation of the L-shape domain, cf. Figure I.26. In Figure I.47the 4/3π-section for a non-symmetric triangulation and the two-element-layer correctiondomain are depicted.

Figure I.47: Left: Non-symmetric mesh for the 4/3π-section; Right: Two-element-layercorrection for the 4/3π-section.

In this case the energy-correction function is the same as for linear finite elements innon-symmetric triangulation, namely, (I.43). Thus, the one-step Newton method is givenby (I.45).

In Table I.34 the roots of the energy-correction function are presented for the 4/3π-section. Further, we derive parameters computed by a one-step Newton method for oneand two parameter.


Table I.34: Optimal energy-correction parameters, correction parameters obtained by theone-step Newton method for one- and two parameter in the 4/3π-section fornon-symmetric triangulations (Second order FEM).

The convergence analysis in Table I.35 for the 4/3π-section shows that optimal conver-gence rates are only achieved by a two parameter correction. The manual chosen valuesyield an improved accuracy but only optimal rates when they are close to the optimalones.




Table I.35: Errors 103×‖u−uh(γ1, γ2)‖0,α with α ≈ 0.38; Data of the convergence study ofthe PDE (I.82) in the 4/3π-section for non-symmetric triangulations (Secondorder FEM).

For the L-shape domain we observe that a two parameter is not sufficient to obtainthe optimal rates. Nevertheless, the energy-correction methods yield to an improvedconvergence. The optimal parameters for a two parameter correction are presented inTable I.36 and the convergence analysis in Table I.37.

l γopt,1 γopt,22 0.06210 -0.024003 0.06208 -0.024354 0.06216 -0.024485 0.06221 -0.02452

Table I.36: Optimal energy-correction parameters for the L-shape for non-symmetric tri-angulations (Second order FEM).

l \ γ (0.0000, 0.0000) rate (γopt,1, γopt,2) rate2 4.32600 — 5.62850 —3 1.54510 1.49 0.72367 3.964 0.61024 1.34 0.09465 2.935 0.24320 1.33 0.00134 2.85

Table I.37: Errors 103×‖u− uh(γ1, γ2)‖0,α with α ≈ 0.38; Data of the convergence studyof the PDE (I.82) in the L-shape domain with a non-symmetric triangulation(Second order FEM).

In this section we saw that optimal rates are recovered by the energy-correction methodfor non-symmetric triangulations in the case of interior angle ≤ 4

3π. Again, the correction

parameters can be computed by a one-step Newton method instead of a full-Newtonmethod. For greater angles a two parameter correction cannot guarantee an optimalerror reduction.


I.4.2.3. Energy-Correction Method vs. Adaptivity

In this section we will compare the energy-correction method and the adaptivity for sec-ond order finite element methods. In the above sections we saw that we can regain optimalconvergence with the energy-correction method and similar for the adaptivity in SectionI.2. Here, we compare the methods with respect to the accuracy in weighted L2-normsand give application preference.

We see the convergence analysis of the energy-correction method (green) and of the adap-tive method (blue) for symmetric triangulations in Figure I.48 for the 5/4π-section, inFigure I.49 for the L-shape domain, in Figure I.50 for the 7/4π-section and in Figure I.51.We observe that the energy-correction method is in all four case more accurate than theadaptive approach when we measure the error in weighted L2-norms

101 102 103 104 105

DOFs

10−7

10−6

10−5

10−4

10−3

Figure I.48: Approximation errors of the PDE (I.82) derived by the energy-correctionmethods and adaptivity in the 5/4π-section in the weighted L2-norm withα ≈ 1.25; Green: Energy-correction; Blue : Adaptivity.


101 102 103 104 105

DOFs

10−7

10−6

10−5

10−4

10−3

Figure I.49: Approximation errors of the PDE (I.82) derived by the energy-correctionmethods and adaptivity in the L-shape domain in the weighted L2-normwith α ≈ 1.38; Green: Energy-correction; Blue : Adaptivity.

101 102 103 104 105

DOFs

10−7

10−6

10−5

10−4

10−3

10−2



101 102 103 104 105

DOFs

10−6

10−5

10−4

10−3

10−2

Figure I.51: Approximation errors of the PDE (I.82) derived by the energy-correctionmethods and adaptivity in the slit domain in the weighted L2-norm withα ≈ 1.55; Green: Energy-correction; Blue: Adaptivity.

Finally, in Figure I.52 we see the convergence analysis of the energy-correction and adap-tive method in a non-symmetric triangulations of the 4/3π-section. Also, here the energy-correction yields a more accurate approximation than the adaptivity for the same numberof DOFs.

101 102 103 104 105

DOFs

10−7

10−6

10−5

10−4

10−3

10−2


In this section we compared the energy-correction method and the adaptivity in re-entrantcorners by neglecting the computational costs and time. In all cases of symmetric andnon-symmetric triangulations we observe a higher accuracy of the energy-correction ap-proximation than of the adaptivity in weighted L2-norms. Therefore, we can recommendan application of the energy-correction method in the case of re-entrant corner. Note,that we measure the error in weighted in L2-norms, cf. Section I.3.2.2.


When we reconsider the experiments above we see that a high accurate solution in thecase of adaptivity needs many re-computation steps of the FE solution (indicated by theblue dots), whereas in the energy-correction method parameters for the correction arecalculated in a pre-process step. Thus, both methods may cause a lot computation costand time which depends on the underlying PDE.

In this part of the thesis we considered the energy-correction method for second orderFEM. We saw that by the theory optimal rates can be recovered under certain condi-tions. We summarized this in Theorem I.2. Then, we verified our theoretical resultsby numerical experiments for our four test scenarios - the 5/4π-section, the L-shape do-main, the 7/4π-section and the slit domain - in symmetric triangulations. We applied atwo parameter energy-correction and obtained optimal rates in weighted L2-norms for allfour geometries. The parameters were computed by a full-Newton method or a one-stepNewton method for one resp. two parameters. The one-step Newton method gives agood approximation of the exact values. For interior angle ≤ 4

3π one parameter and for

greater angles a two parameter correction is necessary for optimal rates. In the case ofnon-symmetric triangulations we only obtained optimal rates for angles ≤ 5

4when ap-

plying a two parameter correction. Finally, we compared the energy-correction methodand the adaptivity by ignoring computational cost and time and observed that the energycorrection method yields to a more accurate approximation for a fixed number of DOFs.Thus, we can recommend the application of the energy-correction in the case of re-entrantcorner.


Part II.The Diffusion Problem: JumpingCoefficients

In this part we will consider the diffusion equation for jumping coefficients. Due to thesejumps the singularities are of type rε with 0 < ε 1. In this situation the convergenceof standard FEM is even worse than for re-entrant corners, cf. [24–26,43].We will expand the solution in a special case of jumping coefficients and further, con-sider two approaches to regain optimal convergence rates - the adaptivity and the energycorrection.

II.1. Singular Functions

In the following we consider an expansion of the solution of a second order linear ellip-tic operator, the diffusion equation with jumping coefficients. Similar to the case of thePoisson equation the solution consists, in general, of singular functions and a regular part.

We consider the second order elliptic problem with homogeneous boundary conditions

− div(K∇u) = f in Ω, u = 0 on ∂Ω, (II.1)

where 0 < K0 ≤ K ∈ L∞(Ω) is a known coefficient, e.g. a diffusivity or the permeabilityof a porous medium. We call this problem diffusion equation with jumping coefficients,jumping coefficients problem or interface problem. Further, let Ω ⊂ R2 be a boundedopen polygonal and convex domain.

Remark II.1. We assume that Ω is convex to exclude the case of singularities occurringdue to re-entrant corners of the boundary. This case can also be handled by the presentedmethod and the knowledge of re-entrant corners.

For simplicity assume Ω = (−1, 1)2 and open polygonal subdomains Ωj ⊂ Ω for i = 1, ..., 4such that

Ωi ∩ Ωj = ∅ for i 6= j and4⋃j=1

Ωj = Ω.

We restrict ourselves to the case of

Ω1 = (0, 1)2, Ω2 = (−1, 0)× (0, 1), Ω3 = (−1, 0)2, Ω4 = (0, 1)× (−1, 0)

and choose the coefficient K such that

K(x) = Kj for x ∈ Ωj.

Remark II.2. For the choice of K1 = K2 = K3 = K4 the problem (II.1) reduces to thePoisson equation on (−1, 1)2.


K1K2

K3 K4

Figure II.1: Right: Domain Ω with subdomains Ωj and coefficientsKj; Left: Triangulationassociated with the subdomains.

In Figure II.1 the above made definitions are summarized. For our finite element ap-proximation we choose a triangulation T1 which is aligned with the boundaries of thesubdomain. By regular refinement we get the same property for the family of triangula-tions Thh≥1, cf. Figure II.1 and [11]. Then, the standard Galerkin method of (II.1) hasthe form: Find u ∈ V 1

h such that

a(u, v) = (f, v) for all V 1h , (II.2)

where V 1h is the linear FE-space defined in Section 3 and the bilinear form

a(u, v) =

∫Ω

K∇u · ∇v dx. (II.3)

Remark II.3. Under our assumptions the bilinear form (II.3) is symmetric, continuous,H1(Ω) -coercive, and thus, by the the Lax-Milgram lemma [21, p. 215] we have a uniquesolution in H1(Ω), and also a unique FE-solution in V 1

h .

Let us consider an example of the homogeneous diffusion equation (II.1) with Dirichletboundary condition and coefficients K1 = K3 = 1 and K2 = K4 = 1000. We choosethe boundary condition such that it coincides with the solution in Ω (namely, the firstsingular function, cf. (II.14)).In Figure II.1 the convergence analysis for this example is presented. By this experimentwe observe that for interface problems we have, in general, a reduced convergence rate forjumps in the coefficients, i.e., when K is discontinuous.


101 102 103 104 105

DOFs

10−5

10−4

10−3

10−2

10−1

Figure II.2: Convergence analysis of the diffusion equation with coefficients K1 = K3 = 1and K2 = K4 = 1000; Blue: Error of the FE-approximation in the L2-norm;Dashed red: Reference line h2.

For the explanation of this reduce convergence rate we want to derive an expansion of thesolution of (II.1). We follow the ideas of Strang and Fix [43, Sec. 8.1].In the general setting of jumping coefficients we only have a singular point at a vertex,where K is discontinuous and the interface is not a straight line, e.g. K1 6= K2, K1 6= K4,cf. Figure II.3 and [43, p. 251]. Similar to the Poisson case we seek for a solution in theneighborhood of singular point, here, the origin. So we define

Ω = (r, ϕ) : 0 < r < r0, ωi < ϕ < ωi+1 for i = 1, .., 4 ⊂ Ω,

where (r, ϕ) are polar coordinates centered at the origin, ω1 = 0, ω2 = π2, ω3 = π, ω4 = 3π

4,

ω5 = 2π and a small r0 > 0, see Figure II.3.

r = r0

ω1,5 = 0, 2π

ω2 = π2

ω3 = π

ω4 = 3π4

Ω

Figure II.3: Definition of Ω1 and ωi for i = 1, ..., 5.

By Weyl’s lemma [21, p. 199] we know that the solution u of (II.1) is analytical in Ω(expect in the origin). Therefore, we can apply the ansatz - separation of variables in


each Ωi = Ωi ∩ Ω and an interface condition. Thus, we can write the solution u of (II.1)

u(r, ϕ) = R(r)Θ(ϕ)

with a function R, Θ which only depends on the radius, angle, respectively. Further, byvariation of the constant we first consider the homogeneous problem of (II.1). We rewritethe problem (II.1) on each Ωi in homogeneous form

−Ki∆ui = 0 in Ωi, (II.4)

ui = uj on ∂Ωi ∩ ∂Ωj, (II.5)

Ki∂

∂nui = Kj

∂

∂nuj on ∂Ωi ∩ ∂Ωj (II.6)

for i, j = 1, ...4, i 6= j, where ui = u∣∣Ωi

and ∂∂n

denotes the partial derivative of the exteriornormal at the interface. Then, we plug in the ansatz of the separation of the variables,we derive for each Ωi

−Ki(R′′i (r)Θi(ϕ) +

1

rR′i(r)Θ(ϕ) +

1

r2Ri(r)Θ

′′(ϕ)) = 0,

where we write Ri = R∣∣Ωi

and Θi = Θ∣∣Ωi

. By the same argumentation as for the Poissonequation in Section I.1 we obtain the following systems of ordinary differential equations(ODEs)

Θ′′i (ϕ) + λ2iΘi(ϕ) = 0 (II.7)

R′′i (r) +1

rR′i(r)− λ2

i

1

r2Ri(r) = 0 (II.8)

for a constant λi ∈ R for Ωi, cf. [24]. We can separately derive a solution for these bothODEs.First, we consider (II.8) which has the solution

Ri(r) = rλi . (II.9)

Remark II.4. In (II.9) we can exclude the negative exponents, since we know by Weyl’slemma that the solution is analytical in Ω.

By the interface condition (II.5), we obtain

ui(r, ϕ) = rλiΘi(ϕ) = rλjΘj(ϕ) = uj(r, ϕ)

⇒ rλi

rλj=

Θj(ϕ)

Θi(ϕ)= const.

⇒ λi = λj

(II.10)

for each interface Ωi ∩ Ωj. Similar argumentation holds for (II.6). Thus, we set

λ = λ1 = ... = λ4.

It remains to solve the ODE (II.7) which is a discontinuous Sturm-Liouville eigenvalueproblem

Θ′′i (ϕ) + λ2Θi(ϕ) = 0 for ωi < ϕ < ωi+1 (II.11)


for i = 1, ..., 4 and the interface conditions

limϕωi

Θi−1(ϕ) = limϕωi

Θi(ϕ)

limϕωi

Ki−1Θ′i−1(ϕ) = limϕωi

KiΘ′i(ϕ)

(II.12)

where and Θ0 = Θ4. We call λ eigenvalue to the corresponding eigenfunction Θ.

Remark II.5. In a general setting we cannot solve the Sturm-Liouville problem (II.12)with interface conditions (II.12) analytically, but the problems are one dimensional, hence,easily computed and the computational costs are negligible, cf. [11].

In our setting Kellog worked out results for the eigenfunctions and eigenvalues in [24–26],which we summarize in the next Lemma.

Lemma II.1. The problem (II.11) with interface conditions (II.12) has a countable num-ber of increasing eigenvalues λmm≥0 (0 = λ0 < λ1 ≤ λ2 ≤ ...) to corresponding eigen-functions Θm such that Θmm≥0 is complete in L2(0, 2π) and satisfy∫ 2π

0

KΘmΘn dϕ =

1, m = n0, m 6= n

.

Furthermore, the following relation hold for the first two non-trivial eigenvalues

λ1 + λ2 = 2.

Remark II.6. The proof of Lemma II.1 is in Lemma 3.3 and Lemma 4.2 of [24]. Theresults there also state the exact formulation of the the first two non-trivial eigenfunctions.

Thus, we combine the results of Lemma II.1 and (II.9) and derive by superposition thehomogeneous solution in the neighborhood of the origin

u(r, ϕ) =∞∑k=0

αkrλkΘk(ϕ), (II.13)

with Fourier-coefficients αk determined by u(r0, ϕ). The solution fulfilling the inhomoge-nous problem (II.1) is obtained by variation of the constant similar to the Poisson casewhich is worked out in [43, p. 258]. Nevertheless, the first two leading r-terms are alreadydetermined by the homogeneous problem, which is sufficient to know for our considera-tions.Before stating the lemma which summarizes the above obtained expansion of the solution,let us define the singular functions in the case of jumping coefficients for n ∈ N

sn(r, ϕ) = rλnΘn(ϕ)

andsn(r, ϕ) = η(r)sn(r, ϕ)

with a smooth cut-off function η(r) which is identical to one in the neighborhood of thesingularity Ω′ ⊂ Ω; for details see [34]. Due to the expansion in the neighborhood ofjumps and the fact the away of the singularity the solution is smooth the solution is acombination of singular function close to the jumps and a regular remainder. We refer tothe work of Nicaise and Sandig [34–36].


Lemma II.2. (Theorem 4.2 [35]). Let β < 1 and λn the eigenvalues of Sturm-Liouville problem (II.11) with 1 − β 6= λn for all n ∈ N. Then, for f ∈ H0

β(Ω) exists

a unique solution u ∈ H1(Ω) of the interface problem (II.1) and admits the followingexpansion

u =∑

λ∈(0,1−β)

knsn + U

with a smooth remainder U ∈ H2β(Ω) and coefficient kn ∈ R. Moreover, it holds∑λ

|kn|+ ‖U‖2,β . ‖f‖0,β.

Remark II.7. Note, we obtain the Lemma II.2 by setting γ = β, n = 2, p = 2, m = 1,k = 0 and the operator AS(ζ) describes the Sturm-Liouville problem (II.11), which has inour case only real eigenvalues, in Theorem 4.2 of [35]. By a close look in the proof wecan extend the lemma by using the embedding theorem of weighted norms [30, Prop. 6.5,Cor. 6.7]

H0β(Ω) → L2(Ω)

for β < 1.

In the next lemma we will study the regularity of the singular function and see that thesmoothness strongly depends on the eigenvalue λ of the Sturm - Liouville problem (II.11).

Lemma II.3. For any α > 1 − λn for all n ∈ N there holds sn ∈ H2−α(Ω), but sn /∈H1+λn(Ω). For the weighted Sobolev spaces similar sn ∈ H2,α(Ω), but sn /∈ H2,1−λn(Ω).

Proof. The proof follows directly by considering the Poisson equation case of Lemma I.2and replace nπ

ωby λ.

Further, we consider the weighted L2-error of the standard Galerkin approximation andsee that even if we measure in weighted norms the error is sub-optimal. We obtained asimilar result for re-entrant corner in Lemma I.3, the proof here is analogously.

Lemma II.4. (Pollution Effect). Let u be the solution of (II.1) with f ∈ H0−α(Ω) for

α > −1. If u /∈ H2,1−λ1(Ω), then

‖u− uh‖0,α ≥ ‖∇(u− uh)‖20 & hλ.

Finally, we consider again the numerical example of the homogeneous diffusion equationwith coefficients K1 = K3 = 1 and K2 = K3 = 1000 and

− div(K∇u) = 0 in Ω, u = s1 on ∂Ω, (II.14)

with solution s1. We analyze the equation in weighted L2- and H1-norms. In a preprocesswe solve the Sturm-Liouville problem (II.11) and obtain the eigenvalue

λ1 ≈ 0.0402499566073.


We approximate the solution of (II.14) by the standard Galerkin approach and obtaina reduced convergence rate in the convergence analysis presented in the Table II.1. Wemeasure the error in a weighted L2-norm.

l 100× ‖u− uh‖0,0.96 rate1 3.06250 —2 2.46880 0.313 1.98150 0.324 1.64070 0.275 1.39060 0.246 1.19830 0.21

Table II.1: Convergence analysis of the diffusion equation (II.14).

II.2. Adaptivity

In this section we consider a standard approach to recover the full convergence rate in caseof interface problems. In Section I.2 we introduced the adaptive method for the Poissonequation and saw that it is necessary to define an error estimator, a refining strategy anda mesh refining strategy. Only the error estimator has to be adapted for a new problemclass, since refining and mesh refining strategy are defined independently of the PDE.Error estimators for this kind of problems are well-research and studied for residual-basedestimator in [4, 6, 12,38].We will state a residual-based error estimator and verify optimal convergence rate in nu-merical examples.

We consider the diffusion equation with jumping coefficient (II.1). In Section II.1 wesaw that in the neighborhood of the intersecting point of the coefficient K singular func-tions occur. These cause problems for the regularity of the solution and thus, in theconvergence rate of FE-methods. Hence, we expect from a adaptive method that theerror estimator detects this singularity and refines there.

A residual-based error estimator for this purpose is

η2T =

∑e∈∂T

η2e , η2

e =∑T∈Ωe

‖hTK1/2f‖20,T + ‖heK1/2

e [K∇uh · ne]‖20,e, (II.15)

where Ωe is the union of the elements T ∩ T = e, hT is the diameter of the element T , heis the diameter of the edge e, Ke = maxT∈Ωe K [6].

Note, that the error estimator (II.15) fulfills the error estimator condition (I.23) of SectionI.2.A problem which arises by growing jumps in the coefficient K is a bad influence on the er-ror estimation of the estimator for the standard approach of residual-based adaptivity [4].The choice of (II.15) counteracts this, cf. [6, 12,38].


110

1 10

1100

1 100

11000

1 1000

Figure II.4: Coefficients of the interface problem (II.1): The 1 − 10 jump, the 1 − 100jump and the 1− 1000 jump.

We consider numerical experiments which illustrate optimal convergence rate. Let ustreat the diffusion equation with homogeneous boundary conditions (II.14) with s1 asthe Dirichlet boundary condition. The boundary condition is chosen such that it is theanalytical solution of the problem. We consider the three different case of distributionsof K, cf. Figure II.4 and refer to them as 1− 10 jump, 1− 100 jump and 1− 1000 jump.

In Figure II.5, Figure II.6 and Figure II.5 the convergence analysis is depicted for the1−10 jump, the 1−100 jump and the 1−1000 jump. The green line denotes the error ofthe standard Galerkin method, the blue one the error of the adaptive method with esti-mator (II.15) and the red dashed one is the h2 reference line. In all three cases we observethat the adaptivity regain the optimal convergence rate and improves the accuracy of theFE-solution.

100 Adaptivity

101 102 103 104 105 106

DOFs

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Figure II.5: Convergence analysis of the interface problem (II.1) for the 1−10 jump; Errormeasure in the weighted L2 norm with α ≈ 0.66; Solid dotted green: StandardFE-solution; Solid blue: Adaptive method; Dashed red: h2 reference line.

101 102 103 104 105 106

DOFs

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Figure II.6: Convergence analysis of the interface problem (II.1) for the 1 − 100 jump;Error measure in the weighted L2 norm with α ≈ 0.92; Solid dotted green:Standard FE-solution; Solid blue: Adaptive method; Dashed red: h2 referenceline.


101 102 103 104 105 106

DOFs

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Figure II.7: Convergence analysis of the interface problem (II.1) for the 1 − 1000 jump;Error measure in the weighted L2 norm with α ≈ 0.96; Solid dotted green:Standard FE-solution; Solid blue: Adaptive method; Dashed red: h2 referenceline.

II.3. Energy-Correction Method for Linear FiniteElements

In this section we consider the energy-correction method for scalar diffusion equations toregain optimal convergence rate. The energy correction method for linear FE-methodswas introduced in [19, 40] and further studied in [41] for the Poisson equation. Scalarelliptic problems are considered in [22]. We study the theoretical background which is notpresented in [22] and more evolving experiments.

First, we study the theory of energy-correction of diffusion equations and obtain a theo-rem that poses optimal convergence rates for weighted L2- and H1-norms. In the proofswe apply several times the Aubin-Nitsche trick for the deviation of the results. We ana-lyze numerical experiments which verify our theoretical results. Further, we compare theenergy-correction method with the adaptivity method of Section II.2.

Let us consider the diffusion equation with jumping coefficient (II.1) and recall somedefinition of the energy-correction method of Section I.3. The modified (linear) Galerkinapproximation for (II.1) reads: Find umh ∈ V 1

h such that

ah(umh , v) = (f, v) for all v ∈ V 1

h

with the mesh-dependent bilinear form ah(·, ·) defined by

ah(u, v) = a(u, v)− ch(u, v),

where

a(u, v) =

∫Ω

K∇u · ∇v dx

and ch(·, ·) a local modification supported in the neighborhood Ωh of the singularity.In addition, we assume some basic properties of the modification (C1), (C2) and (C3),namely, continuity, H1-coerzivity and symmetry. This definitions are similar to theenergy-correction in the case of the Poisson equation, see Section I.3.

II.3.1. Theory of the Energy-Correction Method

In the following we use some results which we have shown for the case of the Poissonequation or are, in general, valid for (weighted) Sobolev spaces and, therefore, can alsobe used here. These are the interpolation Lemma I.7, inverse inequality Lemma I.9, themodified Galerkin orthogonality and the minimization property of Lemma I.5.

The goal of this section is to derive an optimal weighted L2- and H1-approximationresult for the modified Galerkin approximation. Therefore, we prove some lemmas whichfinally yield our main theorem. Note, that the singular functions derived from the Sturm-Liouville problem (II.11) play an important role in our consideration and in this contextthe exponents of the r -term, cf. Lemma II.1. Let us start with an a priori estimate forthe usual Sobolev norms.


Lemma II.5. Let (C1) - (C3) be satisfied and assume u ∈ H2α(Ω) ∩ H1(Ω) for some

0 ≤ α < 1. Then, it holds for all 1− λ1 < β < 1

‖∇(u− umh )‖0 . h1−α‖u‖2,α and ‖u− umh ‖ . h2−α−β‖u‖2,α.

Proof. By similar argumentation as in Lemma I.11 we obtain the stated results.

For our later estimates we need an additional a priori results for mesh-dependent analogsof the weighted norm ‖ · ‖0,α. In [9] the Lemma was determined for the Poisson equationfor re-entrant corners.

Lemma II.6. Define ρ = (r2 + Θh2)1/2 for some sufficient large but fixed Θ > 0. Thenwe have for any exponent 1− λ1 < α < 1

‖ρ−α∇(u− uh)‖0 . ‖ρ−α∇(u− I1hu)‖0 + h−1‖ρ−α(u− I1

hu)‖0

and‖ρ−α(u− uh)‖0 . h‖ρ−α∇(u− I1

hu)‖0 + ‖ρ−α(u− I1hu)‖0.

Proof. For simplifying our notation set e ≡ u − uh. Let us start with the L2-estimateand, hereby, consider the dual problem

−div(K∇v) = ρ−2αI1he in Ω, u = 0 on ∂Ω.

Note, that the right hand side is ρ−2αI1he is in H2

α(Ω), and, thus by Lemma II.2 we have

‖∇2v‖0,α . ‖ρ−2αI1he‖0,α . ‖ρ−αI1

he‖0

by using r ≤ ρ. For the bound, we employ the definition of the dual problem, Green’sformula, Cauchy-Schwarz inequality and continuity of a(·, ·)

‖ρ−αe‖20 = (ρ−αe, ρ−αe) = (ρ−αe, ρ−α(u− I1

hu)) + (e, ρ−2αI1he)

= (ρ−αe, ρ−α(u− I1hu)) + a(e, v) = (ρ−αe, ρ−α(u− I1

hu)) + a(e, v − I1hv)

≤ ‖ρ−αe‖0‖ρ−α(u− I1hu)‖0 + ‖ρ−α∇e‖0‖ρα∇(v − I1

hv)‖0.

By a similar argumentation as in the interpolation estimate of Lemma I.7 we determine

‖ρα∇(v − I1hv)‖0 . h‖rα∇2v‖0

and, thus, by the regularity estimate of the dual problem

‖ρ−α∇(v − I1hv)‖0 . h‖ρ−αI1

he‖0.

Now, we obtain a preliminary L2-estimate by combining the above results, ‖ρ−αI1he‖0 .

‖ρ−αe‖0 and dividing by ‖ρ−αe‖20

‖ρ−αe‖20 . ‖ρ−α(u− I1

hu)‖0 + h‖ρ−αe‖0. (II.16)


In the next step we determine the H1-bound. In the following equality we use Galerkinorthogonality by observing I1

h

(ρ−2αI1

hu)∈ V 1

h

‖ρ−α∇e‖20 = (∇e, ρ−2α∇e) = (∇e, ρ−2α∇(u− I1

hu)) + (∇e, ρ−2α∇I1hu)

= (∇e, ρ−2α∇(u− I1hu)) + (∇e,∇(ρ−2αI1

hu))− (∇e,∇(ρ−2α)I1hu)

= (∇e, ρ−2α∇(u− I1hu))

+ (∇e,∇(ρ−2αI1

hu− I1h

(ρ−2αI1

hu))

)− (∇e,∇(ρ−2α

)I1hu).

Applying the Cauchy-Schwarz inequality and dividing by ‖ρ−α∇e‖0 yield

‖ρ−α∇e‖0 . ‖ρ−α∇(e− I1he)‖0 +‖ρα∇

(ρ−2αI1

he− I1h

(ρ−2αI1

he))‖0 +‖ρ−α−1I1

he‖0. (II.17)

Using again an interpolation result similar to Lemma I.7 for the ρ-norm, there holds

‖ρα∇(ρ−2αI1

he− I1h

(ρ−2αI1

he))‖0 . h‖ρα∇2

(ρ−2αI1

he)‖0

. h(‖ρ−α−2I1

he‖0 + ‖ρ−α−1∇I1he‖0

).

(II.18)

We employ estimate (II.17) by (II.18), inverse inequality of Lemma I.9,

‖ρ−αI1he‖0 . ‖ρ−αe‖0, ‖ρ−α∇I1

he‖0 . ‖ρ−α∇e‖0,

and the preliminary L2-bound (II.16) we have

‖ρ−α∇e‖0 . ‖ρ−α∇(u− I1hu)‖0 + h

(‖ρ−α−2I1

he‖0 + ‖ρ−α−1∇I1he‖0

)+ ‖ρ−α−1I1

he‖0

. ‖ρ−α∇(u− I1hu)‖0 + ‖ρ−α∇I1

he‖0 + ‖ρ−α−1I1he‖0

. ‖ρ−α∇(u− I1hu)‖0 + ‖ρ−α−1e‖0

. ‖ρ−α∇(u− I1hu)‖0 + h−1

(‖ρ−α(u− I1

hu)‖0 + h‖ρ−αe‖0

). ‖ρ−α∇(u− I1

hu)‖0 + h−1‖ρ−α(u− I1hu)‖0.

That concludes the H1-estimate. For the remaining L2-bound we use the (II.16) and theH1-result.

First, we consider the weighted error of the standard Galerkin method for smooth ap-proximations and see optimal convergence rates in this case.

Lemma II.7. Let U ∈ H2−α(Ω) ∩ H1(Ω) with 1− λ1 < α < 1. Then, it follows

‖∇(U − Uh)‖0,−α . h‖U‖0,−α and ‖U − Uh‖0,−α . h2‖U‖2,−α.

Proof. In the first part of the proof we concentrate on the H1-bound. This is analogousto the linear case of the Poisson equation [19] and to the second order case in Lemma I.12


of Section I.4. Nevertheless, we carry out the ideas for the diffusion equation.

Let us proceed in an element-by-element fashion, starting with an T not attached tothe singular point:

‖∇(U − Uh)‖0,−α,T . ‖ρ−α∇(U − Uh)‖0,T

. ‖ρ−α∇(u− I1hu)‖0,T + h−1‖ρ−α(u− I1

hu)‖0,T

. ‖∇(U − I1hU)‖0,−α,T + h−1‖U − I1

hU‖0,−α,T

. h‖U‖2,−α,T .

Here, we used that for elements not attached to the singular point we have r−α ≈ ρ−α,the a priori bound of Lemma II.6 and the interpolation estimates of Lemma I.7.

For elements T , which have one vertex as the singular point, we obtain by the trian-gle inequality

‖∇(U − Uh)‖0,−α,T ≤ ‖∇(U − IhU)‖0,−α,T + ‖∇(IhU − Uh)‖0,−α,T . (II.19)

The first term can be treated by the interpolation bound of Lemma I.7. For the secondterm we exploit that it is discrete and therefore, IhU − Uh is linear on each element. Byα < 1 exact integration of ‖r−α‖0,T , r . h on T , and, thus, ρ . h yield

‖∇(IhU − Uh)‖0,−α,T = ‖r−α‖0,T |∇(IhU − Uh)|

. h1−α|∇(IhU − Uh)|

= ‖h−α(∇(IhU − Uh)‖0,T

. ‖ρ−α(∇(IhU − Uh)‖0,T .

Now, we use the triangle inequality, the a priori bound of the ρ-norm in Lemma II.6 andthe fact r ≤ ρ

‖ρ−α∇(IhU − Uh)‖0,T ≤ ‖ρ−α∇(IhU − U)‖0,T + ‖ρ−α∇(U − Uh)‖0,T

. ‖ρ−α∇(IhU − U)‖0,T + ‖ρ−α∇(U − IhU)‖0,T

+ h−1‖ρ−α(U − IhU)‖0,T

. ‖r−α∇(IhU − U)‖0,T + h−1‖r−α(U − IhU)‖0,T .

(II.20)

Combining the bounds for the first term in (II.19) and for the second term in (II.20) wederive by using the interpolation results of Lemma I.7.

‖∇(U − Uh)‖0,−α,T . ‖∇(U − IhU)‖0,−α,T + ‖r−α(∇(IhU − U)‖0,T

+ h−1‖r−α(U − IhU)‖0,T

. h‖U‖2,−α,T .


By summing over all elements T we determine the H1-bound.

For the L2 estimate we consider the dual problem

−div(K∇v) = r−2α(U − Uh) in Ω, u = 0 on ∂Ω.

Note, that the right hand side r−α(U − Uh) is in H0α(Ω), since U ∈ H0

−α(Ω) ⊂ H2−α(Ω).

By Lemma II.2 we have the regularity estimate

‖v‖2,α . ‖r−2α(U − Uh)‖0 = ‖U − Uh‖0,−α.

We determine by the definition of the dual problem, Green’s formula, Galerkin orthogo-nality, Cauchy-Schwarz inequality, the above H1-estimate and the interpolation result ofLemma I.7

‖U − Uh‖20,−α = (U − Uh, r−2α(U − Uh)) = (U − Uh,−div(K∇v)) = a(U − Uh, v)

= a(U − Uh, v − Ihv) ≤ ‖∇(U − Uh)‖0,−α‖v − Ihv‖0,α

. h‖U‖2,−αh‖v‖2,α . h2‖U‖2,−α‖U − Uh‖0,−α.

By dividing by ‖U − Uh‖0,−α we derive our L2-result.

Next, we determine a weighted error bound for the smooth remainder of the modifiedGalerkin approximation. Here, we use the result about the standard Galerkin bound inLemma II.7.

Lemma II.8. Let (C1)- (C3) be satisfied. For U ∈ H2−α(Ω)∩H1(Ω) with 1−λ1 < α < 1

it holds

‖∇(U − Umh )‖0,−α . h‖U‖2,−α and ‖U − Um

h ‖0,−α . h2‖U‖2,−α.

Proof. We start with the weighted L2-bound and, therefore, consider the dual problem

−div(K∇z) = r−2α(U − Umh ) in Ω, u = 0 on ∂Ω.

The right hand side r−2α(U −Umh ) is in H0

α(Ω), since U ∈ H0−α(Ω) ⊂ H2

−α(Ω). By LemmaII.2 we have the regularity estimate

‖z‖2,α . ‖r−2α(U − Umh )‖0,α = ‖U − Um

h ‖0,−α.

The modified approximation of the dual problem is denoted by zmh ∈ V 1h and defined by

ah(zmh , v) = a(z, v) for all v ∈ V 1

h .

We used here the relation

(r−2α(U − Umh ), v) = (−div(K∇z), v) = a(z, v),


where we employed in the last step Green’s formula. Thus, we determine by the definitionof the dual problem, Green’s formula, modified Galerkin orthogonality, standard Galerkinorthogonality and Green’s formula that

‖U − Umh ‖2

0,−α = (U − Umh , r

−2α(U − Umh )) = (U − Um

h ,−div(K∇z))

= a(U − Umh , z) = a(U − Um

h , z − zmh )− ch(Umh , z

mh )

= a(U − Uh, z − zmh )− ch(Uh, zmh ) = a(U − Uh, z)− ch(Uh, zmh )

= (r−2α(U − Uh),−div(K∇z))− ch(Uh, zmh ).

By applying the triangle inequality, Cauchy Schwarz inequality and condition (C2)we get

‖U − Umh ‖2

0,−α ≤ ‖U − Uh‖0,−α‖div(K∇z)‖0,α + ‖∇Uh‖0,Ωh‖∇zmh ‖0,Ωh

We consider each of these terms separately: The first term is bounded by Lemma II.7

‖U − Uh‖0,−α . h2‖U‖2,−α.

For the second term we use the definition of the dual problem

‖div(K∇z)‖0,α = ‖U − Umh ‖0,−α.

Then, by assumption (I.28), triangle inequality, r . h on Ωh and Lemma II.7 we have

‖∇Uh‖0,Ωh . hα‖∇Uh‖0,−α,Ωh . hα(‖Uh − U‖1,−α,Ωh + ‖U‖1,−α,Ωh) . h1+α‖U‖2,−α.

Finally, the fourth term is bounded by

‖∇zmh ‖0,Ωh . ‖∇(z − zmh )‖0,Ωh + ‖∇z‖0,Ωh . h1−α‖z‖2,α . h1−α‖U − Umh ‖0,−α.

We used here the triangle inequality, a priori result of Lemma II.5 and the regularityestimate of the dual problem. These four estimates yield our weighted L2-bound.

For the H1-result we use the triangle inequality, the inverse inequality of Lemma I.9, andthe L2-estimate

‖∇(U − Umh )‖0,−α . ‖∇(U − IhU)‖0,−α + ‖∇(IhU − Um

h )‖0,−α

. ‖∇(U − IhU)‖0,−α + h−1‖IhU − Umh ‖0,−α

. ‖∇(U − IhU)‖0,−α + h−1(‖IhU − U‖0,−α + ‖U − Umh ‖0,−α)

. h2‖U‖2,−α.

This concludes the proof.

Before formulating the general theorem about the a priori error bound of the weightedL2- and H1-norm a lemma is proven with weaker assumptions as necessary.


Lemma II.9. Let (C1)- (C3) hold for ch(·, ·) and, in addition, let

a(si − si,h, sj − sj,h)− ch(si,h, sj,h) = O(h2) (II.21)

be satisfied for 1 ≤ i, j ≤ 2. Then, for f ∈ H0−α(Ω) with 1− λ1 < α < 1 we have

‖U − Umh ‖0,α . h2‖f‖0,−α and ‖∇(U − Um

h )‖0,α . h‖f‖0,−α.

Proof. Let us start with the L2 estimate and consider the dual problem

−div(K∇v) = r2α(u− uh) in Ω, u = 0 on ∂Ω.

The right hand side r2α(u− uh) is in H0−α(Ω), since u ∈ H0

α(Ω) ⊂ H2α(Ω) and denote the

modified Galerkin solution of the dual problem by vmh ∈ Vh. By setting β = −α in theLemma II.2 of the solution of the diffusion equation problem, we obtain

u =2∑i=1

λisi + U and v =2∑j=1

µj sj + V

with the smooth remainder U, V ∈ H2−α(Ω) and the regularity estimates

2∑i=1

|λi|+ ‖U‖2,−α . ‖f‖0,−α and2∑j=1

|µj|+ ‖V ‖2,−α . ‖u− umh ‖0,α.

We bound the L2-error by the plugging in the expansion of u and v

‖u− umh ‖20,α = (u− umh , r2α(u− umh ))

= a(u− umh , v) = a(u− umh , v − vmh )− ch(umh , vmh )

=2∑

i,j=1

λiµi(a(si − si,h, sj − sj,h)− ch(si,h, sj,h))

+2∑j=1

µi(a(U − Umh , sj − sj,h)− ch(Um

h , sj,h))

+2∑i=1

λi(a(si − si,h, V − V mh )− ch(si,h, V m

h ))

+ a(U − Umh , V − V m

h )− ch(Umh , V

mh ).

(II.22)

We consider each of the terms separately. The first one can be bounded by the assumptionthat the energy defect is O(h2) and the two regularity bounds such that

λiµi(a(si − si,h, si − si,h)− ch(si,h, si,h)) . ‖f‖0,−α‖u− umh ‖0,αh2.

For the second terms we apply the modified Galerkin orthogonality, Green’s formula,Cauchy Schwarz inequality, results for the smooth remainder in Lemma II.8 and regularity


estimates

µi(a(U − Umh , si − si,h)− ch(Um

h , si,h))

= µia(U − Umh , si) = µi(U − Um

h ,−div(K∇si))

≤ |µi| ‖U − Umh ‖0,−α‖div(K∇si)‖0,α . ‖u− umh ‖0,αh

2‖U‖2,−α

. ‖u− umh ‖0,αh2‖f‖0,−α.

The third term follows by similar argumentation, i.e.

λi(a(si − si,h, V − V mh )− ch(si,h, V m

h ))

= λia(si, V − V mh ) = λi(−div(K∇si), V − V m

h )

≤ |λi| ‖div(K∇si)‖0,α‖V − V mh ‖0,−α

. ‖f‖0,−αh2‖V ‖2,−α . ‖f‖0,−αh

2‖u− umh ‖0,α.

Finally, we estimate the energy defect of the smooth remainder.

a(U − Umh , V − V m

h )− ch(Umh , V

mh )

= a(U − Umh , V ) = (U − Um

h ,−div(K∇V ))

≤ ‖U − Umh ‖0,−α‖ − div(K∇V )‖0,α

. ‖U − Umh ‖0,−α‖V ‖2,α

. h2‖U‖2,−α‖u− umh ‖0,α

We used here the modified Galerkin orthogonality, Green’s formula, Cauchy-Schwarz in-equality, regularity estimates, the triangle inequality and Lemma II.8. By summarizingthe arguments of the four terms and dividing by ‖u− umh ‖0,α one derives the L2-bound.

For the H1-estimate we employ the triangle inequality, the inverse inequality of LemmaI.9, the L2-result and interpolation estimates of Lemma I.7.

‖∇(U − Umh )‖0,α ≤ ‖∇(U − IhU)‖0,α + ‖∇(IhU − Um

h )‖0,α

. ‖∇(U − IhU)‖0,α + h−1‖IhU − Umh ‖0,α

. ‖∇(U − IhU)‖0,α + h−1(‖IhU − U‖0,α + ‖U − Umh ‖0,α)

. h‖U‖2,−α . h‖U‖0,−α.

Now, we concentrate on the condition (II.21) of Lemma II.9 in particular on the combi-nations (i, j) = (1, 2), (2, 1), (2, 2) which are automatically fulfilled. This is show in thenext two lemmas.

Lemma II.10. Let (C1)- (C3) hold for ch(·, ·), then,

a(s2 − s2,h, s2 − s2,h)− ch(s2,h, s2,h) = O(h2)


Proof. We treat the term by the Cauchy-Schwarz inequality, (C2), the triangle inequalityand similar bound like in the proof of Lemma I.11

a(s2 − sm2,h, s2 − sm2,h)−ch(sm2,h, sm2,h) ≤ ‖s2 − sm2,h‖21 + ‖∇sm2,h‖2

0,Ωh

≤ ‖s2 − sm2,h‖21 + ‖∇(s2 − sm2,h)‖2

0,Ωh+ ‖∇s2‖2

0,Ωh

. ‖s2 − sm2,h‖21 + ‖∇s2‖0,Ωh . h2λ2 = h4−2λ1 = O(h2),

since λ1 < 1 by Lemma II.1. We used here explicit integration of the term ‖∇s2‖0,Ωh anda similar interpolation result as in Lemma I.8.

Lemma II.11. Let (C1)- (C3) hold for ch(·, ·), then,

a(si − si,h, sj − sj,h)− ch(si,h, sj,h) = O(h2)

for (i, j) = (1, 2), (2, 1).

Proof. We may restrict ourselves to the case (i, j) = (1, 2), since the other case followsby symmetry of a(·, ·) and ch(·, ·).

Let us define 1− λ1 < α < 1 and a piecewise linear function Ichs1 by

Ichs1(p) =

s1(p) for all vertices p /∈ Ωh,0 elsewhere on Ωh.

Then, we obtain by the modified Galerkin orthogonality, the standard Galerkin orthogo-nality and Cauchy Schwarz inequality

a(s1 − sm1,h, s2 − sm2,h)− ch(sm1,h, sm2,h) = a(s1, s2 − sm2,h)= a(s1 − Ichs1, s2 − sm2,h) ≤ ‖∇(s1 − Ichs1)‖0,α‖∇(s2 − sm2,h)‖0,−α.

Now, consider both terms separately.

‖∇(s1 − Ichs1)‖0,α . ‖∇(s1 − Ihs1)‖0,α + ‖∇(Ihs1 − Ichs1)‖0,α

. h+ ‖∇(Ihs1 − Ichs1)‖0,α . hαh−1‖Ihs1 − Ichs1‖0

. h+ hα‖s1‖∞ . h+ hαhλ1 = O(h).

For the second term we note that s2 ∈ H2−α(Ω) ∩ H1(Ω) with 1 − λ2 = 1 − (2 − λ1) =

−1 + λ1 < α < 1. Thus, by Lemma II.8 we have

‖∇(s2 − sm2,h)‖0,−α . h‖s2‖2,−α

and obtain the required result.

Thus, we summarize all our above results in the next theorem.


Theorem II.1. Let (C1)- (C3) hold for ch(·, ·) and, in addition, let

a(s1 − s1,h, s1 − s1,h)− ch(s1,h, s1,h) = O(h2) (II.23)

be satisfied for 1 ≤ i, j ≤ 2. Then, for f ∈ H0−α(Ω) with 1− λ1 < α < 1 we have

‖U − Umh ‖0,α . h2‖f‖0,−α and ‖U − Um

h ‖0,α . h‖f‖0,−α.

By Theorem II.1 we can derive optimal convergence for a proper local modification of thebilinear from. It remains to define a sufficient modification. Motivated by [19] we choose

ch(v, w) = γdh(v, w) = γ

∫Ωh

K∇v · ∇w dx

for γ ∈ [0, 1). It is obvious that the modification fulfills the conditions (C1), (C2) and(C3) for any γ ∈ [0, 1). Further, for the choice of γ = 0 we obtain the standard Galerkinmethod. By a similar argumentation as in [19, Lem. 5.1., Lem. 5.2.] a correction param-eter γ ∈ [0, 1) can be guaranteed in γ ∈ [0, 1), which satisfies condition (II.23).

Similar, to the case of re-entrant corners we restricted ourselves to problems with onesingularity, thus, the computation of the correction parameter we can replace s1 by s1

and sm1,h by sm1,h in (II.23). Hence, we can solve the following subproblem: Find γ ∈ [0, 1)such that

g(γ) = a(s1 − sm1,h, s1 − sm1,h)− γdh(sm1,h, sm1,h) = 0, (II.24)

where sm1,h is the modified FE-solution of the problem

−div(K∇u) = 0 in Ω, u = s1 on ∂Ω.

In the case of more singularities we proceed similar as described in Remark I.8. By The-orem II.1 it is not necessary to obtain the exact root of (II.24), but a close (i.e. in O(h2)neighborhood ) approximation. This motivates to derive a nested-Newton method on afamily of uniformly refined meshes Th similar to [41].

Given the initial guess γ0 = 0 ∈ [0, 1) on the coarse mesh T1, we set for l = 0, 1, ....

γl+1 = g′(γl)−1 · (a(s1,h(γl), s1,h(γ

l))− a(s1, s1)) (II.25)

withg′(γ) = dh(s1,h(γ

l), s1,h(γl)).

Note, that s1,h(γ) denotes the modified Galerkin approximation obtained with γ and theindex l the refinement level. On each level typically only one Newton step is carried out.A detail analysis of this algorithm is not a central goal of this thesis and therefore weconcentrate on the experiments derived by it but similar to [41].


II.3.2. Numerical Examples

Let us now illustrate some experiments which verify our theoretical results in the caseof one and several jumps and, finally, compare the energy-correction method with theadaptivity of Section II.2.

II.3.2.1. Energy-Correction for One Singular Point

We consider the homogeneous diffusion equation (II.14) with Dirichlet boundary condi-tion s1 for the 1− 10 jump, 1− 100 jump and the 1− 1000 jump, cf. Figure II.4.

It remains to choice a proper correction domain Ωh before applying the energy-correctionmethod. Motivated by the choice in the case of re-entrant corner singularities we chooseΩh = ωh1 , cf. (I.47). The visualization of the correction domain is depicted in Figure II.8

Figure II.8: Correction domain Ωh in case of one singularity.

For the first example - the 1 - 10 jump - we derive the following correction parameters by afull, i.e., the root of the energy-correction function (II.24), and a one-step Newton method.The values are depicted in Table II.2. In the convergence analysis for the 1 − 10 jump

l γopt γoN1 0.38848 0.413742 0.39879 0.408333 0.40349 0.407614 0.40554 0.407325 0.40643 0.407196 0.40681 0.40714

Table II.2: Correction parameters for the 1 - 10 jump obtained by a full and a one-stepNewton method.

we compare the energy-correction method with different correction parameters accordingto their convergence rate and accuracy in weighted L2-norms for a sequence of uniformlyrefined meshed. The eigenvalue of Sturm-Liouville problem (II.11) is ≈ 0.38996. Hence,we choose the α = 1.05− λ1 ≈ 0.66004 in the weighted L2-norm. In Table II.3 the valuesof the method for the full- and one-step Newton method are presented, both yield optimal


convergence rates and similar errors. Further, we choose two static parameters (0.4, 0.5)for the correction. Here, we observe an improved rate and accuracy in comparison to thestandard Galerkin method for γ = 0, but not optimal rates.

l \ γ 0.0000 rate γopt rate γoN rate 0.4400 rate 0.4500 rate1 3.06090 — 2.72860 — 2.70740 — 2.68590 — 2.67790 —2 1.38890 1.14 0.71951 1.92 0.71091 1.93 0.68923 1.96 0.68482 1.973 0.69770 0.99 0.18359 1.97 0.18191 1.97 0.18110 1.93 0.18554 1.884 0.37922 0.88 0.04636 1.99 0.04605 1.98 0.05608 1.69 0.06395 1.545 0.21420 0.82 0.01170 1.99 0.01163 1.98 0.02441 1.20 0.03090 1.056 0.12297 0.80 0.00295 1.99 0.00294 1.99 0.01333 0.87 0.01743 0.83

Table II.3: Errors 100× ‖u− uh(γ)‖0,α with α ≈ 0.66004; Data of the convergence studyof the PDE (II.14) for the 1 -10 jump.

For the 1 -100 jump and the 1 - 1000 jump the correction parameters of the full- andone-step Newton method are depicted in Table II.4 and Table II.5.

The convergence study for the 1 -100 jump in Table II.6 and 1 - 1000 jump in TableII.7 yields similar results as in the first experiment. For the optimal roots and the param-eters which are obtained by the one-step Newton method, we observe optimal convergencerates of order 2.0 and a high accuracy. The eigenvalue of the 1-100 jump is λ1 ≈ 0.12690and of the 1 -1000 jump λ1 ≈ 0.04025, thus, the weights in the L2-norms are α ≈ 0.92310and α ≈ 0.96745 For the static choices of the correction parameter (0.7960, 0.8000 for the1 - 100 jump and 0.9360, 0.9400 for the 1 -1000 jump), we only obtain optimal convergencerates and an improved accuracy, when the parameters are close enough at the optimalones.

l γopt γoN1 0.79391 0.848932 0.79554 0.799033 0.79606 0.796324 0.79622 0.796305 0.79627 0.796296 0.79629 0.79629



l γopt γoN1 0.93576 0.999992 0.93604 0.940843 0.93609 0.936154 0.93611 0.936115 0.93611 0.936116 0.93611 0.93611


l \ γ 0.0000 rate γopt rate γoN rate 0.7960 rate 0.8000 rate1 1.47840 — 1.22470 — 1.20660 — 1.22390 — 1.22260 —2 0.85842 0.78 0.32825 1.90 0.32724 1.88 0.32812 1.90 0.32697 1.903 0.58509 0.55 0.08423 1.96 0.08420 1.96 0.08424 1.96 0.08392 1.964 0.43313 0.43 0.02130 1.98 0.02130 1.98 0.02131 1.98 0.00215 1.965 0.33299 0.38 0.00537 1.99 0.00537 1.99 0.00537 1.99 0.00063 1.786 0.26147 0.35 0.00135 1.99 0.00135 1.99 0.00137 1.97 0.00030 1.05

Table II.6: Errors 100 × ‖u − uh(γ)‖0,α with α ≈ 0.92; Data of the convergence study ofthe PDE (II.14) for the 1 -100 jump.

l \ γ 0.0000 rate γopt rate γoN rate 0.9360 rate 0.9400 rate

1 0.51892 — 0.41774 — 0.41093 — 0.41771 — 0.41725 —2 0.32494 0.68 0.11332 1.88 0.11290 1.86 0.11332 1.88 0.11297 1.893 0.24001 0.44 0.02917 1.96 0.02917 1.95 0.02917 1.96 0.02912 1.964 0.19177 0.32 0.00738 1.98 0.00738 1.98 0.00738 1.98 0.00763 1.935 0.15902 0.27 0.00186 1.99 0.00186 1.99 0.00186 1.99 0.00262 1.546 0.13485 0.24 0.00047 1.99 0.00047 1.99 0.00047 1.99 0.00176 0.57

Table II.7: Errors 100 × ‖u − uh(γ)‖0,α with α ≈ 0.96; Data of the convergence study ofthe PDE (II.14) for the 1 -1000 jump.


II.3.2.2. Energy-Correction for Several Singular Points

In the last section we verified our theoretical result for simple examples. Going forwardwe consider two more advanced example of the jumping coefficient problem in Ω = (0, 4)2

−div(K∇u) = 0 in Ω,

u = 0 on Γ1,

u = 1 on Γ2,

∂u

∂n= 0 on Γ3,

(II.26)

with Γ1 = [0, 4]× 4, Γ2 = [0, 4]× 0, Γ3 = 0, 4 × [0, 4].

In the first experiment we study a problem with several singularities but the same shape,i.e. for the i = bxc and j = byc we set

K =

1000 if (−1)i+j > 0,1 if (−1)i+j < 0,

(II.27)

also compare Figure II.9. We identify 9 singularities which are corrected as depicted inFigure II.9. Due to the construction of the coefficient K each of the 9 singularities has thesame first eigenvalue as in the 1 - 1000 example, i.e. λ1 ≈ 0.04025. We can simply choosethe 1 − 1000 jump as local problem, and, thus, we set γ = 0.93611 for all mesh levels.First, we compute a reference solution for our convergence study with a high resolution,since we have no exact solution at hand. In Figure II.10 we plotted once the approx-imation obtained by the energy-correction method and the other time by the standardGalerkin approach. By the plot one can already see the higher accuracy of the energy-correction approximation, since at the 9 singularity points a steep ascent can be observed.

For our analysis we weight the circle with radius 1 of each singularity, i.e.

w = Π9c=1 min1, rαcc , with rc = |x− xc|, (II.28)

where xc denotes the coordinates of the singularity. In this case the weighted L2 norm is

‖uh − u‖0,w = ‖w · (uh − u)‖0.

Since all singularities are the same, even the exponents are, namely, αc ≈ 0.96475.


1 1000 1 1000

1000 1 1000 1

1 1000 1 1000

1000 1 1000 1

Figure II.9: Defintion of coefficient K, the initial mesh T1, correction domains.

In the convergence analysis of Table II.8 we see that for the energy corrected approxima-tion we derive optimal convergence rates and a more accurate solution in the weightedL2-norm (II.28). A poor rate is observed for the standard Galerkin method.

l \ γ γopt rate 0.00000 rate1 0.50254 — 1.15720 —2 0.09690 2.07 0.92582 0.323 0.02472 1.97 0.77083 0.264 0.00636 1.96 0.65581 0.23

Table II.8: Error 100 · ‖u − uh‖0,w; Data of the convergence analysis of problem (II.26)and jumps (II.27).


Figure II.10: FE-solution of the problem (II.26) and jumps (II.27); Top figure: Energycorrected approximation; Below figure: Standard Galerkin approximation;Both with 262,144 DOFs.


Further, we consider the problem (II.26) with more different jumps. The distribution ofthe coefficient K is depicted in Figure II.11, the initial mesh and the correction domainsare the same as for the above example, cf. Figure II.9.

1000 10 100 1

10 1000 1 100

100 1 1000 10

1 100 10 1000

Figure II.11: Defintion of coefficient K.

In a preprocess we determine for each of the 9 singular points the first eigenvalue, thecorrection parameter and local weights for the weighted L2 norm. The values are presentedTable II.9.

Singular point λ1 γopt αc(1, 1) 0.12690 0.79625 0.92310(2, 1) 0.21511 0.65599 0.83489(3, 1) 0.12690 0.79625 0.92310(2, 1) 0.21511 0.65599 0.83489(2, 2) 0.04025 0.93611 0.96475(2, 3) 0.21511 0.65599 0.83489(3, 1) 0.12690 0.79625 0.92310(3, 2) 0.21511 0.65599 0.83489(3, 3) 0.12690 0.79625 0.92310

Table II.9: Data to the 9 singular point of problem (II.26) and with different jumps in K.

Similar as in the above example we have no analytic solution at hand, thus, we computea reference solution with 262,144 DOFs. In Figure II.12 the approximations obtained bythe energy-correction method and standard Galerkin method are depicted. One alreadyobserves that the energy corrected solution has a much higher accuracy than the standardone, since the ascent at the singular point is steeper.In Table II.10 we present the convergence analysis for this experiment and observe thatthe energy-correction method yield optimal convergence rates and high accurate solutions.Note, that the error is measure in the weight norm (II.28) and the error is evaluated bytreating the reference solution as the analytic one.


Figure II.12: FE solution of the problem (II.26) and different jumps; Top figure: Energycorrected approximation; Below figure: Standard Galerkin approximation;Both with 262,144 DOFs.

l \ γ γopt rate 0 rate1 0.62844 — 1.89360 —2 0.16315 1.95 1.77170 0.103 0.04206 1.96 1.65830 0.104 0.01061 1.99 1.55090 0.10

Table II.10: Error 10 · ‖u − uh‖0,w; Data of the convergence analysis of problem (II.26)and different jumps in K.


II.3.2.3. Energy-Correction Method vs. Adaptivity

In the two above sections we verified that by the energy-correction method the poorconvergence rate of interface problems can be improved and is even optimal. In Sec-tion II.2 similar effect could be observed for the adaptive method. In this subsection wewill compare both methods according to their convergence to the exact solution. We ne-glect computation cost and time of both approaches and only concentrate on the accuracy.

Let us again consider the homogeneous interface problem (II.14) with Dirichlet data andanalytic solution s1 for the 1 - 10 jump, the 1 - 100 jump and the 1 - 1000 jump, cf. FigureII.4. The convergence plots are depicted in Figure II.3.2.3 for the 1 - 10 jump, FigureII.3.2.3 for the 1 - 100 jump and Figure II.3.2.3 for the 1 - 1000 jump.

101 102 103 104 105

DOFs

10−6

10−5

10−4

10−3

10−2

10−1

100

Figure II.13: Comparison of the energy-correction method and adaptivity for the 1 - 10jump measured in the weighted L2-norm with α ≈ 0.66; Green: Energy-correction; Blue: Adaptivity; Dashed red: h2 reference line.

101 102 103 104 105 106

DOFs

10−6

10−5

10−4

10−3

10−2

10−1

100



101 102 103 104 105 106

DOFs

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100


In all three cases the approximation obtained by the energy-correction method is moreaccurate than the one of the adaptive approach. It can also be observed that the per-formance of the energy-correction in comparison to the adaptive methods increase as thejump does, i.e., as more singular the problem becomes. One reason for this is that theadaptivity refines the mesh at the singularity, where the weighted norm relaxes the solu-tion. One can prefer the usage of the energy-correction method instead of adaptivity inthe case of jumping coefficients by neglecting the computational cost and time.When we reconsider the problems and their computational cost and time, we need manyand fine refining steps to obtain optimal convergence optimal and an accurate solution,thus, the computational cost and time become very expansive, since we have to resolvethe problem very often. The reason for this is that the refining takes only place in a smallneighborhood of the singularity.Despite that for the energy-correction method we have to determine a proper correctionparameter, which is derived by a pre-process that uses the singular function. Thus, weneed the explicit formula for the singular function which depends on the problem howexpansive the computations are and rises by the number of singularities in the problem.

In this section we saw that in the case of interface problems optimal convergence ratescan be obtained by using the energy-correction methods. In Theorem II.1 we proved theoptimal approximation result in the weighted L2- and H1-norm. Then, we introduced anexplicit modification of the bilinear form which we applied in our numerical experimentsand stated a nested-Newton method for the calculation of the correction parameter. Weverified our theoretical results by numerical examples and compared the convergence ofthe energy-correction method with different parameters. The parameters obtained bythe one-step Newton method are sufficient for the correction, also static parameters canbe used. Further, we considered more advanced examples for the verification. Finally,we compared the adaptivity with the energy-correction method and saw that a higheraccuracy is obtained by the energy-correction in case of weighted norms.

Part III.Conclusion and Outline

In this thesis we considered the Poisson problem in domains with re-entrant corners andthe diffusion equation with jumping coefficients. In both cases we saw that the solutioncan be extended into singular functions and a smooth remainder. In the standard ap-proach of the FE method these singular functions cause problems in the accuracy of theapproximation of the solution, and a reduced rate in the convergence analysis. This iscalled the pollution effect.

In the first part we considered the Poisson problem. Using the adaptive method werecovered optimal rates and high accurate approximations for our experiments indepen-dent of the initial triangulation and the order of finite element space V p

h . Another methoddealing with this effect is the energy correction method, which we studied for linear andsecond order FE. For the linear case, we obtained h2 convergence rates for optimal pa-rameters as well as parameters computed by a one-step Newton method in weightedSobolev norms for symmetric and non-symmetric triangulations. For the symmetric casea one-parameter correction is sufficient for the correction at the singular point. Fornon-symmetric meshes with re-entrant corner angle > 3

2π, two correction parameters are

necessary, which modifies the bilinear form in a one- or two-element layer around the sin-gularity. A comparison between the adaptivity and the energy correction methods shows,by neglecting the computational cost and time, that the energy correction yields a higheraccurate approximation than the adaptive method in weighted L2 norms.For second-order finite elements, we started with the theory of the energy correctionmethod and showed, under certain correction conditions, optimal a priori estimates inweighted Sobolev norms. Also the numerical experiments verified the optimal conver-gence and exact approximations. The one-step Newton method is sufficient to computethe correction parameters. The parameters modify the bilinear form in a two-elementlayer situated at the singular point. The energy correction method yield more accurateapproximation than the adaptivity when we neglect computational cost and time.

In the second part we considered the diffusion equation with jumping coefficients. Weused a residual-based adaptive method to recover optimal convergence rates with a veryfine refinement at the singularity necessitating many re-computations. For the energy cor-rection, we proved an optimal approximation result and stated an explicit modificationof the bilinear form which varies the elements attached to the singular point sufficiently.Further, a one-step Newton method was developed. Through experimentation, we showedthat the by parameters provided by the one-step Newton method are good approximationsof the optimal values and yield optimal convergence rates. Furthermore, the examplesshowed that optimal convergence rates and high accurate approximations are obtained bythe energy correction method for jumping coefficients. Finally, we compared the adap-tivity with the energy correction and saw that in weighted Sobolev norms the energycorrection yields a more accurate approximation than the adaptive approach for the samenumber of DOFs, when neglecting computational costs and time. Therefore, the use of

124

the energy correction method can be recommended in the case of jumping coefficients.

To summarize using the energy correction we obtained optimal convergence rates for thePoisson equation for the linear and second-order FE method for domains with re-entrantcorners. Furthermore, for the diffusion equation in the case of jumping coefficients wealso achieved optimal rates using the energy correction method.

Future studies of the energy correction method might apply it to general elliptic prob-lems, linear elasticity and the Navier-Stokes equations. Also a closer consideration ofnested-Newton methods and their theoretical study in the case of more energy-defectconditions might be of interest. Finally, an extension from two dimensional problemsto three dimensions is an important direction for future study of the energy correctionmethod.

References

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[2] M. Ainsworth and J. Oden. A posteriori error estimation in finite element analysis.Wiley, 2000.

[3] T. Apel, A. Sandig, and J. Whiteman. Graded mesh refinement and error estimatesfor finite element solutions of elliptic boundary value problems in non-smooth do-mains. Math. Methods Appl. Sci., 19:63–85, 1996.

[4] I. Babuska and W.C. Reinboldt. Error estimates for adaptive finite element compu-tations. SIAM Journal Numerische Analysis, 15:735–754, 1978.

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