Coupling of the Finite Volume Method and the Boundary

89069 Ulm | Germany

Fakultät für Mathematik und Wirtschaftswissenschaften

Institut für Numerische Mathematik

Dissertation

Coupling of the Finite Volume Method

and the Boundary Element Method

Theory, Analysis, and Numerics

zur Erlangung des Doktorgrades Dr. rer. nat.

der Fakultät für Mathematik und Wirtschaftswissenschaften

der Universität Ulm

vorgelegt von

Christoph Erath

aus Schlins, Österreich

April 2010

Amtierender Dekan: Prof. Dr. Werner Kratz (Universität Ulm, D)

1. Gutachter: Prof. Dr. Stefan A. Funken (Universität Ulm, D)

2. Gutachter: Prof. Dr. Karsten Urban (Universität Ulm, D)

3. Gutachter: Prof. Dr. Dirk Praetorius (Technische Universität Wien, A)

4. Gutachter: Prof. Dr. Helmut Harbrecht (Universität Stuttgart, D)

Tag der Promotion: 21. Juli 2010

Abstract

We develop a discretization scheme for the coupling of the finite volume method and the

boundary element method in two dimensions, which describes, for example, the transport

of a concentration in a fluid. The discrete system maintains naturally local conservation. In

a bounded interior domain we approximate a diffusion convection reaction problem either

by the finite volume element method or by the cell-centered finite volume method, whereas

in the corresponding exterior domain the Laplace problem is solved by the boundary

element method. On the coupling boundary we have appropriate transmission conditions.

A weighted upwind scheme guarantees the stability of the method also for convection

dominated problems. We show existence and uniqueness of the continuous system and

provide an a priori analysis for the coupling with the finite volume element method. For

both coupling systems we derive residual-based a posteriori estimates, which give upper

and lower bounds for the error between the exact solution and the approximate solution.

These bounds measure the error in an energy (semi-) norm and are robust in the sense

that they do not depend on the variation of the model data. The local contributions of the

a posteriori estimates are used to steer an adaptive mesh-refining algorithm. Numerical

experiments show that our adaptive coupling is an efficient method for the numerical

treatment of transmission problems, which exhibit local behavior.

Kurzfassung

Wir entwickeln ein Diskretisierungsschema für die Kopplung der Finiten Volumen Meth-

ode mit der Randelemente Methode für den zweidimensionalen Raum. Ein Modellproblem

hierfür ist der Transport einer Konzentration in einer Flüssigkeit. Lokale Konservativität

bleibt dabei auch für das diskrete System erhalten. Wir approximieren ein Diffusions-

Konvektions- Reaktions- Problem in einem beschränkten Innengebiet entweder mit der

Finiten Volumen Elemente Methode oder mit der zellenorientierten Finiten Volumen

Methode. Im dazugehörigen unbeschränkten Außenraum lösen wir das Laplace Problem

mit der Randelemente Methode, während wir auf dem Kopplungsrand geeignete Kop-

plungsbedingungen definieren. Mit Hilfe einer gewichteten Upwind Methode garantieren

wir die Stabilität des Systems auch für konvektionsdominante Probleme. Wir zeigen Ex-

istenz und Eindeutigkeit des Modellproblems und beweisen a priori Aussagen für die Kop-

plung mit der Finiten Volumen Elemente Methode. Für beide Kopplungsmethoden leiten

wir residualbasierte a posteriori Fehlerschätzer her, welche eine obere und eine untere

Schranke für den Fehler zwischen der exakten und approximativen Lösung liefern. Diese

Schranken messen den Fehler in einer Energie(halb)norm und sind robust gegenüber Vari-

ationen der Modelldaten. Die lokalen Beiträge der a posteriori Abschätzungen können zur

Steuerung eines adaptiven Algorithmus verwendet werden. Numerische Beispiele zeigen

schließlich, dass unsere adaptive Kopplung ein effizientes Verfahren zur Behandlung von

Problemen ist, deren Lösungen lokales Verhalten aufweisen.

Version: 28. April 2010

c© 2010 Christoph Erath

Typeset: LATEX 2ε

Contents

Introduction iii

1 Analytical Basics and Notation 1

1.1 Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Boundary Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.1 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.2 Representation Formula and Calderón System . . . . . . . . . . . . . 11

1.3 Triangulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.3.1 The Primal Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.3.2 The Dual Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.3.3 Normal, Tangential Vectors and Patches . . . . . . . . . . . . . . . . 17

1.4 Discrete Spaces on the Primal and the Dual Mesh . . . . . . . . . . . . . . 18

1.5 Some Inequalities and Definitions . . . . . . . . . . . . . . . . . . . . . . . . 21

2 The Coupling Problem 23

2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2 The Weak Form of the Model Problem . . . . . . . . . . . . . . . . . . . . . 26

2.3 Coupling with the Finite Volume Element Method . . . . . . . . . . . . . . 29

2.3.1 Discretization in a Finite Volume Element Sense . . . . . . . . . . . 29

2.3.2 The Discrete Problem with an Upwind Approximation . . . . . . . . 31

2.3.3 An A Priori Convergence Result . . . . . . . . . . . . . . . . . . . . 35

2.4 Coupling with the Cell-Centered Finite Volume Method . . . . . . . . . . . 44

2.4.1 Discretization in a Cell-Centered Finite Volume Sense . . . . . . . . 44

2.4.2 Approximation of the Boundary Values and the Fluxes . . . . . . . . 47

3 A Posteriori Error Estimates 53

3.1 Estimation for the Coupling with the Finite Volume Element Method . . . 53

3.1.1 The Piecewise Constant Diffusion Coefficient and Quasi-Monotonicity 55

3.1.2 Residual-Based Error Estimation . . . . . . . . . . . . . . . . . . . . 59

3.1.3 Reliability of the Error Estimator . . . . . . . . . . . . . . . . . . . . 65

3.1.4 Efficiency of the Error Estimator . . . . . . . . . . . . . . . . . . . . 71

3.2 Estimation for the Coupling with the Cell-Centered Finite Volume Method 83

i

ii Contents

3.2.1 The Morley Interpolant . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.2.2 Reliability of the Error Estimator . . . . . . . . . . . . . . . . . . . . 90

3.2.3 Local Efficiency of the Error Estimator . . . . . . . . . . . . . . . . 93

4 Numerical Experiments 95

4.1 Implementation Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.1.1 The Discrete Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.1.2 Implementation of the Error Estimators . . . . . . . . . . . . . . . . 100

4.1.3 Implementation of the Energy Norm . . . . . . . . . . . . . . . . . . 101

4.1.4 Adaptive Algorithm and Mesh-Refinement . . . . . . . . . . . . . . . 102

4.2 Examples for the Coupling with the Finite Volume Element Method . . . . 103

4.2.1 Diffusion Reaction Problem with a Generic Singularity . . . . . . . . 103

4.2.2 Diffusion Convection Problem . . . . . . . . . . . . . . . . . . . . . . 110

4.2.3 Convection Dominated Problem . . . . . . . . . . . . . . . . . . . . 113

4.3 Examples for the Coupling with the Cell-Centered Finite Volume Method . 118

4.3.1 L-Shaped Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

4.3.2 Diffusion Reaction Problem . . . . . . . . . . . . . . . . . . . . . . . 121

4.3.3 Problem with Convection . . . . . . . . . . . . . . . . . . . . . . . . 124

Conclusion 127

Bibliography 129

Index of Notation 135

List of Figures 141

Curriculum Vitæ 143

Introduction

The problem we explore in this thesis is if and how we can simulate a boundary value

problem with a possibly convection dominated elliptic equation in a bounded interior do-

main while the problem in the exterior domain is governed by a diffusion equation.

This problem can describe the transport of a concentration in a fluid or the heat prop-

agation in a bounded interior domain by a diffusion convection reaction equation and a

homogeneous diffusion process in an unbounded exterior domain that can only be solved

by a numerical scheme. Therefore, a method which ensures local conservation and stabil-

ity with respect to the convection term is preferable.

The finite volume method is a well-adapted method for the discretization of various par-

tial differential equations in bounded domains. In particular, it is well-established in the

engineering community (fluid mechanics) because of its conservative properties of the nu-

merical fluxes and the natural formulation of an upwind scheme, which ensures stability for

the convection part. In addition, it is stable with respect to a reaction dominated problem

and it is applicable to problems with inhomogenous material properties. The boundary

element method can, however, be applied to the most important linear partial differential

equations with constant coefficients in bounded and also in unbounded domains and in

a sense it features local conservation, as well. The coupling of the finite volume method

and the boundary element method combines the advantages of both methods. While a

diffusion convection reaction process is modeled by the finite volume method, the pure

diffusive transport (in a possibly unbounded domain) is solved by using the boundary

element method. We stress that, for example, the finite element method does not provide

local conservation of numerical fluxes in general.

There exist several different finite volume schemes. In this thesis we develop discretization

schemes for the coupling of the finite volume element method and the boundary element

method and for the coupling of the cell-centered finite volume method and the boundary

element method in two dimensions. For both coupling methods we provide a posteriori

estimates of residual type, which allow adaptive mesh-refinement in order to efficiently

treat problems that exhibit local behavior. Numerical experiments show the applicability

of our theoretical results.

iii

iv Introduction

ΩC

Ω

b

b

nΓout

Γin

Figure I. Domains and notation for the model problem with the boundary Γ = Γin ∪ Γout.

Model Problem

Let Ω ⊂ R2 be a bounded and connected domain with polygonal Lipschitz boundary Γ,

see Figure I. We call Ω the interior domain. In Ω we consider the following stationary

diffusion convection reaction problem: Find u such that

div(−A∇u+ bu) + cu = f in Ω,

where A is a symmetric diffusion matrix, b is a possibly dominating velocity field, c is a

reaction function and f is a source term. In the complement ΩC = R2\Ω, the so called

exterior domain, we seek uc such that

∆uc = 0 in ΩC

together with the radiation condition

uc(x) = a∞ + b∞ log |x| + o(1) for |x| → ∞.

We can fix either a∞ ∈ R or b∞ ∈ R and calculate the other one, that means uc behaves

asymptotically like the fundamental solution of the Laplace operator, see [58]. Note that

the radiation condition ensures existence and uniqueness of the problem. Both problems

are coupled on the interface Γ = ∂Ω = ∂ΩC , which is closed and has positive surface

measure. The coupling boundary Γ is divided in an inflow and outflow part, namely

Γin :=x ∈ Γ

∣∣b(x) · n(x) < 0

and Γout :=x ∈ Γ

∣∣b(x) · n(x) ≥ 0, respectively, where

n is the normal vector on Γ pointing outward with respect to Ω. We allow prescribed

jumps u0 and t0 on Γ and demand therefore

u = uc + u0 on Γ,

(A∇u− bu) · n =∂uc

∂n+ t0 on Γin,

(A∇u) · n =∂uc

∂n+ t0 on Γout.

Introduction v

x1

x2

−0.25 −0.1 0 0.1 0.25−0.25

0

0.05

0.2

0.25

(a) Mesh, vector field b and volume force f . (b) Adaptively generated mesh with

4661 elements.

(c) Interior and exterior solution. (d) Contour lines.

Figure II. Data and solution for a convection dominated problem: Subfigure (a) shows the

domain Ω, which is L-shaped. The convection field b has a clockwise rotation and we have a

volume force f in the gray shaded interior rectangle. With the coupling of the finite volume

element method and the boundary element method we generated the adaptive refined mesh

in (b), the interior and exterior solution in (c) and the contour lines in (d).

In Figure II we see the solution of the coupling of the finite volume element method and

the boundary element method for a diffusion convection reaction problem in the interior

domain, where we use a weighted upwind method to discretize the convection flux. The

L-shaped domain is given in Figure II(a). We choose a diffusion matrix A = αI with

α = 0.2, a constant reaction function c = 0.01, the jumps u0 = 0, t0 = 0 and the radiation

condition at infinity b∞ = 0. The volume force f is 5 in the interior (gray) rectangle of Ω,

otherwise 0. The convection vector b (velocity field), see Figure II(a), is given as solution

of a Stokes problem on the same domain with div b = 0, a constant volume force 1 and

homogeneous boundary condition. The rotation is clockwise and supx∈Ω

|b(x)| = 32, thus we

have a convection dominated problem in the interior domain. In Figure II(b) we see an

adaptively generated mesh with 4661 elements. The refinement follows along the strongest

vi Introduction

K

(a) Cell-vertex approach.

K

(b) Cell-centered approach.

Figure III. The meshes for the finite volume element method (a) and the cell-centered finite

volume method (b) with the corresponding control volume K (gray). The filled circles present

the location of the unknowns.

transport (through b) starting from the rectangle (f = 5). We observe a finer resolution

also in the reentrant corner, where we expect to have a singularity. In Figure II(c)–(d)

we plot the solution and the contour lines, respectively. We see that the transport follows

along b and that the solution is continuous on the boundary Γ. Since we only consider a

diffusion process in the exterior domain, the contour lines are circular outside of Ω.

The Finite Volume Method and the Boundary Element Method

A finite volume method works just like a finite element method on a triangulation or mesh

of the problem domain Ω. If the mesh is refined, we expect a better approximation of the

solution. This method is locally conservative because it is based on a balance approach.

For each control volume K ⊂ Ω of the mesh we get the balance equation

∫

∂K(−A∇u+ bu) · n ds+

∫

Kcu dx =

∫

Kf dx,

where n denotes the unit normal vector pointing outward of K. We discretize the diffusive

and convective fluxes on the boundary ∂K and the reaction term with respect to the dis-

crete unknowns, which leads to a system of linear equations. In this thesis, we distinguish

two types of finite volume schemes on a mesh T , which always consists of triangles.

First, each discrete unknown is associated to a node of the mesh T . This is known as

the cell-vertex approach, where we will consider the finite volume element method or

box method in this thesis. Here, we build a dual mesh T ∗ from the primal mesh T .

The control volume K coincides with an element of the dual mesh T ∗, see Figure III(a),

where the gray area represents the control volume K and the filled circles show the lo-

cation of the unknowns. We approximate u by T -piecewise affine globally continuous

functions. To guarantee the stability of the method also in the convection dominated

case, we approximate the convective flux by a weighted upwind scheme [65, 52]. There

exist many publications on the analysis of the finite volume element method, which are

based on its relationship to the finite element method for piecewise affine linear functions,

see [7, 49, 55, 78].

Introduction vii

For the second type the unknown is located in the interior of an element of T . This is

known as the cell-centered approach. In the cell-centered finite volume method the control

volumes K coincide with the elements of T , see Figure III(b). Then u is approximated

by a T -piecewise constant function. We approximate the convective flux by a weighted

upwind scheme [65, 52], as well. The approximation of the diffusion flux is more com-

plicated. The classical choice is based on an admissibility condition of the triangulation

T in the sense of [45], where a detailed representation and analysis of the cell-centered

finite volume method on admissible meshes for elliptic equations can be found. However,

locally refined meshes are usually not admissible and therefore another choice has to be

made, namely the diamond-path technique for triangles. We stress that this only has been

mathematically analyzed for rectangular meshes in [34, 35].

In the exterior domain we will apply the boundary element method. In order to do this,

we rewrite the exterior problem into an integral equation on the boundary Γ, which is a

representation by the Cauchy data (the trace and the conormal derivative of uc) of the

model problem. Then the discretization follows from a Galerkin scheme and leads to the

boundary element method, see also [67, 71]. On the coupling boundary we use piecewise

affine globally continuous functions to approximate the trace of uc and piecewise constant

functions to approximate the conormal derivative of uc.

Discretization of the Model Problem

In the literature one can find various forms of the coupling of the finite element method

and the boundary element method in order to solve the above problem. Usually, the finite

element method is applied for the interior problem in Ω and the boundary element method

is used to solve the exterior problem in ΩC . The first significant results concerning the

theoretical justification of a coupling procedure can be found in [12, 51]. The convergence

result of this type was recently extended to Lipschitz coupling interfaces in [68]. But a

disadvantage of this coupling method is that it produces a non-symmetric coefficient matrix

of the system of linear equations, even if a symmetric discretization scheme (Galerkin

scheme) is applied. A coupling method, which avoids this contradiction, can be found, for

example, in [31]. For an overview of further theoretical developments in context of the

coupling of the finite element method and the boundary element method we refer to [50].

To the best of the author’s knowledge, there is no theoretical justification for any coupling

of a finite volume method and the boundary element method available yet.

A Posteriori Error Estimates

The numerical discretization of partial differential equations may lead to linear systems

with millions of unknowns. Here, the question arises whether the available computational

resources are used efficiently. This becomes even more important when the solution of

a problem exhibits local behavior, for example due to singularities, discontinuities of the

viii Introduction

coefficients or localized sources. If the nature and the position of this singular behavior

are known a priori, the mesh-refinement can reflect this. Otherwise, we have to determine

these regions with an a posteriori error estimator. Reliability and local efficiency of such

an estimator ensure an upper and lower bound of the error up to a certain constant, re-

spectively. The estimator involves only known data, more precisely it is built local from

the discrete solution and the model data. An adaptive mesh-refining algorithm generates

a new finer mesh based on the estimates of the current mesh.

For the coupling of the finite element method and the boundary element method a posteri-

ori error estimates are well-known, e.g. [25, 14] for the coupling with conforming elements

and [19] for non-conforming elements to mention only a few but not all. In these works

results and ideas from the theory of the finite element method and boundary element

method are used. A posteriori estimates for the finite element method are well-known

since the pioneering work [6], see also the survey in [73, 2]. One particular issue is to

show the robustness of the estimates with respect to the model data A, b and c. Robust-

ness means that the constants in the bounds do not depend on the values or ratio of the

model data. In the context of the finite element method such estimators can be found

in [9, 63, 47] for discontinuous diffusion coefficients and [4, 74] for convection or reaction

dominated problems. A first general framework for a posteriori error estimates for the

boundary element method can be found in [24].

In this work we take the same approach. More precisely we combine results from the the-

ory of the finite volume method and the boundary element method. An early work for an

a posteriori error analysis for the finite volume element method on Voronoi meshes can be

found in [3]. In [22] a similar path is used, namely the relation and similarities of the finite

volume element method and the finite element method in order to derive an estimator,

but unfortunately the estimates are not robust with respect to the model data. The a pos-

teriori theory for cell-centered schemes is much less developed. We mention [60, 42] and

emphasize [61, 76] for robust estimates with respect to the model data in a natural energy

norm, where the piecewise constant solution is replaced by a post-processed approximation

to gain an estimator. This motivates us to develop a posteriori error estimates for the

coupling of the finite volume method and the boundary element method. In particular,

we focus on bounds for the error measured in the natural energy (semi-) norm and the

boundary element energy norm to achieve robustness.

Outline and Main Results

This thesis consists of four chapters.

In the first chapter we summarize the necessary analytical background of function spaces

and introduce the notation that is used throughout this thesis. In particular, we give a

brief overview of boundary integral equations applied to the Laplace equation in both the

interior and exterior domain. Furthermore, we define a regular triangulation T of triangles

Introduction ix

and construct the dual mesh T ∗.

The second chapter defines the coupling problem in a weak sense with a special focus on

a convection field in Ω. In the weak representation of the interior problem we replace

the conormal derivative of u by the conormal derivative of uc. For our model problem

the convection vector b divides the boundary naturally in an inflow and outflow part. If

b = (0, 0)T we get the coupling conditions stated in [25]. The Calderón system presents

the exterior problem in a weak form by an integral equation ansatz, where we replace the

exterior trace of uc by u. In contrary to the weak formulations in [50, 25] we introduce

the trace of uc as an unknown, which will be mandatory for the error analysis of the dis-

crete system in a finite volume sense. A similar approach was used in [18] for the Poisson

problem approximated by non-conforming finite elements. We use the Poincaré Steklov

operator in a similar way as in [25] to define another equivalent weak formulation, which

allows us to show existence and uniqueness by the Lax-Milgram Theorem. If we replace

the interior weak form through a finite volume element discretization and simultaneously

approximate the boundary element part by the usual Galerkin method (namely replace

the continuous function spaces by discrete ones), we get a 3 × 3 discrete block system of

linear equations. For this coupling we prove a convergence result and for sufficient regular

solutions an a priori estimate of order O(h), where h is the maximal mesh size. The analy-

sis makes use of a comparison of the standard discrete finite element and the finite volume

element bilinear form. After that the Galerkin orthogonality for the boundary element

method part and some standard estimates conclude the proof. We use the convergence

result to gain the existence and the uniqueness of a discrete solution. We also define a

discrete system with a weighted upwind scheme in the sense of [65, 52] and then use an

estimate between the standard and the upwind discrete finite volume element bilinear form

to get convergence and an a priori estimate. Existence and uniqueness follow again from

this result.

The situation for the coupling with the cell-centered finite volume method is more com-

plicated. Keeping in mind that we need an approximation on the boundary nodes for the

diamond-path (diffusion fluxes) and the Morley interpolant (a post-processed interpolant

for the a posteriori estimator), we interpolate appropriate values on the boundary node

from the piecewise constant interior solution and the approximated conormal derivate of

uc. We refer to the work of the author [42], where a similar approach is used to handle

Neumann boundary conditions. We approximate the convection fluxes by a weighted up-

wind scheme. The resulting system of linear equations is now a 4×4 block-system instead

of a 3 × 3 system for the coupling with the finite volume element method. There is no

a priori analysis known for this coupling.

The regularity assumptions for attaining a convergence rate of O(h) are usually not met

in practice. Thus, the third chapter provides a posteriori estimates of residual type for

both coupling systems, which are suitable to steer an adaptive mesh-refining algorithm to

x Introduction

recover the optimal order of convergence in the case of non-smooth solutions. We prove

an a posteriori error estimator for the coupling of the finite volume element method and

the boundary element method for a T -piecewise constant diffusion coefficient, which has

a quasi-monotone distribution, and a possibly dominating convection vector or a possibly

dominating reaction function. The constants in the upper and lower bound are robust

with respect to the ratio of the piecewise constant α and with respect to a small diffu-

sion compared to the convection field b or the reaction term c. This result can also be

used for the pure finite volume element method. Following the ideas of [22] we prove

an L2-orthogonality property of the residual for our system. Furthermore, we use a ro-

bust interpolation operator with respect to the piecewise constant diffusion coefficient,

see [63, 47]. Roughly speaking, the analytical idea is to extend an estimate (coming from

the energy norm error) by the L2-orthogonality property and by this interpolation oper-

ator, which further allows estimates similar to the well-known a posteriori error analysis

from the context of the finite element method and estimates between discrete error terms.

With the Galerkin orthogonality and localization techniques for the Sobolev norms on the

boundary of [15], we can prove a robust a posteriori estimator with respect to the model

data. Besides the usual residual and a normal jump term, where the part of the normal

jump on the coupling boundary is replaced by the coupling condition, we additionally have

a term from the boundary element method measuring the error of the Cauchy data through

the Calderón system. A tangential jump measures the error in the tangential direction on

the coupling boundary. If we apply an upwind method, we also derive a quantity, which

measures the amount of upwinding. We give an alternative proof to [22], which does not

use a continuous bilinear form of the finite volume element scheme. To get local efficiency

and thus a lower bound, we estimate the finite volume element quantities by the standard

arguments of [73, 74], which consist of bubble functions and an edge lifting operator. For

the tangential jump on the boundary we use results from the theory of the a posteriori

error estimate in the context of the non-conforming finite element method. Our proof can

also be used to estimate a similar tangential component in [19], where the proof contains

a mistake. Collecting all the results we show that the estimator is local and, in case of a

quasi-uniform mesh on the boundary, also generically efficient.

The a posteriori error estimate for the coupling with the cell-centered finite volume method

makes use of a post-processed interpolation of the interior piecewise constant solution.

Following [61], we introduce a Morley-type interpolant which belongs to a certain H1(Ω)-

conforming finite element space. The definition is a generalization of the definition in [61]

to the case of the coupling boundary and we use a different approach in the case of a pure

diffusion problem. We do the analysis only for diffusion or diffusion reaction problems in

the interior domain. We stress that we can apply and extend the analytical ideas of [61]

for convection problems too. Unfortunately, a reproduction of the results in [61] fails for

diffusion convection problems. In other words, we do not get experimental convergence

in the H1-seminorm for the Morley error. We point out that there is no theoretical con-

Introduction xi

vergence result for the Morley interpolant in [61]. The analytical idea is to ensure that

the Morley interpolant has enough orthogonality properties, which can be shown by the

conservation properties of the interpolant. This allows us to adapt the a posteriori error

analysis from the coupling with the finite volume element method and we get similar terms

for the residual error estimator. We stress that the estimator is built from the Morley in-

terpolant and gives us a robust upper and lower bound with respect to the energy norm

of the Morley interpolant and the Cauchy data.

In the fourth chapter we provide numerical experiments. First, we introduce the used

numerical methods with a special focus on assembling the system of linear equations.

We calculate six examples, where we observe the usability of our coupling approach. We

confirm the reliability and efficiency and in particular the robustness of our a posteri-

ori estimators for diffusion and/or reaction problems for both coupling systems and for

problems with convection for the coupling with the finite volume element method. The

proposed strategy for adaptive mesh-refinement recovers the optimal convergence rate for

solutions with a singular behavior and leads to better absolute error values in the case of

convection or reaction dominated problems, even if the solution is supposed to be smooth.

In Figure IV we see a road map and the main results of this thesis.

Acknowledgments

This work is dedicated in memoriam of my father.

First and foremost, I would like to express my gratitude to Prof. Dr. Stefan A. Funken,

who supported my research and provided me with the opportunity of working in his team.

In particular, his great knowledge on implementation aspects helped me a lot. I appreciate

the readiness of Prof. Dr. Karsten Urban to examine this work. Special thanks go also to

Prof. Dr. Dirk Praetorius from the Vienna University of Technology, Austria, who always

was interested in my research progress, supported me and who serves as external reviewer.

Furthermore, I also want to thank Prof. Dr. Helmut Harbrecht from the University of

Stuttgart, Germany, for examining this thesis as an additional external reviewer.

I am grateful to Mario Rometsch and Markus Bantle for proofreading this thesis. I also

appreciate the good atmosphere in the Institute for Numerical Mathematics at the Uni-

versity of Ulm and want to thank all my colleagues, especially Manuel Landstorfer, Mario

Rometsch and my roommate Timo Tonn. I would also like to thank our secretary Petra

Hildebrand.

Last but not least, I owe a lot to my mother and my family for supporting me all the

years, in particular Lea for managing my life the last months.

This thesis has been gratefully supported by a postgraduate scholarship issued by the

federal state Baden-Württemberg, Germany.

xii Introduction

Definition 2.0.2, p. 24:Model problem.

Definition 2.2.2, p. 26:

Weak form.

Theorem 2.2.7, p. 29:

Existence and uniqueness.

Discretization.


Coupling of the finite volume el-

ement method and the boundary

element method (with upwind-

ing in Definition 2.3.7, p. 35).


Coupling of the cell-centered

finite volume method and the

boundary element method.

Theorem 2.3.10, p. 38:

A priori convergence result (with

upwinding in Theorem 2.3.15, p. 43).

Corollary 2.3.12, p. 41:

Existence and uniqueness (with

upwinding in Corollary 2.3.17, p. 43).

Theorem 3.1.19, p. 66:Reliability in the energy er-ror Eh with appropriate re-

finement indicators ηT (3.20):

Eh ≤ Crel

(∑T ∈T

η2T

)1/2.

Theorem 3.1.23, p. 70:

Reliability for upwinding with

additional indicators ηT,up (3.23)

Theorem 3.1.38, p. 81:

Local efficiency gives a local lower

bound in the energy error Eh with

the refinement indicators ηT (3.20).

Subsection 3.2.1, p. 84:

Definition of a post-processed

approximation of Morley type.

Theorem 3.2.15, p. 91:Reliability in the Morley en-

ergy error Em with appropriaterefinement indicators for theMorley interpolant ηT (3.63):

Em ≤ Crel

(∑T ∈T

η2T

)1/2.

Theorem 3.2.17, p. 93:

Local efficiency gives an lo-

cal lower bound of the Morley

energy error Em by the re-

finement indicators ηT (3.20).

Section 4.2, p. 103:

Numerical experiments.

Section 4.3, p. 118:

Numerical experiments.

Figure IV. Road map and main results of this thesis.

Chapter 1

Analytical Basics and Notation

In this chapter we will give a summary of function spaces and notation used in this

work. Furthermore, a short introduction on boundary integral equations provides the

main results applied on the Laplace operator followed by the definition and notation of

the used triangulations of the considered domain and the corresponding discrete function

spaces. The last section states inequalities, which are mainly used in the a posteriori

analysis.

1.1 Function Spaces

In this section we assume that Ω is a (possibly unbounded) Lipschitz domain in R2 with

boundary Γ := ∂Ω = Ω ∩ (R2\Ω). The main property of a Lipschitz domain is that Ω is

locally only on one side of the boundary Γ and that the boundary Γ is locally the graph

of a Lipschitz continuous function. We refer to [58, 67] for a detailed definition. Thus, we

can define an outer normal vector n = n(x) for almost every x ∈ Γ.

Remark 1.1.1. If Ω is a bounded Lipschitz domain, then R2\Ω is an unbounded Lipschitz

domain. Note that the interior and exterior domain in the model problem are Lipschitz

domains.

To avoid any ambiguities we will give a brief collection of spaces of continuous functions

and Lebesgue spaces and also of Sobolev spaces Hm(Ω), which are based on L2(Ω), see [1].

Based on these we construct Sobolev spaces on boundaries and define trace operators.

Spaces of Continuous Functions and Lebesgue Spaces. Let us write Ck(Ω) for the

space of k ∈ N0 times continuously differentiable functions on Ω and the space of infinitely

differentiable functions is

C∞(Ω) :=⋂

k∈N0

Ck(Ω).

This leads us to the definition of

Ck(Ω) :=v|Ω

∣∣ v ∈ Ck(R2)

for k ∈ N0 ∪ ∞.

1

2 Chapter 1. Analytical Basics and Notation

The space of all Ck(Ω)(C∞(Ω)

)functions with compact support in Ω is denoted by

Ckc (Ω)

(C∞c (Ω)

). We define the Lebesgue spaces Lp(Ω) with 1 ≤ p < ∞ for the class

of all measurable functions v : Ω → R, which satisfy∫

Ω|v(x)|p dx < ∞.

The space Lp(Ω) is equipped with the norm

‖v‖Lp(Ω) :=( ∫

Ω|v(x)|p dx

)1/pfor 1 ≤ p < ∞.

By L∞(Ω) we denote the space consisting of all functions v that are essentially bounded

on Ω and the corresponding norm is given by

‖v‖L∞(Ω) := ess supx∈Ω |v(x)| = infM > 0

∣∣ |v| ≤ M almost everywhere in Ω.

Theorem 1.1.2. The space L2(Ω) defines a Hilbert space with the L2 scalar product

(v, w)Ω := (v, w)L2(Ω) :=

∫

Ωvw dx =

∫

Ωv(x)w(x) dx for all v, w ∈ L2(Ω).

This scalar product induces the norm

‖v‖2L2(Ω) := (v, v)Ω .

The space of all measurable and locally integrable functions on Ω is defined by

Lpℓoc(Ω) :=

v : Ω → R measurable

∣∣ for all K ⊂ Ω compact holds v|K ∈ Lp(K)

for 1 ≤ p < ∞.

The notation Lp(Ω)2 for 1 ≤ p ≤ ∞ defines the space of Lp functions in Ω with values in

R2 and is equipped with the norm

‖v‖2Lp(Ω) := ‖v1‖2

Lp(Ω) + ‖v2‖2Lp(Ω) for all v = (v1, v2)T ∈ Lp(Ω)2,

where we use the same notation as for the norm for scalar functions. The above notation

is also valid for Lp space on the boundary Γ.

Sobolev Spaces on Domains. Before we define Sobolev spaces on domains we define

the weak partial derivative of a function.

Definition 1.1.3. A function v ∈ L1ℓoc(Ω) is weakly differentiable, if there exists a

function ∂jv ∈ L1ℓoc(Ω) (j = 1, 2), such that the integration by parts formula with smooth

test functions is satisfied, i.e. there holds∫

Ωv(∂jw) dx = −

∫

Ω(∂jv)w dx for all w ∈ C∞

c (Ω).

1.1. Function Spaces 3

Remark 1.1.4. The notation (∇v,∇w)Ω is an abbreviation for

(∇v,∇w)Ω =

∫

Ω∇v · ∇w dx =

∫

Ω

( ∂v∂x1

∂w

∂x1+

∂v

∂x2

∂w

∂x2

)dx.

Definition 1.1.5 (Sobolev Spaces). We identify the Sobolev space H0(Ω) with L2(Ω).

Furthermore, we define

H1(Ω) :=v ∈ L2(Ω)

∣∣ v weakly differentiable with ∇v ∈ L2(Ω)2,

which is associated with the scalar product

(v, w)H1(Ω) := (v, w)Ω + (∇v,∇w)Ω .

This scalar product induces the norm

‖v‖2H1(Ω) := ‖v‖2

L2(Ω) + ‖∇v‖2L2(Ω).

Higher order Sobolev spaces of integer order m ∈ N may be defined inductively by

Hm(Ω) :=v ∈ L2(Ω)

∣∣ v is weakly differentiable with ∇v ∈ Hm−1(Ω)2

with the associated scalar product and norm

(v, w)Hm(Ω) := (v, w)Ω + (∇v,∇w)Hm−1(Ω) ,

‖v‖2Hm(Ω) := ‖v‖2

L2(Ω) + ‖∇v‖2Hm−1(Ω).

Definition 1.1.6. For a real number 0 < s < 1 we define the Sobolev Slobodeckij

seminorm

|v|2s,Ω :=

∫

Ω

∫

Ω

|v(x) − v(y)|2|x− y|2+2s

dy dx.

This seminorm with m ∈ N0 defines the fractional order Sobolev spaces

Hm+s(Ω) :=v ∈ Hm(Ω)

∣∣ |Dmv|s,Ω < ∞

equipped with the norm

‖v‖2Hm+s(Ω) := ‖v‖2

Hm(Ω) + |Dmv|2s,Ω.

Here, Dm denotes the mth (weak) derivative of v.

This leads to the set of all local Hm functions

Hmℓoc(Ω) :=

v : Ω → R

∣∣ for all K ⊂ Ω compact holds v|K ∈ Hm(K).

The next three theorems represent important properties of Sobolev spaces.

Theorem 1.1.7 ([1, Theorem 3.5]). For s ≥ 0, Hs(Ω) is a Hilbert space.

Theorem 1.1.8 ([11, Theorem 1.3.4]). For each m ≥ 0, C∞(Ω) ∩Hm(Ω) is a dense

subspace of Hm(Ω). Moreover, C∞(Ω) is a dense subspace of Hm(Ω).


Theorem 1.1.9. For m > 1 there holds Hm(Ω) ⊂ C(Ω) with continuous embedding,

i.e. ‖v‖L∞(Ω) ≤ C‖v‖Hm(Ω) for all v ∈ Hm(Ω), where the constant C > 0 does not depend

on v.

We define the dual space Hm(Ω)∗ of Hm(Ω) by the extended L2 scalar product. The next

lemma gives us a mathematically basis for this.

Lemma 1.1.10. Let X and Y be real Hilbert spaces with continuous inclusion X ⊆ Y .

Then, the Riesz mapping

JY : Y → Y ∗, JY y := (y, ·)Y

is well-defined as operator JY ∈ L(Y ;X∗) and JY (Y ) is a dense subspace of X∗.

Note that we can apply this lemma in the case of Sobolev spaces by X = Hm(Ω) and Y =

L2(Ω). Then L2(Ω) is a dense subspace of Hm(Ω)∗ and the duality brackets 〈v, w〉L2(Ω)

for v ∈ Hm(Ω) and w ∈ Hm(Ω)∗ coincide with the L2 scalar product (v, w)L2(Ω) provided

that w ∈ L2(Ω). This leads us to the following definition of Sobolev spaces with negative

order.

Definition 1.1.11. We denote by H−m(Ω) for m ≥ 0 the dual space of Hm(Ω) with

respect to the extended L2 scalar product denoted by 〈·, ·〉Ω. The norm on the dual space

is given by

‖v‖H−m(Ω)

:= sup06=w∈Hm(Ω)

| 〈v, w〉Ω |‖w‖H1(Ω)

.

For 1 ≤ p ≤ ∞ we denote by W 1,p(Ω) an additional Sobolev space of Lp(Ω) functions,

whose gradient are in Lp(Ω)2. The corresponding norm is

‖v‖pW 1,p(Ω) = ‖v‖p

Lp(Ω) + ‖∇v‖pLp(Ω) for 1 ≤ p < ∞,

‖v‖W 1,∞(Ω) = max‖v‖L∞(Ω), ‖∇v‖L∞(Ω)

for p = ∞.

In W 1,∞ are exactly the Lipschitz continuous functions [43, §5.8, Theorem 4].

Sobolev Spaces on Boundaries. The Sobolev spaces on boundaries are defined through

a local parametrization. The main idea is to define these spaces by Sobolev spaces on pa-

rameter domains by lifting up. We will not go into details, refer to [58, 67] and remark only

some important facts. Since the boundary of a Lipschitz domain is locally (parametrized)

the graph of a Lipschitz function, we only can construct Sobolev spaces Hm(Γ) for |m| ≤ 1.

To construct Sobolev spaces on the boundary of higher order we need a smoother bound-

ary.

Definition 1.1.12. A function g : Ω → R is Hölder continuous of order (k, λ), if g ∈ Ck(Ω)

and all kth derivatives of g with multi indices ι ∈ N20, 0 < λ ≤ 1 and x, y ∈ Ω satisfy

sup|ι|=k

supx 6=y

|∂ιg(x) − ∂ιg(y)||x− y|λ < ∞.

1.1. Function Spaces 5

The space of all Hölder continuous functions on Ω is denoted by Ck,λ(Ω). The domain Ω is

a Ck,λ domain, if it is a Lipschitz domain with a Hölder continuous local parametrization

of Γ of order (k, λ).

This allows us to define Sobolev spaces Hm(Γ) through parametrization for all m and

the following theorem states that the Sobolev spaces do not depend on the choice of the

parametrization.

Theorem 1.1.13 ([77, Theorem 4.2]). Let Ω be a Ck−1,1 domain, k ≥ 1 and 0 < m ≤k. The Sobolev space Hm(Γ) is uniquely defined for any parametrization of Γ considered as

a set and the corresponding norms are equivalent. In particular, there holds L2(Γ) = H0(Γ)

with equivalent norms. Hm(Γ) is a Hilbert space and the inclusion Hs(Γ) ⊂ Hm(Γ) for

m < s ≤ k is continuous.

Remark 1.1.14. Similar to Theorem 1.1.9 there holds Hm(Γ) ⊂ C(Γ) for m > 1/2.

Note that we can apply Lemma 1.1.10 for X = Hm(Γ) and Y = L2(Γ) here as well to

define the dual spaces.

Definition 1.1.15. Let Ω be a bounded Ck−1,1 domain, k ≥ 1 and 0 < m ≤ k. Then we

define the Sobolev space H−m(Γ) as the dual space of Hm(Γ) with respect to the extended

L2(Γ) scalar product 〈·, ·〉Γ.

Remark 1.1.16. In this work we will only need Hm(Γ) for |m| ≤ 1. These spaces are well-

defined on C0,1 domains, which are Lipschitz domains. Thus, our domains with polygonal

boundary used in the model problem fit these theoretical settings. The space Hm(Γ) is

associated with the norm ‖ · ‖Hm(Γ), which we will never need explicitly.

We define Hm∗ (Γ) as the Sobolev space with integral mean zero on the boundary, i.e.

Hm∗ (Γ) :=

ψ ∈ Hm(Γ)

∣∣ 〈ψ, 1〉Γ = 0.

Trace Operators. For sufficient smooth boundaries of an open Lipschitz domain Ω we

can define traces on Sobolev spaces Hm(Ω) to get an analytical description of Sobolev

functions on boundaries.

Theorem 1.1.17 ([58, Theorem 3.37]). If Ω is a bounded Ck−1,1 domain and if12 < m ≤ k, then there is a unique bounded linear operator

γ0 : Hm(Ω) → Hm−1/2(Γ)

such that γ0v = v|Γ for any v ∈ C∞(Ω). For one side traces, i.e. traces coming from Ω

and R2\Ω, respectively, we have

γint0 : Hm(Ω) → Hm−1/2(Γ) and γext

0 : Hmℓoc(R

2\Ω) → Hm−1/2(Γ).

We just write γ0 for the one side trace, if it is clear from which side the trace is taken.


Remark 1.1.18. For Lipschitz domains the boundedness of the trace operator γ0 :

Hm(Ω) → Hm−1/2(Γ) remains true for 1/2 < m < 3/2, see [32].

Next we define the conormal derivative for the elliptic operator

Lv := div(−A∇v + bv) + cv

with sufficient smooth A, b and c. Let

HmL (Ω) :=

v ∈ Hm

ℓoc(Ω)∣∣Lv ∈ L2

ℓoc(Ω) in a weak sense

and the bilinear form be

BL(v, w) := (A∇v,∇w)Ω + (div(bv), w)Ω + (cv, w)Ω for all v, w ∈ H1(Ω).

The conormal derivative arises naturally via the following lemma, known as the first Green

identity.

Lemma 1.1.19 ([58, Lemma 4.3]). Let Ωint be a bounded Lipschitz domain, and Ω be

either Ωint or R2\Ωint. Then the mapping

γ1 :H1L(Ω) → H−1/2(Γ),

v 7→ 〈γ1v, γ0w〉L2(Γ) := σΩ

(BL(v, w) − (Lv,w)Ω

)for all w ∈ H1

ℓoc(Ω)

is continuous. Here, σΩ = 1 for Ω = Ωint and σΩ = −1 otherwise.

We write γint1 and γext

1 , respectively, to distinguish if γ1 is applied in Ω or in R2\Ω. If it

is clear from which side the trace is taken we will just write γ1.

1.2 Boundary Integral Equations

In order to rewrite the exterior problem into an integral equation we need some analytical

basics to apply the boundary element method, which is applicable to problems for which

a fundamental solution can be calculated. The Malgrange-Ehrenpreis Theorem [38, 57]

states that every non-zero partial differential operator with constant coefficients has a

fundamental solution. For many partial differential equations the fundamental solution is

explicitly known, e.g. Laplace equation, Helmholtz equation, Lamé equation to mention

only a few but not all. For the Laplace operator in two dimensions the fundamental

solution (or Newton kernel) reads

− 1

2πlog |x|.

In this section we introduce some basic integral operators and the representation formula

based on this fundamental solution. Although we apply the boundary element method in

this work only for exterior problems, we will provide the theory for solving the Laplace

equation in the interior domain Ω, as well. Here, Ω is a bounded Lipschitz domain with

boundary Γ. Therefore, an exterior problem is defined on the unbounded Lipschitz domain

ΩC = R2\Ω.

1.2. Boundary Integral Equations 7

1.2.1 Operators

We collect here some important operators with their properties. We refer to the relevant

literature [58, 67, 71] for details. Based on the fundamental solution of the Laplace operator

we define the Newton potential for g : R2 → R, which has compact support in R2, by

(N g)(x) := − 1

2π

∫

R2g(y) log |x− y| dy for all x ∈ R

2, (1.1)

the single layer potential for ψ : Γ → R by

(Vψ)(x) := − 1

2π

∫

Γψ(y) log |x− y| dsy for all x ∈ R

2\Γ, (1.2)

and the double layer potential for θ : Γ → R by

(Kθ)(x) := − 1

2π

∫

Γθ(y)

∂

∂nylog |x− y| dsy for all x ∈ R

2\Γ. (1.3)

Here, ny is a normal vector with respect to the variable y. The following theorems gather

the most important properties of the operators N , V and K.

Theorem 1.2.1 ([58]). The Newton potential defines an operator N ∈L(H−1

c (R2);H1ℓoc(R

2))

and for all g ∈ H−1c (R2) there holds

−∆(N g) = g weakly in R2,

where H−1c (R2) denotes the space of all H−1(R2) functions with compact support in R

2.

In particular, the operators

γint0 N ∈ L

(H−1(Ω);H1/2(Γ)

)and γext

0 N ∈ L(H−1

c (ΩC);H1/2(Γ))

as well as

γint1 N ∈ L

(H−1(Ω);H−1/2(Γ)

)and γext

1 N ∈ L(H−1

c (ΩC);H−1/2(Γ))

are well-defined and there hold the following jump relations:

[[γ0N g]] := γext0 (N g) − γint

0 (N g) = 0,

[[γ1N g]] := γext1 (N g) − γint

1 (N g) = 0.

Theorem 1.2.2 ([58, Theorem 6.11]). The single layer potential defines an operator

V ∈ L(H−1/2(Γ);H1

ℓoc(R2))

and for all ψ ∈ H−1/2(Γ) there holds

−∆(Vψ) = 0 weakly in R2\Γ.


γint0 V ∈ L

(H−1/2(Γ);H1/2(Γ)

)and γext

0 V ∈ L(H−1/2(Γ);H1/2(Γ)

)


as well as

γint1 V ∈ L

(H−1/2(Γ);H−1/2(Γ)

)and γext

1 V ∈ L(H−1/2(Γ);H−1/2(Γ)

)


[[γ0Vψ]] := γext0 (Vψ) − γint

0 (Vψ) = 0,

[[γ1Vψ]] := γext1 (Vψ) − γint

1 (Vψ) = −ψ.(1.4)

A similar theorem holds for the double layer potential.

Theorem 1.2.3 ([58, Theorem 6.11]). The double layer potential defines an operator

K ∈ L(H1/2(Γ);H1(Ω) ∩H1

ℓoc(ΩC))

and there holds for all θ ∈ H1/2(Γ)

−∆(Kθ) = 0 weakly in R2\Γ.


γint0 K ∈ L

(H1/2(Γ);H1/2(Γ)

)and γext

0 K ∈ L(H1/2(Γ);H1/2(Γ)

)

as well as

γint1 K ∈ L

(H1/2(Γ);H−1/2(Γ)

)and γext

1 K ∈ L(H1/2(Γ);H−1/2(Γ)

)


[[γ0Kθ]] := γext0 (Kθ) − γint

0 (Kθ) = θ,

[[γ1Kθ]] := γext1 (Kθ) − γint

1 (Kθ) = 0.(1.5)

The above two theorems motivate us to define the following integral operators almost

everywhere on the boundary Γ:

• the single layer operator V := γint0 V,

• the double layer operator K := 1/2 + γint0 K,

• the adjoint double layer operator K∗ := −1/2 + γint1 V,

• the hypersingular integral operator W := −γint1 K.

Because of the jump relations (1.4) and (1.5) we obtain

γext0 V = V and γext

1 V = −1

2+ K∗

as well as

γext0 K =

1

2+ K and − γext

1 K = W.


Ω

Γ

tr

tl

ϕx

Figure 1.1. The notation for Lemma 1.2.4 which defines the trace of K and conormal deriva-

tive of V. The two tangential vectors tl and tr define the angle ϕ in the corner x ∈ Γ.

We stress that the definitions for V and W even hold for all x ∈ Γ, whereas for K and K∗

only holds for almost every x ∈ Γ. That means that Γ is smooth in a neighborhood of x,

i.e. differentiable. For all x ∈ Γ, e.g. if x is in a corner, we need the following result for Kand K∗.

Lemma 1.2.4 ([53, 71]). For simplicity, we assume that our boundary Γ has straight

lines in a neighborhood of each possible corner. We denote the interior angle of the inter-

section of the two tangential vectors in a point x ∈ Γ (coming from left and right) by ϕ,

where 0 < ϕ < 2π, see Figure 1.1. Then there holds for the trace of K with θ ∈ C(Γ)

(Kθ)(x) := γint0 (Kθ)(x) +

(1 − ϕ

2π

)θ(x), (1.6)

(Kθ)(x) := γext0 (Kθ)(x) − ϕ

2πθ(x). (1.7)

For the conormal derivative of V there holds

(K∗θ)(x) := γint1 (Vθ)(x) − ϕ

2πθ(x), (1.8)

(K∗θ)(x) := γext1 (Vθ)(x) −

( ϕ2π

− 1)θ(x). (1.9)

Remark 1.2.5. Note that for our Γ the formulae (1.6)–(1.9) coincide with the first defi-

nition for almost every x ∈ Γ, since ϕ = π. In our analysis we will never need (1.6)–(1.9)

with an angle ϕ 6= π.

This leads us to the following theorem.

Theorem 1.2.6 ([32, Theorem 1] or [58, Theorem 7.1]). The boundary integral

operators

V : Hs−1/2(Γ) 7→ Hs+1/2(Γ), K : Hs+1/2(Γ) 7→ Hs+1/2(Γ),

K∗ : Hs−1/2(Γ) 7→ Hs−1/2(Γ), W : Hs+1/2(Γ) 7→ Hs−1/2(Γ)

are linear and bounded for any s ∈ [−1/2, 1/2].

The next theorem states that the above operators have a integral representation. Note

that a discretization is usually based on piecewise polynomials.


Theorem 1.2.7 ([58, 71]). Let Γ be piecewise smooth and x ∈ Γ. Then we get for

ψ ∈ L∞(Γ)

(Vψ)(x) := − 1

2π

∫

Γψ(y) log |x− y| dsy.

We additionally define U(x, ε) :=y ∈ Γ

∣∣ |x− y| < ε

with ε > 0. Then we get for θ ∈H1/2(Γ) and ψ ∈ H−1/2(Γ)

(Kθ)(x) := − 1

2πlimε→0

∫

Γ\U(x,ε)θ(y)

∂

∂nylog |x− y| dsy,

(K∗ψ)(x) := − 1

2πlimε→0

∫

Γ\U(x,ε)ψ(y)

∂

∂nxlog |x− y| dsy,

(Wθ)(x) := −γ1(Kθ)(x).

Here, nx and ny are the normal vectors with respect to the variable x and y, respectively.

Beside these, we have some further properties of the operators V, K, K∗ and W, which

are well-stated in the literature [58, 67, 71].

• The operator V is symmetric, i.e.

〈Vψ, θ〉Γ = 〈ψ,Vθ〉Γ for all ψ, θ ∈ H−1/2(Γ).

• If diam(Ω) < 1, the operator V is H−1/2-elliptic, i.e.

〈Vψ,ψ〉Γ ≥ C‖ψ‖2H−1/2(Γ) for all ψ ∈ H−1/2(Γ)

for a constant C > 0.

• The operator W is symmetric, i.e.

〈Wψ, θ〉Γ = 〈ψ,Wθ〉Γ for all ψ, θ ∈ H1/2(Γ).

• The operator W is H1/2∗ -elliptic, i.e.

〈Wψ,ψ〉Γ ≥ C‖ψ‖2H1/2(Γ) for all ψ ∈ H

1/2∗ (Γ)

for a constant C > 0.

• The operator K∗ is the adjoint of K, i.e.

〈K∗ψ, θ〉Γ = 〈ψ,Kθ〉Γ for all ψ ∈ H−1/2(Γ) and θ ∈ H1/2(Γ).

Remark 1.2.8. The H−1/2-ellipticity of V even holds, if cap(Γ), the capacity of Γ, is

less than 1. For a detailed definition of cap(Γ) we refer to [70] and mention only that

diam(Ω) < 1 implies cap(Γ) < 1, which can always be achieved by scaling.


The next theorem states the relation between single layer and hypersingular integral op-

erator.

Theorem 1.2.9 ([58, Theorem 9.15]). Let ψ, θ ∈ H1/2(Γ). Then we have the identity

〈Wψ, θ〉Γ =

⟨V ∂ψ∂s,∂θ

∂s

⟩

Γ,

where ∂/∂s is the derivative along Γ with respect to the arc length.

1.2.2 Representation Formula and Calderón System

Now we are ready to formulate the representation formula for both the interior and ex-

terior domain. It states that the solution of a Laplace problem is uniquely determined

by its Cauchy data, i.e. the trace and the normal derivative of v on the boundary of the

corresponding domain.

Theorem 1.2.10 (Interior Representation Formula, [67, Theorem 3.1.6]). For

v ∈ H1(Ω) with −∆v = f ∈ H−1(Ω) there holds

v = Nf + V(γint1 v) − K(γint

0 v) almost everywhere in Ω. (1.10)

Taking the trace γint0 and the conormal derivative γint

1 in (1.10) and writing for the Cauchy

data ψ = γint0 v ∈ H1/2(Γ) and θ = γint

1 v ∈ H−1/2(Γ), respectively, we get by use of the

jump properties (1.4) and (1.5) the Calderón system

(ψ

θ

):=

(1/2 − K V

W 1/2 + K∗

)(ψ

θ

)+

(γint

0 Nf

γint1 Nf

). (1.11)

A similar result holds for the exterior domain with an appropriate radiation condition.

Theorem 1.2.11 (Exterior Representation Formula, [33, Lemma 3.5]). For

v ∈ H1ℓoc(ΩC) with −∆v = 0 and v(x) = a∞ + b∞ log |x| + o(1) for |x| → ∞, a∞, b∞ ∈ R

there holds

v = −V(γext1 v) + K(γext

0 v) + a∞ almost everywhere in ΩC . (1.12)

Applying γext0 and γext

1 to (1.12) the exterior Calderón system reads

(ψ

θ

):=

(1/2 + K −V

−W 1/2 − K∗

)(ψ

θ

)+

(a∞

0

)(1.13)

with the Cauchy data ψ = γext0 v ∈ H1/2(Γ) and θ = γext

1 v ∈ H−1/2(Γ), respectively. We

can fix either a∞ or b∞. For more details on this choice we refer to the next paragraph

and [58, Theorem 8.9].

The next theorem describes the relationship between arbitrary functions (ψ, θ) ∈H1/2(Γ) ×H−1/2(Γ) and the Calderón systems.


Theorem 1.2.12 ([33, Theorem 3.11]). The following two statements for (ψ, θ) ∈H1/2(Γ) ×H−1/2(Γ) are equivalent:

(a) ψ, θ are Cauchy data of the problem v ∈ H1(Ω) with −∆v = f in Ω(or v ∈ H1

ℓoc(ΩC)

with −∆v = 0 in ΩC and v(x) = a∞ + b∞ log |x| + o(1) for |x| → ∞, a∞, b∞ ∈ R).

(b) There holds (1.11)(or (1.13)

).

Remark 1.2.13 (Boundary Element Method). In Theorem 1.2.10 or Theorem 1.2.11 a

partial differential equation has been formulated as a boundary integral equation. The

discretization of the boundary integral equation on the boundary through a Galerkin

method leads to the boundary element method. We often can reduce a problem to calculate

the missing Cauchy data. For example, the Laplace equation −∆v = 0 with Dirichlet

boundary conditions v = vD leads to Symm’s integral equation VvN = (K + 1/2)vD,

which is discretized through the boundary element method to get the Neumann data

vN . Once we have the Cauchy data, i.e. the Dirichlet and Neumann data on the whole

boundary of the above problem, we can use the representation formula again to calculate

(numerically) the solution at any desired point of the solution domain.

Behavior at Infinity. Let us consider the exterior problem

−∆v = 0 on ΩC , and v prescribed on Γ,

v(x) = a∞ + b∞ log |x| + o(1) for |x| → ∞(1.14)

with v ∈ H1ℓoc(ΩC) and a∞, b∞ ∈ R. The following lemma gives us an additional property.

Lemma 1.2.14 ([48, Lemma 2.1]). Let v ∈ H1ℓoc(ΩC) be the solution of the exterior

problem (1.14). Then there holds

∫

Γγext

1 v ds = 2πb∞. (1.15)

Proof. We enclose Ω by a ball UR with R > diam(Ω) and get for the bounded domain

ΩC ∩ UR

0 =

∫

ΩC∩UR

∆v dx = −∫

Γγext

1 v ds+

∫

∂UR

∂v

∂nds.

We use the asymptotic behavior of v to get

∂v

∂n=b∞R

+ O(R−2)

which concludes the proof.

That means we can either fix a∞ or b∞. If we fix a∞ we simply can work with the

representation formula (1.12) and the corresponding Calderón system (1.13) and then


use (1.15) to calculate b∞. If we want to fix b∞, e.g. b∞ = 0 if v should be bounded, (1.15)

gives us an additional constraint and a∞ is an additional unknown.

The Poincaré Steklov Operator. Since V is positive definite for diam(Ω) < 1 and

thus invertible we may consider the interior/exterior Poincaré Steklov operator S int/ext :

H1/2(Γ) → H−1/2(Γ), defined as

S int := W + (1/2 + K∗)V−1(1/2 + K),

Sext := W + (1/2 − K∗)V−1(1/2 − K).

It is easy to see that these definitions are motivated by inserting the first line of (1.11) in

the second line of (1.11) (with f = 0) and the first line of (1.13) (with a∞ = 0) in the

second line of (1.13), respectively. Thus, S int/ext defines a Dirichlet to Neumann map, e.g.

for v ∈ H1ℓoc(R

2) with R2 = Ω ∪ Γ ∪ ΩC with −∆v = 0 and v(x) = b∞ log |x| + o(1) for

|x| → ∞, b∞ ∈ R there holds

γint1 v = S intγint

0 v and γext1 v = −Sextγext

0 v,

respectively. If it is clear from the context, we only write S for both, the interior

and exterior Poincaré Steklov operator. The next two lemmas state properties of the

Poincaré Steklov operator.

Lemma 1.2.15 ([25, Lemma 4.]). The operator S : H1/2(Γ) → H−1/2(Γ) is linear,

bounded and symmetric. If V is positive definite, e.g. diam(Ω) < 1, Sext is additionally

elliptic, i.e.

⟨Sextψ,ψ

⟩Γ

≥ C‖ψ‖2H1/2(Γ) for all ψ ∈ H1/2(Γ).

Lemma 1.2.16 ([14, Lemma 2.1]). Let v ∈ H1ℓoc(ΩC) satisfy −∆v = 0 and v(x) =

b∞ log |x| + o(1) for |x| → ∞, b∞ ∈ R, then

γext1 v = −Sextγext

0 v.

Conversely, for ψ ∈ H1/2(Γ) there exists a unique function v ∈ H1ℓoc(ΩC) satisfying −∆v =

0 and v(x) = b∞ log |x| + o(1) for |x| → ∞, b∞ ∈ R and

γext0 v = ψ and γext

1 v = −Sextψ.

The function v is given by the representation formula

v(x) = − 1

2π

∫

ΓSextψ log |x− y| dsy − 1

2π

∫

Γψ

∂

∂nylog |x− y| dsy

for x ∈ ΩC .


1.3 Triangulation

In this thesis we apply a finite volume scheme, which is a discretization method for partial

differential equations, in the interior domain Ω. One of the most characteristic of finite

volume schemes is a triangulation or mesh of the domain. The words triangulation and

mesh are used as synonyms of each other. Thus, we introduce a partition T of Ω into some

cells, called primal mesh. This mesh is sufficient to approximate the interior problem by

the cell-centered finite volume method. From the primal mesh we can generate a second

mesh, the dual mesh, where we can apply the finite volume element method.

1.3.1 The Primal Mesh

In this work we only consider a decomposition of Ω into triangles T ∈ T , which are non-

degenerated and considered to be closed, but we mention that finite volume schemes are

not limited to this kind of elements. Throughout, we write N and E for the set of all

nodes and edges of the triangulation T . Here, an edge E ∈ E is a straight line of the

boundary ∂T of an element T ∈ T . With hT := diam(T ) := supx,y∈T |x − y| we denote

the Euclidean diameter of T ∈ T . Moreover, hE denotes the length of an edge E ∈ E .

The global mesh size functions hT , hE ∈ L∞(Ω) are defined by

hT : Ω → [0,∞), x 7→

hT for x ∈ int(T ), T ∈ T ,

0 otherwise,

and

hE :⋃

E∈EE → [0,∞), x 7→

hE for x ∈ int(E), E ∈ E ,

0 otherwise,

respectively. For the maximal mesh size we write h := maxT ∈T hT .

Nodes. In the following, we introduce a partition of all nodes of T

N = NΓ ∪ NI

into coupling and interior nodes (free nodes), respectively: First, let NΓ :=a ∈ N

∣∣ a ∈ Γ

be the set of all nodes that belong to the coupling boundary. The set of interior nodes is

NI := N \NΓ. For an element T ∈ T we denote by NT the set of nodes of T , i.e. |NT | = 3

for T being a triangle. Let us further define xEm as the midpoint of an edge E ∈ E , NM

as the set of all midpoints, i.e. NM :=xEm

∣∣xEm is a midpoint of an edge E ∈ E. Note

that NM ∩ N = ∅. Additionally, we need for a vertex ai ∈ N the index set Ni of all

neighbors of ai in N , i.e. all vertices which are connected to ai by an edge E ∈ E .

Edges. For the edges we introduce a partition

E = EΓ ∪ EI

1.3. Triangulation 15

into coupling and interior edges, respectively: First, we define EΓ :=E ∈ E

∣∣E ⊂ Γ.

Second, the interior edges EI := E\EΓ. Finally, for an element T ∈ T , we denote by

ET ⊂ E the set of all edges of T , i.e. ET :=E ∈ E

∣∣E ⊂ ∂T.

The notation in this thesis is consistent in the sense that E inΓ ⊂ EΓ denotes all coupling

edges on Γin and so on. Furthermore, we denote by #T the number of elements of Tand #E the number of edges of E . Moreover, we write #N for the number of nodes. The

notation is valid for indices too. Note that #EΓ = #NΓ, since Γ is closed. This leads us

to the definition of a regular triangulation in the sense of Ciarlet [28]:

Definition 1.3.1 (Regular Triangulation). We say that the triangulation T with

non-degenerate triangles T is regular if the following hold:

• We cover the closure of Ω, i.e. Ω =⋃

T ∈T T .

• For distinct T1, T2 ∈ T (T1 6= T2) we have int(T1) ∩ int(T2) = ∅.

• The boundary conditions are resolved, i.e. each edge E ∈ E with E ∩ Γ 6= ∅ satisfies

either E ∈ E inΓ or E ∈ Eout

Γ .

• The intersection T1 ∩ T2 of two elements T1, T2 ∈ T with T1 6= T2 is either empty or

a node a ∈ N or an edge E ∈ E .

Remark 1.3.2 (Shape Regularity Constant). The regularity of T implies that the ele-

ments are shape regular, i.e. the ratio of the diameter hT of any element T ∈ T to the

diameter of its largest inscribed ball is bounded by a constant independent of hT . We call

this constant shape regularity constant and say that this constant depends on the shape

of the elements in T .

Remark 1.3.3 (Quasi-Uniform on the Boundary). In this thesis we use the term quasi-

uniform for the mesh on the coupling boundary, i.e. edges E ∈ EΓ, in the sense that the

ratio of the longest edge to the shortest edge for a sequence of meshes is bounded by a

constant, which does not depend on the size of the elements. We will explicitly mention

when we use this property.

Additionally, we assume that the triangulation T is aligned with the discontinuities of the

coefficients of the differential equation (if any), the data f , u0 and t0, and the interfaces

between Γin and Γout.

Remark 1.3.4. The elements T ∈ T are the control volumes for the cell-centered finite

volume method.

1.3.2 The Dual Mesh

For the finite volume element method we need a second mesh, which is built from the

primal mesh T . We denote by xT the center of gravity of an element T ∈ T . If we connect


xT1

xT2

xEm

V1

V2

V3

V4

V5

V6

V7

(a) The dual mesh T ∗.

V

(b) The elements of EV .

Figure 1.2. Subfigure (a) shows the construction of the dual mesh T ∗ from the primal mesh

T . The dashed lines are the new control volumes of T ∗. In (b) the set EV consists the bold

lines, which are parts of edges in E .

the midpoint xEm ∈ NM of an edge E ∈ E with the center of gravity xT of all T ∈ T ,

which share the edge E, we get a second mesh, the dual mesh T ∗. Figure 1.2(a) shows

the construction, where a representative xEm with the associated xT1 and xT2 is drawn.

The dashed lines are the new boxes (elements) from the dual mesh T ∗. These boxes

V ∈ T ∗ are considered to be closed and non-degenerated since T consists only elements,

which are non-degenerated. A box associated with a vertex ai ∈ N (from the primal mesh,

i = 1 . . .#N , which lies in the box) is denoted by Vi ∈ T ∗. Note that this vertex is unique.

Furthermore, we denote by hV := diam(V ) the Euclidean diameter of V and define the

set EV := E ∩ V for all V ∈ T ∗, see Figure 1.2(b). Note that⋃

E∈E E =⋃

V ∈T ∗ EV . The

interface between two control volumes Vi and Vj of the dual mesh with Vi ∩ Vj 6= ∅ is

denoted by τij , i.e. τij = Vi ∩ Vj , whereas τTij = Vi ∩ Vj ∩ T is the part of τij , which

lies in the corresponding T ∈ T , see Figure 1.3(a). Through construction, τij and τTij ,

respectively, can not be a single point. Additionally for all T ∈ T , we define the set

DT :=τT

ij

∣∣ τTij = Vi ∩ Vj ∩ T for Vi, Vj ∈ T ∗ with Vi 6= Vj

. Note that |DT | = 3 for all

T ∈ T . In the analysis we further need a partition of a control volume V ∈ T ∗. The

intersection V ∩ T 6= ∅ for all T ∈ T defines a partition of V in quadrilaterals, see

Figure 1.3(a). Each quadrilateral can itself be divided into two triangles and we denote

the partition of V into such triangles by ZV , see Figure 1.3(b). The triangles Z ∈ ZV are

considered to be closed. Note that there holds

Ω =⋃

V ∈T ∗

⋃

Z∈ZV

Z.

and that all T ∈ T can be split into 6 smaller triangles Zi ∈ ⋃V ∈T ∗ ZV , i = 1 . . . 6 with

Zi ∈ ZV , int(Zi) ∩ int(Zj) = ∅, i 6= j and T =⋃6

i=1 Zi.

1.3. Triangulation 17

V1

T1T1 ∩ V1

V2

T2

τ12

V3

V4

τT2

13 τT2

14

τT2

34

(a) Construction of τ12 and DT2 and the

quadrilateral of T ∈ T and V ∈ T ∗.

Z1 Z2

Z3

Z4

Z5

Z6

Z7Z8

Z9

Z10

Z11

Z12

(b) Zi build the set ZV .

Figure 1.3. Subfigure (a) shows the interface τ12 = V1 ∩V2 and the set DT2 = τT2

13 , τT2

14 , τT2

34 .

The intersection of T ∈ T and V ∈ T ∗ is either empty or a quadrilateral. In (b) each V ∈ Tcan be split into a set of triangles ZV .

Remark 1.3.5. The boxes V ∈ T ∗ will play the rule of the control volumes for the dis-

cretization of the interior problem by the finite volume element method, see Remark 1.3.4.

Definition 1.3.6 (Star-Shaped Domain). Let K ⊂ R2 be a closed Lipschitz domain.

We call K a star-shaped domain, if there exists a z ∈ K such that for all x ∈ K the

closed convex hull conv(x, z) is a subset of K. We say, the domain K is star-shaped

with respect to the point z.

Remark 1.3.7. A convex domain is star-shaped with respect to each of its points.

The elements V ∈ T ∗ of the dual elements are star-shaped, which follows from the regu-

larity of T and the choice of T ∗, [22, Lemma 3.2].

1.3.3 Normal, Tangential Vectors and Patches

In this subsection we define the normal and tangential vectors and recall the definition of

the patches, which are well-known from the finite element literature, e.g. [73]. We refer

also to Figure 1.4.

Normal and Tangential Vectors. We have already defined the unit normal vector n

for the boundary Γ. This is consistent in the sense that for all edges E ∈ EΓ, n points

outwards with respect to Ω. For the remaining edges, namely EI we may choose the

orientation of n arbitrarily. In general, if n appears in a boundary integral, it denotes

the unit normal vector to the boundary pointing outward the domain. The notation nx

means that the normal vector depends on the argument x. Finally, the tangential vector

t of an edge E ∈ E is always chosen orthogonal to n in mathematically positive sense.


a

(a) ωa.

E

(b) ωE .

E

(c) ωE .

T

(d) ωT .

T

(e) ωT .

Figure 1.4. The five patches introduced in Subsection 1.3.3.

Patch of a Node. For a ∈ N , the patch is given by

ωa =⋃

T ∈ωa

T, where ωa :=T ∈ T

∣∣ a ⊂ ∂T.

Patch of an Edge. For an edge E ∈ E , the patch is given by

ωE :=⋃

T ∈ωE

T, where ωE :=T ∈ T

∣∣E ⊂ ∂T,

and if we add the nodes of E

ωE :=

⋃

T ∈ωE

T, where ωE :=

T ∈ T

∣∣E ∩ ∂T 6= ∅ .

Patch of an Element. The patch of an element T ∈ T is defined by

ωT :=⋃

T ∈ωT

T, where ωT :=T ′ ∈ T

∣∣T ∩ T ′ ∈ E ,

and if we add the nodes of T

ωT :=

⋃

T ∈ωT

T, where ωT :=

T ′ ∈ T

∣∣T ∩ T ′ 6= ∅ .

1.4 Discrete Spaces on the Primal and the Dual Mesh

In this section we will provide suitable discrete function spaces to approximate partial

differential equations by the finite volume schemes and the boundary element method.

Additionally, we introduce an interpolation operator to work with the dual mesh.

Function Spaces on the Meshes. We define the piecewise affine and global continuous

function space on T by

S1(T ) :=v ∈ C(Ω)

∣∣ v|T affine for all T ∈ T .

1.4. Discrete Spaces on the Primal and the Dual Mesh 19

T

ai

I∗hvh|Vi

vh|T

Vi

Figure 1.5. The value of vh on T ∈ T and I∗hvh = vh(ai) on V ∈ T ∗ with ai ∈ N .

Note that S1(T ) = spanηi

∣∣ ai ∈ N, where ηi is the standard nodal linear basis function

associated with the node ai. On the dual mesh T ∗ we provide

P0(T ∗) :=v ∈ L2(Ω)

∣∣ v|V constant V ∈ T ∗,

where P0(T ∗) = spanχ∗

i

∣∣ ai ∈ N. Here, χ∗i denotes the characteristic function of the

volume Vi and we write for v∗ ∈ P0(T ∗)

v∗ :=∑

xi∈Nv∗

i χ∗i .

The spaces S1(EΓ) and P0(EΓ) are equivalently defined as above related to Γ. In general

Pp(T ) and Pp(EΓ) denote the space of T - and EΓ-piecewise polynomials of degree p ∈ N0,

respectively. We also need the ‘broken Sobolev space’ on EΓ, i.e.

Hm(EΓ) :=v ∈ L2(Γ)

∣∣ v|E ∈ Hm(E) for all E ∈ EΓ

for m ∈ N.

Interpolation Operator. We define the T ∗-piecewise interpolation operator

I∗h : C(Ω) → P0(T ∗), I∗

hv :=∑

ai∈Nv(ai)χ

∗i (x). (1.16)

Figure 1.5 shows the operator I∗h applied on a function vh ∈ S1(T ). We will need the

following properties of this operator for analyzing the finite volume element part in the

coupling.


Lemma 1.4.1. Let T ∈ T and E be an edge of T , i.e. E ∈ ET . For vh ∈ S1(T ) there

holds∫

T(vh − I∗

hvh) dx = 0, (1.17)∫

E(vh − I∗

hvh) ds = 0, (1.18)

‖vh − I∗hvh‖L2(T ) ≤ C1hT ‖∇vh‖L2(T ), (1.19)

‖vh − I∗hvh‖L2(E) ≤ C2h

1/2E ‖∇vh‖L2(T ), (1.20)

where the constants C1, C2 > 0 depend only on the shape regularity constant.

Proof. The proof is based on the construction of T ∗ and is simple, since vh is a piecewise

linear function on T , see [27, 78].

Equation (1.19) even holds for a small triangle Z ∈ ZV with V ∈ T ∗, which will be needed

in the a posteriori error analysis.

Lemma 1.4.2. Let vh ∈ S1(T ), V ∈ T ∗. For all triangles Z ∈ ZV we have

‖vh − I∗hvh‖L2(Z) ≤ hZ‖∇vh‖L2(Z), (1.21)

where hZ defines the diameter of Z. Additionally, we get

‖vh − I∗hvh‖L2(Z) ≤ C‖vh‖L2(Z) (1.22)

with a constant C > 0, which neither depends on the size of Z nor on the shape regularity

constant.

Proof. Let ai ∈ N be the unique point associated with the control volume Vi ∈ T ∗. Since

Z ⊂ T for a T ∈ T the function vh is linear on Z and there holds for x ∈ Z

vh(x) − vh(ai) = (x− ai) · ∇vh(x).

If we use the Cauchy-Schwarz inequality we get

|vh(x) − vh(ai)| = |(x− ai) · ∇vh(x)| ≤ |(x− ai)||∇vh(x)| ≤ hZ |∇vh(x)|,

and this leads to (1.21). To prove (1.22) one may use (1.21) for a reference triangle Z,

e.g. with the corner points (0, 0), (1, 0) and (0, 1), in order to get ‖vh − I∗hvh‖

L2(Z)≤

√2‖∇vh‖

L2(Z). Since all norms are equivalent on a finite dimensional space we get

‖∇vh‖L2(Z)

≤ ‖vh‖H1(Z)

≤ C‖vh‖L2(Z)

. Thus, we have ‖vh − I∗hvh‖

L2(Z)≤

√2C‖vh‖

L2(Z)

and a standard scaling argument leads to (1.22) with a constant C > 0, which neither

depends on the size of Z nor on the shape regularity constant. A different way to prove in-

equality (1.22) is to use (1.21) and apply the inverse inequality, e.g. [10, §2, Theorem 6.8],

but then the constant depends on the shape regularity constant.

1.5. Some Inequalities and Definitions 21

1.5 Some Inequalities and Definitions

In this section we collect some well-known results and definitions from the literature.

Poincaré and Trace Inequalities. We use the symbol . if an estimate holds up to

a multiplicative constant, which depends only on the shape regularity constant. On a

domain K ⊂ R2 we define the integral mean over K for a function v ∈ L2(K) by

vK :=1

|K|

∫

Kv dx.

Then, for v ∈ H1(T ) with T ∈ T there holds the Poincaré inequality

‖v − vT ‖2L2(T ) ≤ CP,Th

2T ‖∇v‖2

L2(T ), (1.23)

where CP,T = 1/π2, when T is convex [62, 8]. Furthermore, for v ∈ H1(V ) with V ∈ T ∗

the Poincaré inequality

‖v − vV ‖2L2(V ) . h2

V ‖∇v‖2L2(V ) (1.24)

is valid in this case because the volumes V are star-shaped with respect to a ball of

radius ∼ hV , see Definition 1.3.6. The trace inequality [11, Theorem 1.6.6] and a scaling

argument lead to

h1/2E ‖v‖L2(E) . ‖v‖L2(T ) + hE‖∇v‖L2(T ) (1.25)

for v ∈ H1(T ) and edges E of an element T ∈ T . In [74, Lemma 3.1] another trace

inequality is proved. For all T ∈ T , E ∈ ET and v ∈ H1(T ) there holds

‖v‖2L2(E) . h−1

T ‖v‖2L2(T ) + ‖v‖L2(T )‖∇v‖L2(T ). (1.26)

The trace inequality for E ∈ E , E ∈ ET and v ∈ H1(T ) reads

‖v − vE‖2L2(E) . hT ‖∇v‖2

L2(T ). (1.27)

The constant can be calculated explicitly, which is given in [60] for triangles T , i.e.

‖v − vE‖2L2(E) ≤ CEh

−1E h2

T ‖∇v‖2L2(T )

with CE ≈ 1.55416.

Jump Terms, Bubble Functions and Extended Operator. For T ∈ T , E ∈ ET ,

and ϕ ∈ H1(T ), let ϕ|E,T denote the trace of ϕ on E. Let E ∈ EI now be an interior edge

and T ′ and T the unique elements with E = T ′ ∩ T . For a T ′, T-piecewise H1 function

ϕ, the jump of ϕ on E is defined by

[[ϕ]]E := ϕ|E,T ′ − ϕ|E,T .


(a) bT . (b) bE . (c) bE .

Figure 1.6. Bubble functions: bT on an element T ∈ T in (a), bE on an interior edge E ∈ EI

with support on ωE in (b) and bE on a boundary edge E ∈ EΓ in (c).

Note that [[ϕ]]E = 0 provided ϕ ∈ H1(T ′∪T ). Moreover, for a T ′, T-piecewise polynomial

ϕ, the jump on E with the normal vector n pointing from the element T to T ′ reads

[[ϕ]]E(x) := limt→0+

ϕ(x+ tn) − limt→0+

ϕ(x− tn) for all x ∈ E.

Bubble functions give us an efficient tool to prove the efficiency of a residual-based a pos-

teriori error estimator. Furthermore, we need the bubble function to define an interpolant

in context with the cell-centered finite volume scheme. Thus, we want to recall the defi-

nition of bubble functions bT on a triangle T ∈ T and bE on an edge E ∈ E from [73]. If

T = conva1, a2, a3 ⊂ R2 is a non-degenerate triangle T ∈ T we easily define the bubble

function on T by the standard nodal linear basis functions η1, η2 and η3, i.e.

bT := 27η1η2η3. (1.28)

For E ∈ EI with ωE = T1 ∪ T2 we order the nodal basis η1,Tj and η2,Tj for j = 1, 2 such

that η1,T1(a1) = η1,T2(a1) = 1 and η2,T1(a2) = η2,T2(a2) = 1 hold, where a1, a2 ∈ N are

the nodes of the edge E. Thus, we define

bE := 4η1,Tjη2,Tj on Tj , j = 1, 2. (1.29)

For E ∈ EΓ we define

bE := 4η1,T η2,T on T with E ∈ ET . (1.30)

The bubble functions have the properties

bT = 0 on ∂T, 0 ≤ bT ≤ 1, maxx∈T

bT (x) = 1,

bE = 0 on ∂T\E with T ∈ ωE , 0 ≤ bE ≤ 1, maxx∈E

bE(x) = 1.

Note that there holds bT = 0 in Ω\T and bE = 0 in Ω\ωE . Finally, the edge lifting

operator

Fext : Pp(E) → H1(ωE) (1.31)

extends a polynomial on an edge E ∈ E to the patch ωE .

Chapter 2

The Coupling Problem

In this chapter we introduce the coupling problem, our model problem, in a weak sense,

where we show the existence and uniqueness of the interior and exterior solution. We write

the exterior problem as an integral equation on its boundary, which allows us to solve the

problem on the unbounded domain. A coupling formulation as well as its discrete form

for the coupling of the finite volume element method and the boundary element method

is provided with an a priori convergence proof and an existence and uniqueness proof.

The same stays valid if we use an upwind discretization for the convection term, which is

necessary to get a stable solution for convection dominated problems. Finally, we introduce

a discretization of the coupling problem, if we apply the cell-centered finite volume method

in the interior domain.

But first, we specify our model problem, where we use the model formulation introduced

in the introduction. We stress that our analysis holds for b constant and c = 0 as well.

Assumption 2.0.1 (Model Data). For the model data we request the following as-

sumptions:

(a) The diffusion matrix A = A(x) is bounded, symmetric and uniformly positive definite,

i.e. there exist positive constants CA,1 and CA,2 with

CA,1|v|2 ≤ vT A(x)v ≤ CA,2|v|2

for all v ∈ R2 and almost every x ∈ Ω. The entries of A(x) are either W 1,∞(Ω)

functions or T -piecewise constant functions.

(b) The convection vector function satisfies b ∈ W 1,∞(Ω)2.

(c) The reaction function satisfies c ∈ L∞(Ω).

(d) The convection vector and reaction function satisfy

1

2div b(x) + c(x) ≥ Cbc,1 ≥ 0 for almost every x ∈ Ω

with the constant Cbc,1 ≥ 0.

23

24 Chapter 2. The Coupling Problem

(e) For the right-hand side data we have f ∈ L2(Ω), the coefficients of the radiation

condition are a∞ ∈ R and b∞ ∈ R, and for the jump terms we suppose u0 ∈ H1/2(Γ)

and t0 ∈ H−1/2(Γ).

(f) We assume diam(Ω) < 1 to ensure H−1/2-ellipticity of V. This can always be achieved

by scaling of Ω, see [70].

We are now able to formulate our model problem in a weak sense:

Definition 2.0.2 (Model Problem). For the given data f ∈ L2(Ω), u0 ∈ H1/2(Γ) and

t0 ∈ H−1/2(Γ) we seek u ∈ H1(Ω) and uc ∈ H1ℓoc(ΩC) satisfying

div(−A∇u+ bu) + cu = f in Ω, (2.1a)

∆uc = 0 in ΩC , (2.1b)

uc(x) = a∞ + b∞ log |x| + o(1) for |x| → ∞, (2.1c)

u = uc + u0 on Γ, (2.1d)

(A∇u− bu) · n = ∇uc · n + t0 on Γin, (2.1e)

(A∇u) · n = ∇uc · n + t0 on Γout. (2.1f)

We can fix either a∞ ∈ R or b∞ ∈ R and calculate the other one in the radiation condition,

see Subsection 1.2.2 for more details.

Remark 2.0.3. In the interior domain Ω we also could have Dirichlet and Neumann

boundary conditions, ΓD and ΓN , respectively. Then the Neumann boundary is similarly

divided as the coupling boundary Γ, namely ΓN = ΓinN ∪ Γout

N , ΓinN ∩ Γout

N = ∅, where

ΓinN :=

x ∈ ΓN

∣∣b(x) · n(x) < 0

and ΓoutN :=

x ∈ ΓN

∣∣b(x) · n(x) ≥ 0. Note that Γ ∩

(ΓD ∪ ΓN ) = ∅. Thus, we get the boundary conditions

u = uD on ΓD, (2.1g)

(A∇u− bu) · n = gN on ΓinN , (2.1h)

(A∇u) · n = gN on ΓoutN , (2.1i)

where uD ∈ H1(Γ) and gN ∈ L2(Γ) are given data. Since these mixed boundary conditions

could appear in the interior domain Ω, one can handle them as in the context of pure finite

volume methods. In general we do not consider them for notational reasons. If we refer

to a result, where these conditions appear, the notation is consistent with an additional

index D for Dirichlet and N for Neumann boundaries.

Furthermore, we assume in this chapter a∞ = 0 in the radiation condition and write

for the Cauchy data of the exterior problem ξ := γext0 uc and φ := γext

1 uc, respectively

(φ = ∇uc·n in a trace sense). To fix b∞ in the radiation condition we refer to Lemma 1.2.14.

Additionally, we do not use a notational difference for functions in a domain and its traces,

if it is clear from the context.

2.1. Preliminaries 25

2.1 Preliminaries

For analytical investigations for diffusion convection reaction problems we introduce a

bilinear form on H1(Ω) ×H1(Ω) by

A(v, w) := (A∇v − bv,∇w)Ω + (cv, w)Ω + 〈b · n v, w〉Γout , (2.2)

which induce the natural energy (semi-) norm by

|||u|||2Ω := ‖A1/2∇u‖2L2(Ω) + ‖

(1

2div b + c

)1/2

u‖2L2(Ω) for all u ∈ H1(Ω), (2.3)

which will be motivated by Lemma 2.1.1 below. A similar norm for such kind of problems

was used in [75, 61, 76] to get robust a posteriori error estimates, i.e. the estimate is

independent of the variation of the model data A, b and c. We remark that ||| · |||Ω is only

a norm if 12 div b + c > 0. But this would exclude the important case when b ∈ R

2 is a

constant and c = 0 in the interior domain. Thus, we will always refer ||| · |||Ω as energy

norm. The next lemma shows that the bilinear form A(v, w) is coercive and continuous

on H1(Ω) ×H1(Ω) with respect to the energy norm ||| · |||Ω.

Lemma 2.1.1. The bilinear form A(v, w) is coercive and for Cbc,1 > 0 continuous on

H1(Ω) ×H1(Ω) with respect to the energy norm ||| · |||Ω, i.e.

A(v, v) ≥ |||v|||2Ω, (2.4)

|A(v, w)| ≤ CA,2|||v|||Ω|||w|||Ω, and |A(v, w)| ≤ CA,2′ |||v|||Ω‖w‖H1(Ω). (2.5)

Here, the constants CA,2, CA,2′ > 0 depend on the data A, b and c and on the constant

Cbc,1. If Cbc,1 = 0 we also require div b + c = 0 on Ω and b · n = 0 on Γin. Then A(v, w)

is continuous.

Proof. We can easily verify

A(v, v) ≥ (A∇v,∇v)Ω +1

2(div(b)v, v)Ω + (cv, v)Ω = |||v|||2Ω,

which follows from∫

Γoutb · n v2 ds ≥ 1

2

∫

Γb · n v2 ds =

1

2

∫

Ωdiv(bv2) dx =

1

2(div(b)v, v)Ω + (bv,∇v)Ω

with Assumption 2.0.1(d). For Cbc,1 > 0 we get

|A(v, w)| ≤ ‖A1/2∇v‖L2(Ω)‖A1/2∇w‖L2(Ω) + ‖b‖L∞(Ω)‖v‖L2(Ω)‖∇w‖L2(Ω)

+ ‖c‖L∞(Ω)‖v‖L2(Ω)‖w‖L2(Ω) + ‖b · n‖L∞(Γout)‖v‖L2(Γout)‖w‖L2(Γout)

≤ CA,2|||v|||Ω|||w|||Ω.

Since ‖A1/2∇w‖L2(Ω) ≤ C1/2A,2‖∇w‖L2(Ω) the second inequality of (2.5) follows by similar

steps as above. For Cbc,1 = 0 we calculate

A(v, w) = (A∇v,∇w)Ω + (div(bv), w)Ω + (cv, w)Ω − 〈b · n v, w〉Γin

and get continuity and the upper bound of A(v, w) with the additional assumption.


2.2 The Weak Form of the Model Problem

Our next goal is to formulate the model problem in a weak sense, where we want to write

the exterior problem in an integral equation form. We multiply (2.1a) with v ∈ H1(Ω)

and integrate over Ω to get∫

Ωdiv(−A∇u+ bu)v dx+

∫

Ωcuv dx =

∫

Ωfv dx

for all v ∈ H1(Ω). Integration by parts of the left-hand side leads to

(A∇u− bu,∇v)Ω + (cu, v)Ω − 〈(A∇u− bu) · n, v〉Γ =

∫

Ωfv dx

for all v ∈ H1(Ω). Next we replace (A∇u−bu) ·n and A∇u by using the conditions (2.1e)

and (2.1f), respectively, and get for all v ∈ H1(Ω)

(A∇u− bu,∇v)Ω + (cu, v)Ω + 〈b · nu, v〉Γout − 〈φ, v〉Γ = (f, v)Ω + 〈t0, v〉Γ

with the notation φ := ∇uc ·n. The exterior problem (2.1b)–(2.1c) can be rewritten to the

Calderón system (1.13), see Theorem 1.2.11 with the Cauchy data ξ and φ. We remind

that we assume a∞ = 0 and thus the variational formulation of this system reads

〈ξ, ψ〉Γ = 〈(1/2 + K)ξ, ψ〉Γ − 〈Vφ, ψ〉Γ for all ψ ∈ H−1/2(Γ),

〈φ, θ〉Γ = − 〈Wξ, θ〉Γ + 〈(1/2 − K∗)φ, θ〉Γ for all θ ∈ H1/2(Γ).

If we replace ξ = u− u0 on the left-hand side by the jump condition (2.1d), we can write

− 〈u, ψ〉Γ − 〈Vφ, ψ〉Γ + 〈(1/2 + K)ξ, ψ〉Γ = − 〈u0, ψ〉Γ for all ψ ∈ H−1/2(Γ), (2.6)

〈(1/2 + K∗)φ, θ〉Γ + 〈Wξ, θ〉Γ = 0 for all θ ∈ H1/2(Γ). (2.7)

Remark 2.2.1. From the Calderón system (1.13) we observe with ξ = 1, φ = 0 and

a∞ = 1 that W1 = 0 and (1/2+K)1 = 0. Thus, the variable ξ is determined in (2.6)–(2.7)

up to an additive constant and we fix this constant by 〈ξ, 1〉Γ = 0, i.e. ξ ∈ H1/2∗ (Γ),

see also [20]. Notice that uc is unique because of a∞ = 0 while ξ acts as a layer in the

boundary integral operators and is non-unique, but ξ − γ0uc is constant.

Thus, the weak form of our model problem reads:

Definition 2.2.2 (Weak Formulation). Find u ∈ H1(Ω), ξ ∈ H1/2∗ (Γ), φ ∈ H−1/2(Γ)

such that

A(u, v) − 〈φ, v〉Γ = (f, v)Ω + 〈t0, v〉Γ , (2.8a)

− 〈u, ψ〉Γ − 〈Vφ, ψ〉Γ + 〈(1/2 + K)ξ, ψ〉Γ = − 〈u0, ψ〉Γ , (2.8b)

〈(1/2 + K∗)φ, θ〉Γ + 〈Wξ, θ〉Γ = 0 (2.8c)

for all v ∈ H1(Ω), θ ∈ H1/2∗ (Γ), ψ ∈ H−1/2(Γ) with

A(u, v) = (A∇u− bu,∇v)Ω + (cu, v)Ω + 〈b · nu, v〉Γout .

2.2. The Weak Form of the Model Problem 27

Remark 2.2.3. The coupling is enforced through 〈φ, v〉Γ in (2.8a) and 〈u, ψ〉Γ in (2.8b).

The next theorem states the equivalence of the model problem in Definition 2.0.2 and the

weak formulation in Definition 2.2.2. One can find a similar proof in [25], which is here

extended for the different interior problem and the above weak formulation.

Theorem 2.2.4. If u ∈ H1(Ω), uc ∈ H1ℓoc(ΩC) is a solution of the model problem in

Definition 2.0.2, then u ∈ H1(Ω), ξ ∈ H1/2∗ (Γ), φ ∈ H−1/2(Γ) with ξ := γ0uc and φ := γ1uc

solves the weak formulation in Definition 2.2.2. If, conversely u, ξ, φ is a solution of the

weak form in Definition 2.2.2, then u, uc solves our model problem in Definition 2.0.2 with

uc ∈ H1ℓoc(ΩC) defined by

uc = −V(φ) + K(ξ). (2.9)

Proof. The first direction follows from the above calculations, where we deduced the

weak form in Definition 2.2.2 from the model problem in Definition 2.0.2. Let us prove

the other direction. If uc is defined by (2.9) we know from Theorem 1.2.11 and 1.2.12,

respectively, that uc satisfies (2.1b), (2.1c) and

(γ0uc

γ1uc

):=

(1/2 + K −V

−W 1/2 − K∗

)(γ0uc

γ1uc

). (2.10)

On the other hand, taking the traces of (2.9) we get

(γ0uc

γ1uc

):=

(1/2 + K −V

−W 1/2 − K∗

)(ξ

φ

). (2.11)

The first equation of (2.11) and (2.8b) then show γ0uc = u − u0 and thus (2.1d). From

the second equation of (2.11) and (2.8c) we easily deduce γ1uc = φ and thus from (2.10)

and (2.11) we get γ0uc = ξ. Now we apply integration by parts in (2.8a) to get

(div(−A∇u+ bu), v)Ω + (cu, v)Ω + 〈(A∇u− bu) · nu, v〉Γ

+ 〈b · nu, v〉Γout − 〈φ, v〉Γ = (f, v)Ω + 〈t0, v〉Γ

for all v ∈ H1(Ω). If we choose v ∈ H10 (Ω) =

v ∈ H1(Ω)

∣∣ γ0v = 0

we get directly the

weak form of (2.1a). Hence, if we use (2.1a) in the above equation we get (2.1e) and (2.1f),

respectively, because we already showed γ1uc = φ.

To prove uniqueness of our model problem, it is more convenient to rewrite the weak

formulation in Definition 2.2.2 with the help of the Poincaré Steklov operator S. We

replace φ in (2.8a) by φ = Su−Su0. This is motivated by Lemma 1.2.16 and the coupling

condition (2.1d). Then we define

B(u, v) := A(u, v) + 〈Su, v〉Γ (2.12)

and get the following theorem:


Theorem 2.2.5. The equation

B(u, v) = (f, v)Ω + 〈t0, v〉Γ + 〈Su0, v〉Γ for all v ∈ H1(Ω), (2.13)

where the right-hand side is a linear bounded functional on H1(Ω), defines another weak

form of the model problem in Definition 2.0.2, which is equivalent to the weak formulation

in Definition 2.2.2.

Proof. Note that (2.8b) in connection with (2.1d) is equivalent to φ = V−1(−1/2+K)ξ. If

we insert this in (2.8c) we get φ = −Wξ− (1/2 − K∗)V−1(1/2 − K) = −Sξ and thus (2.8a)

together with (2.1d) leads to the assertion.

The next lemma will be used to show the existence and uniqueness of solutions to our

model problem.

Lemma 2.2.6. The bilinear form B(v, w) is continuous and coercive, i.e. for all v, w ∈H1(Ω) there holds

B(v, w) ≤ Ccont‖v‖H1(Ω)‖w‖H1(Ω) and B(v, v) ≥ Ccoer‖v‖2H1(Ω)

with constants Ccont > 0 and Ccoer > 0.

Proof. The Cauchy-Schwarz inequality leads to

B(v, w) ≤ CA,2‖∇v‖L2(Ω)‖∇w‖L2(Ω) + ‖b‖L∞(Ω)‖v‖L2(Ω)‖∇w‖L2(Ω)

+ ‖c‖L∞(Ω)‖v‖L2(Ω)‖w‖L2(Ω) + 〈b · n v, w〉Γout + 〈Sv, w〉Γ .

For the last two terms we use the Cauchy-Schwarz inequality, the boundedness of S and

the trace theorem [71, Theorem 2.21], i.e. ‖v‖H1/2(Γ) ≤ C‖v‖H1(Ω) with C > 0, to get

〈b · n v, w〉Γout ≤ ‖b · n‖L∞(Γ)‖v‖L2(Γ)‖w‖L2(Γ)

≤ C1‖b · n‖L∞(Γ)‖v‖H1(Ω)‖w‖H1(Ω)

and

〈Sv, w〉L2(Γ) ≤ ‖Sv‖H−1/2(Γ)‖w‖H1/2(Γ) ≤ C2‖v‖H1/2(Γ)‖w‖H1(Ω)

≤ C3‖v‖H1(Ω)‖w‖H1(Ω)

with C1, C2, C3 > 0. Let us now prove the coercivity of B. Similar as in the proof for (2.4)

we conclude

B(v, v) ≥ (A∇v,∇v)Ω +1

2(div(b)v, v)Ω + (cv, v)Ω + 〈Sv, v〉Γ .

Due to the data Assumptions 2.0.1 and since the Poincaré Steklov operator S is positive

definite, we get

B(v, v) ≥ CA,1‖∇v‖2L2(Ω) + Cell‖v‖2

H1/2(Γ)

2.3. Coupling with the Finite Volume Element Method 29

with the constants CA,1, Cell > 0. In the last inequality we use the fact that the right-hand

side defines an equivalent norm on H1(Ω).

We are now in the position to gain existence and uniqueness for our model problem by

the Lax-Milgram Theorem.

Theorem 2.2.7. The problems in Definition 2.0.2 and Definition 2.2.2 and equation (2.13)

have unique solutions.

Proof. Since the bilinear form B is continuous and coercive, existence and uniqueness

of (2.13) follow from the Lax-Milgram Theorem, see e.g. [11, Theorem 2.7.7].

2.3 Coupling with the Finite Volume Element Method

The first subsection gives a discrete formulation in a finite volume element sense. In the

second subsection we show an a priori convergence result, and existence and uniqueness for

our discrete system with respect to the weak solution of the weak form in Definition 2.2.2.

We conclude with a formulation of an upwind scheme for the convection part, which is

recommended to ensure stability, if our model problem is convection dominated in the

interior domain.

2.3.1 Discretization in a Finite Volume Element Sense

For technical reasons we assume that the conormal derivative γ1u ∈ L2(Γ), the Cauchy

data φ ∈ L2(Γ) and the jump t0 ∈ L2(Γ). We stress that these assumptions are only needed

to motivate the coupling with a finite volume scheme. In general a finite volume scheme

integrates the model equation over control volumes and transforms this surface integral

partly into its boundary. Thus, we integrate equation (2.1a) over each dual element V ∈ T ∗

and apply the divergence theorem to get

∫

V(div(−A∇u+ bu) + cu) dx =

∫

∂V(−A∇u+ bu ) · n ds+

∫

Vcu dx =

∫

Vf dx

and thus

∫

∂V \Γ(−A∇u+ bu) · n ds+

∫

Vcu dx−

∫

∂V ∩Γ(A∇u− bu) · n ds =

∫

Vf dx

for all V ∈ T ∗. If we use (2.1e) and (2.1f) we get

∫

∂V \Γ(−A∇u+ bu) · n ds+

∫

Vcu dx

+

∫

∂V ∩Γoutb · nu ds−

∫

∂V ∩Γφds =

∫

Vf dx+

∫

∂V ∩Γt0 ds

(2.14)

for all V ∈ T ∗.


Remark 2.3.1. In general (2.14), (2.6)–(2.7) do not define another weak formulation of

the model problem in Definition 2.0.2. But (2.14) has to be understood as a motivation to

define a finite volume element discretization. Therefore, discrete solutions always have to

be seen as approximation of the weak solution in the weak formulation in Definition 2.2.2.

We can write the finite volume element part of the left-hand side of (2.14) as bilinear form

over H1(Ω) × P0(T ∗). We define for all w ∈ H1(Ω) and for all v∗ ∈ P0(T ∗)

AV (w, v∗) :=∑

ai∈Nv∗

i

(∫

∂Vi\Γ(−A∇w + bw) · n ds+

∫

Vi

cw dx

+

∫

∂Vi∩Γoutb · nw ds

).

(2.15)

Then the right-hand side reads

F (v∗) :=∑

ai∈Nv∗

i

(∫

Vi

f dx+

∫

∂Vi∩Γt0 ds

). (2.16)

Thus, we get for (2.14) the bilinear formulation

AV (u, v∗) − (φ, v∗)Γ := F (v∗) for all v∗ ∈ P0(T ∗).

This motivate us to approximate the solution u ∈ H1(Ω) of (2.1a) in the interior domain

in a conforming finite element space S1(T ) based on the primal mesh T but using a

discretization of an integral formulation of the problem on the boxes of the dual mesh T ∗

as shown in (2.14). Thus, we replace u in (2.14) by uh ∈ S1(T ). The approximation of the

exterior part, namely (2.8b)–(2.8c), is based on the replacement of the continuous spaces

by suitable discrete spaces. Therefore, the usual (discrete) variational formulation reads

− 〈uh, ψh〉Γ − 〈Vφh, ψh〉Γ + 〈(1/2 + K)ξh, ψh〉Γ = − 〈u0, ψh〉Γ for all ψh ∈ P0(EΓ),

〈(1/2 + K∗)φh, θh〉Γ + 〈Wξh, θh〉Γ = 0 for all θh ∈ S1∗ (EΓ).

Here, ξh ∈ S1∗ (EΓ) is the piecewise affine approximation of ξ ∈ H

1/2∗ (Γ) on Γ and φh ∈

P0(EΓ) is the piecewise constant approximation of φ ∈ H−1/2(Γ) on Γ, respectively.

Definition 2.3.2 (Discrete Problem). Additionally to the data Assumption 2.0.1, we

demand t0 ∈ L2(Γ). Find uh ∈ S1(T ), ξh ∈ S1∗ (EΓ) and φh ∈ P0(EΓ) such that

∫

∂V \Γ(−A∇uh + buh) · n ds+

∫

Vcuh dx

+

∫

∂V ∩Γoutb · nuh ds−

∫

∂V ∩Γφh ds =

∫

Vf dx+

∫

∂V ∩Γt0 ds, (2.17a)

− 〈uh, ψh〉Γ − 〈Vφh, ψh〉Γ + 〈(1/2 + K)ξh, ψh〉Γ = − 〈u0, ψh〉Γ , (2.17b)

〈(1/2 + K∗)φh, θh〉Γ + 〈Wξh, θh〉Γ = 0 (2.17c)

for all V ∈ T ∗, θh ∈ S1∗ (EΓ), ψh ∈ P0(EΓ).


Equation (2.17a) is equivalent to

AV (uh, v∗) − (φh, v

∗)Γ := F (v∗) for all v∗ ∈ P0(T ∗) (2.18)

with the bilinear form (2.15) and the right-hand side (2.16). We prove existence and

uniqueness of the system in Definition 2.3.2 with the help of a convergence result in

Corollary 2.3.12.

2.3.2 The Discrete Problem with an Upwind Approximation

For singularly perturbed diffusion convection problems, i.e. the diffusion is small with

respect to the convection vector, the above approximation leads to oscillating numerical

results. This disappointing behavior occurs because such methods lose stability and cannot

adequately approximate solutions inside layers. In the context of finite element methods

the streamline diffusion finite element method (SDFEM) is used whereas in finite volume

methods an upwind scheme naturally appears. We want to give a brief sketch of both

methods and assume therefore a pure Dirichlet problem of a diffusion convection reaction

problem with A = αI, α ∈ R+, i.e.

div(−α∇u+ bu) + cu = f with u = 0 on Γ. (2.19)

Streamline Diffusion Method. This method is also known as the streamline upwind

Petrov Galerkin method (SUPG-method). The basic idea is to add a stabilization term to

the common weak formulation. Thus, assuming that the solution u is more regular in the

sense that div(−α∇u+ bu) + cu = f in L2(T ) for all T ∈ T we conclude that u satisfies

for all v ∈ H1(Ω)

(α∇u− bu,∇v)L2(Ω) + (cu, v)L2(Ω) +∑

T ∈TκT (div(−α∇u+ bu) + cu,b · ∇v)L2(T )

= (f, v)L2(Ω) +∑

T ∈TκT (f,b · ∇v)L2(T )

with the user chosen constant κT . In general, the optimal value for κT is not known. We

remark that with the local Péclet number

Pe|T :=‖b‖L∞(T )hT

α(2.20)

and under the constraint

0 < κT ≤ 1

2min

h2

T

αC2inv

,Cbc,1

‖c‖2L∞(Ω)

we demand

κT :=

κ1hT if Pe|T > 1 (convection dominated),

κ2h2T /α if Pe|T ≤ 1 (diffusion dominated)


ai

ajak

al

am an

aoVi

τij τTij

T

Figure 2.1. From the definition in Subsection 1.3.2 we remind that the index set Ni contains

the indices of aj . . . ao. Additionally, we provide the line segment τij and its part τTij between

ai and aj .

with appropriate positive constants κ1 and κ2. Here, the model data b and c are chosen

such that 12 div b + c > Cbc,1, and the constant Cinv > 0, which is independent of T and

hT , arises from the local inverse inequality

‖∆vh‖L2(T ) ≤ Cinvh−1T ‖∇vh‖L2(T ) for all vh ∈ Pp(T ), p ≥ 1.

Furthermore, the method does not ensure local conservation and there is no general proof

for inverse monotonicity available, see [65, 52].

Remark 2.3.3. Here, an operator L is inverse monotone, if

(Lv)(x) ≥ 0 for all x ∈ Ω

v(x) ≥ 0 for all x ∈ Γ

imply v(x) ≥ 0 for all x ∈ Ω.

Upwind Scheme. For finite volume schemes local conservation appears naturally and

an upwind scheme for the convection part preserves this property and gives the desired

stabilization and, with some constraints, also inverse monotonicity of the entire scheme.

This motivates us to introduce an upwind scheme of first order as in [65, 52]. Therefore,

we rewrite a finite volume element discretization of (2.19) with Dirichlet conditions in the

following: Find uh ∈ S1(T ) for all v ∈ P0(T ∗) such that

∑

ai∈Nv∗

i

(∫

∂Vi\Γ−A∇uh · n ds+

∫

Vi

cuh dx+

∫

∂V ∩Γb · nuh (2.21)

+∑

j∈Ni

∑

τTij ⊂τij

∫

τTij

b · niuTh,ij ds

)=∑

ai∈Nv∗

i

∫

Vi

f dx.

We refer to Figure 2.1 for the notation and remark that there are exactly two τTij ⊂ τij and

ni is the outer unit normal vector to ∂Vi. The difference to the common discretization is


how to define uTh,ij in an upwind sense and we stress that there exist several possibilities

to do upwinding. In order to avoid technicalities we assume that b · ni does not change

sign over τTij . We define

βTij :=

1

|τTij |

∫

τTij

b · ni ds

with τTij = Vi ∩ Vj ∩ T for Vi, Vj ∈ T ∗ and T ∈ T . We write the general approximation by

uTh,ij := λT

ijuh(ai) + (1 − λTij)uh(aj), (2.22)

where we define λTij := Φ(βT

ij |τTij |/αT ). Here, Φ : R → [0, 1] is a weighting function, where

the value depends on the local Péclet number. This function has to fulfill for all t ∈ R:

[1 − Φ(t) − Φ(−t)]t = 0,

[Φ(t) − 1

2

]t ≥ 0,

1 − [1 − Φ(t)

]t ≥ 0.

If we choose now

Φ(t) = (sign(t) + 1)/2, (2.23)

i.e. λTij = 1 for βT

ij ≥ 0 and λTij = 0 otherwise, we get a full upwind scheme for (2.21). For

this method we have the following inverse monotony property.

Theorem 2.3.4 ([52, 65]). Let T be a triangulation of weakly acute type, i.e. the angles

of the triangles T ∈ T are less or equal π/2, and let the weighting function be Φ(t) =

(sign(t) + 1)/2. Assume that the coefficients b and c ≥ 0 and the right-hand side f

of (2.19) are sufficiently smooth. If we replace uh in the reaction integral by uh(ai), the

system (2.21) is inverse monotone.

The full upwind scheme is classical. But in order to reduce the excessive numerical dif-

fusion added by the full upstream weighting while simultaneously guarantee the stability

of the scheme we also introduce an approximation, where we can steer the amount of

upwinding [52]. Therefore, we define the weighting function

Φ(t) :=

min2|t|−1, 1

/2 for t < 0,

1 − min2|t|−1, 1

/2 for t ≥ 0.

(2.24)

In [46, 76] they also used this scheme successfully in numerical examples.

Remark 2.3.5. We stress that (2.24) becomes 1/2 for |t| → 0. Hence, λTij = 1/2 for

|τTij | → 0, and therefore uT

h,ij is the centered value. We can see that βTij and αT together

with the local mesh size |τTij | determine the amount and the sign of βT

ij the direction of

the upstream weighting.


With the α-weighted norm

‖v‖2α := ‖α1/2v‖2

L2(Ω) + ‖∇v‖2L2(Ω)

we have the following error estimation for the finite volume element method (2.21) with

the general upwind function (2.22).

Theorem 2.3.6 ([52, 65]). Let T be a triangulation of weakly acute type and assume

that the data b and c with 12 div b + c ≥ Cbc,1 > 0 of (2.19) are sufficient smooth. Let

f ∈ W 1,q(Ω) with q > 2. Then there holds for the discrete problem (2.21) for sufficient

small h0 > 0 with h ∈ (0, h0)

‖u− uh‖α ≤ Ch√α

(‖u‖H2(Ω) + ‖f‖W 1,q(Ω)

),

where the constant C neither depends on h0 nor on α.

In the standard Galerkin finite element method with u ∈ H2(Ω) and uh ∈ S1(T ), we can

easily prove the energy norm estimate

‖u− uh‖α ≤ Ch‖u‖H2(Ω). (2.25)

But this method is nevertheless unstable for 0 < α ≪ 1. Thus, we pay for the gain in

stability and local conservation by a loss in accuracy if we use upwind schemes.

In [66] the streamline diffusion method and the upwind method are compared, where the

transitions in the finite volume element solutions are sharper and less oscillatory than those

in the streamline diffusion method. But we stress that the upwind scheme is of first order,

i.e. for u ∈ Hk+1(Ω) and uh ∈ Sk(T ) with k ≥ 1 we get O(h), whereas for the streamline

diffusion finite element method there holds O(hk) in the streamline diffusion norm, which

can be found in [65, 52]. That means that the upwind scheme is not appropriate to design

higher order methods.

Coupling with Upwinding. The above advantages of the upwind approximation, es-

pecially the local conservation, motivates us to define a coupling method with an up-

wind finite volume element scheme. The modified bilinear form compared to (2.15) for

uh ∈ S1(T ), φh ∈ P0(EΓ) and v∗ ∈ P0(T ∗) reads

AupV (uh, v

∗) :=∑

ai∈Nv∗

i

(∫

∂Vi\Γ−A∇uh · n ds+

∫

Vi

cu dx

+∑

j∈Ni

∑

τTij ⊂τij

∫

τTij

b · niuTh,ij ds+

∫

∂Vi∩Γoutb · nuh ds

) (2.26)

with uTh,ij defined in (2.22). Thus, the modified coupling discretization in the finite volume

element method sense with upwinding reads:


Definition 2.3.7 (Discrete Problem with Upwinding). Additionally to the data

Assumption 2.0.1, we demand t0 ∈ L2(Γ). Find uh ∈ S1(T ), ξh ∈ S1∗ (EΓ) and φh ∈ P0(EΓ)

such that

AupV (uh, φh, v

∗) − (φh, v∗)Γ = F (v∗) (2.27a)

− 〈uh, ψh〉Γ − 〈Vφh, ψh〉Γ + 〈(1/2 + K∗)ξh, ψh〉Γ = − 〈u0, ψh〉Γ , (2.27b)

〈(1/2 + K∗)φh, θh〉Γ + 〈Wξh, θh〉Γ = 0 (2.27c)

for all V ∈ T ∗, θh ∈ S1∗ (EΓ), ψh ∈ P0(EΓ).

The proof for existence and uniqueness of the system in Definition 2.3.7 can be found in

Corollary 2.3.17.

2.3.3 An A Priori Convergence Result

In this subsection we gain an a priori convergence result for our discrete problem in

Definition 2.3.2 and later for the discrete solution with upwinding in Definition 2.3.7. Let

us assume u ∈ H1(Ω), ξ ∈ H1/2∗ (Γ), φ ∈ L2(Γ) to be the solution of the weak form in

Definition 2.2.2 and uh ∈ S1(T ), ξh ∈ S1∗ (EΓ), φh ∈ P0(EΓ) the solution of the discrete

problem in Definition 2.3.2. Note that for technical reasons φ is slightly more regular, see

Remark 2.3.11. We also assume t0 ∈ L2(Γ). Then we define the error e := u−uh ∈ H1(Ω)

in the interior domain, the trace error δ := ξ − ξh ∈ H1/2(Γ) and the conormal error

ǫ := φ− φh ∈ L2(Γ) of the exterior problem. First, we observe an orthogonality property

for the exterior part because the boundary element method is a Galerkin scheme, i.e. the

discrete system results from the continuous variational formulation. From (2.8b)–(2.8c)

and (2.17b)–(2.17c) we easily verify

− 〈e, ψh〉Γ − 〈Vǫ, ψh〉Γ + 〈(1/2 + K)δ, ψh〉Γ = 0 for all ψh ∈ P0(EΓ),

〈(1/2 + K∗)ǫ, θh〉Γ + 〈Wδ, θh〉Γ = 0 for all θh ∈ S1∗(EΓ).

Therefore, we define

p0 := −e− Vǫ+ (1/2 + K)δ ⊥ P0(EΓ), (2.28)

p1 := (1/2 + K∗)ǫ+ Wδ ⊥ S1∗ (EΓ). (2.29)

Note that

p0 = uh − u0 + Vφh − (1/2 + K)ξh, (2.30)

p1 = −(1/2 + K∗)φh − Wξh, (2.31)

where we have used the Calderón system (1.13) with a∞ = 0 and the jump condition (2.1d).

In the following we get an estimate for the right-hand side f . We remind that I∗h is the

interpolation operator defined in (1.16).


ai aj

ak

Vi Vj

Vk

T

∂T

T ∩ (∂Vi\Γ)

T ∩ (∂Vj\Γ)

T ∩ (∂Vk\Γ)

∂(T ∩ Vi) ∂(T ∩ Vj)

∂(T ∩ Vk): = −

Figure 2.2. Illustration of equation (2.36) for the proof to Lemma 2.3.9. We see that we can

split the boundary T ∩ (∂Vi\Γ) with ai ∈ T in boundaries, which allow to apply integration

by parts.

Lemma 2.3.8. There holds for the right-hand side f

| (f, vh − I∗hvh)Ω | ≤ C

∑

T ∈ThT ‖f‖L2(T )‖∇vh‖L2(T ) (2.32)

for all vh ∈ S1(T ) with a constant C > 0, which depends only on the shape regularity

constant.

Proof. The proof follows exactly [78]. With v∗h = I∗

hvh we simply calculate

| (f, vh − v∗h)Ω | =

∣∣∣∣∣∑

T ∈T

∫

Tf(vh − v∗

h)

∣∣∣∣∣ ≤∑

T ∈T‖f‖L2(T )‖vh − v∗

h‖L2(T )

≤ C∑

T ∈T‖f‖L2(T )hT ‖∇vh‖L2(T ),

where we have used (1.19) for the last inequality.

The next lemma gives us an estimate between the weak and the finite volume element

bilinear form for a function vh ∈ S1(T ). This gives an effective tool to prove an a priori

result.

Lemma 2.3.9. Let us assume that b · n is piecewise constant on Γin, i.e. b · n|Γin ∈P0(E in

Γ ). If Cbc,1 = 0 we also require div b + c = 0 on Ω. For all vh, wh ∈ S1(T ) there

holds

|A(vh, wh) − AV (vh, I∗hwh)| ≤ C

∑

T ∈T

(hT |||vh|||T ‖∇wh‖L2(T )

)(2.33)

with a constant C > 0, which depends only on the model data A, b, c and the shape

regularity constant.

Proof. The proof is similar to [78]. First, we rewrite the bilinear form A(vh, wh) (see (2.2))

by applying integration by parts to

A(vh, wh) =∑

T ∈T

(− (div(A∇vh − bvh), wh)T + (A∇vh · n, wh)∂T

+ (cvh, wh)T + (b · n vh, wh)∂T ∩Γin

).


Let us define w∗h := I∗

hwh. We rewrite the finite volume element bilinear form AV (vh, w∗h)

in a similar way

AV (vh, w∗h) =

∑

T ∈T

[ ∑

ai∈NT

( ∫

T ∩(∂Vi\Γ)(−A∇vh + bvh) · nw∗

h ds

+

∫

T ∩Vi

cvhw∗h ds+

∫

T ∩(∂Vi∩Γout)b · n vhw

∗h ds

)].

(2.34)

Note that with integration by parts, see Figure 2.2, we can write

∑

ai∈NT

∫

T ∩(∂Vi\Γ)(−A∇vh + bvh) · nw∗

h ds

=∑

ai∈NT

∫

∂(T ∩Vi)(−A∇vh + bvh) · nw∗

h ds−∫

∂T(−A∇vh + bvh) · nw∗

h ds

=

∫

Tdiv(−A∇vh + bvh)w∗

h dx−∫

∂T(−A∇vh + bvh) · nw∗

h ds. (2.35)

Thus, (2.34) and the fact that w∗h does not jump across the edges E leads to

AV (vh, w∗h) =

∑

T ∈T

(− (div(A∇vh − bvh), w∗

h)T + (A∇vh · n, w∗h)∂T

+ (cvh, w∗h)T − (b · n vh, w

∗h)∂T ∩Γin

).

(2.36)

Note that if A is T -piecewise constant, all parts with A vanish in A(vh, wh) − AV (vh, w∗h)

because of div(A∇vh) = 0 and (1.18). This is well-known, see [7, 44, 49, 52]. Thus the

following shows the estimation for A with entries in W 1,∞(Ω). Since ∇vh is constant

and div(bvh) = div(b)vh + b · ∇vh on T and by assumption b · n|Γin ∈ P0(E inΓ ) we get

with (1.18)

A(vh, wh) − AV (vh, w∗h)

=∑

T ∈T

((− div(A)∇vh + div(b)vh + b · ∇vh) + cvh, wh − w∗

h)T

+∑

E∈ET

((A − A)∇vh · n, wh − w∗

h

)E

−∑

E∈ET ∩Γin

(b · n(vh − vh), wh − w∗h)E

).

Here, div(A) := (div(a11, a21),div(a12, a22)) with the entries aij of A, A ∈ R2×2 denotes a

constant matrix, where the entries are the integral means of aij over E, and vh ∈ P0(EΓ)

is EΓ-piecewise constant approximation of vh. Next we apply the Poincaré inequality

‖g − g‖L∞(E) ≤ CPhE‖g′‖L∞(E) (CP > 0, g ∈ W 1,∞(E) and g the integral mean of g

over E) for the entries of A − A. If we additionally use the Cauchy-Schwarz inequality


we deduce

|A(vh, wh) − AV (vh, w∗h)|

≤∑

T ∈TCT

(|||vh|||T ‖wh − w∗

h‖L2(T ) +∑

E∈ET

hE‖∇vh‖L2(E)‖wh − w∗h‖L2(E)

+∑

E∈ET ∩EinΓ

‖vh − vh‖L2(E)‖wh − w∗h‖L2(E)

)

with the constant CT > 0, which depends on the shape regularity constant, CA,1 and

Cbc,1, if Cbc,1 > 0. Note that the estimation is valid for Cbc,1 = 0 as well because

then div b + c = 0. A simply calculation proves ‖∇vh‖L2(E) ≤ C1h−1/2E ‖∇vh‖L2(T ) with

C1 > 0. Together with the trace inequality ‖vh − vh‖L2(E) ≤ C2h1/2T ‖∇vh‖L2(T ) with

C2 > 2 and (1.19)–(1.20) we conclude

|A(vh, wh) − AV (vh, w∗h)| ≤ C

∑

T ∈T

(hT |||vh|||T ‖∇wh‖L2(T )

)

with C > 0.

This leads us to the main result in this chapter, a convergence and a priori result for the

discrete solution.

Theorem 2.3.10 (A Priori Convergence Estimate). Let b · n be piecewise constant

on Γin, e.g. b · n|Γin ∈ P0(E inΓ ). For Cbc,1 = 0 we also require div b + c = 0 on Ω and

b · n = 0 on Γin. For the solution u ∈ H1(Ω), ξ ∈ H1/2∗ (Γ), φ ∈ L2(Γ) of our model

problem in Definition 2.0.2 there holds with a discrete solution uh ∈ S1(T ), ξh ∈ S1∗ (EΓ),

φh ∈ P0(EΓ) of the discrete problem in Definition 2.3.2 and hT small enough

|||u− uh|||Ω + ‖ξ − ξh‖H1/2(Γ) + ‖φ− φh‖H−1/2(Γ)

≤ C(‖hT f‖L2(Ω) + |||hT u|||Ω + ‖h1/2

E (t0 − t0)‖L2(Γ) + ‖h1/2E (φ− φ)‖L2(Γ)

+ infvh∈S1(T )

‖u− vh‖H1(Ω) + infξh∈S1

∗(EΓ)

‖ξ − ξh‖H1/2(Γ)

),

where φ and t0 is the EΓ-piecewise integral mean of φ and t0, respectively, and C > 0 is a

constant, which depends on the model data A, b and c and the shape regularity constant.

Furthermore, for u ∈ H2(Ω), ξ ∈ H1∗ (Γ) ∩ H2(EΓ), φ ∈ L2(Γ) ∩ H1/2(EΓ) and t0 ∈

L2(Γ) ∩H1/2(EΓ) we get

|||u− uh|||Ω + ‖ξ − ξh‖H1/2(Γ) + ‖φ− φh‖H−1/2(Γ) = O(h)

with h := maxT ∈T hT .

Proof. From (2.4) we estimate with an arbitrary vh ∈ S1(T ) and e = u− uh

|||e|||2Ω ≤ A(e, u− vh) + A(e, vh − uh).


The first term of the right-hand side can be easily estimated by (2.5)

|A(e, u− vh)| ≤ CA,2′ |||e|||Ω‖u− vh‖H1(Ω) ≤ C1|||e|||2Ω + C−11 ‖u− vh‖2

H1(Ω),

where we can fix C1 > 0 because of Young’s inequality. For the last term we use for

A(u, vh − uh) the identity (2.8a) and insert (2.18) with I∗h(wh − uh) to obtain

A(u, vh − uh) − A(uh, vh − uh)

= (f, vh − uh)Ω + (t0, vh − uh)Γ + 〈φ, vh − uh〉Γ

− A(uh, vh − uh) + AV (uh, I∗h(vh − uh))

− (φh, I∗h(vh − uh))Γ − (f, I∗

h(vh − uh))Ω − (t0, I∗h(vh − uh))Γ .

Summing up some terms we arrive at

A(u, vh − uh) − A(uh, vh − uh)

= (f, vh − uh − I∗h(vh − uh))Ω − A(uh, vh − uh) + AV (uh, I∗

h(vh − uh))

+ (t0, vh − uh − I∗h(vh − uh))Γ + 〈φ− φh, vh − uh〉Γ

+ (φh, vh − uh − I∗h(vh − uh))Γ .

(2.37)

Now we estimate these terms separately. With Lemma 2.3.8 and 2.3.9 we calculate

(f, vh − uh − I∗h(vh − uh))Ω − A(uh, vh − uh) + AV (uh, I∗

h(vh − uh))

≤ C2

∑

T ∈T

(hT ‖f‖L2(T ) + hT |||u|||T + hT |||u− uh|||T

)‖∇(vh − uh)‖L2(T )

≤ C−13

(‖hT f‖2L2(Ω) + |||hT u|||2Ω + |||hT (u− uh)|||2Ω

)

+ C3‖∇(u− vh)‖2L2(Ω) + C3|||u− uh|||2Ω.

We have used Young’s inequality in the last step, where we can fix the constant C3 > 0.

Note that t0 ∈ L2(Γ) thus we get with (1.18) and (1.20)

| (t0, vh − uh − I∗h(vh − uh))Γ | ≤

∑

E∈EΓ

(t0 − t0, vh − uh − I∗h(vh − uh))E

≤∑

E∈EΓ

‖t0 − t0‖L2(E)h1/2E ‖∇(vh − uh)‖L2(TE).

Here, TE is the element associated with E. The same calculations as above lead to

| (t0, vh − uh − I∗h(vh − uh))Γ |

≤ C−14 ‖h1/2

E (t0 − t0)‖2L2(E) + C4‖∇(u− vh)‖2

L2(Ω) + C4|||u− uh|||2Ω,

where we can fix C4 > 0 arbitrary. The Hölder inequality and the trace theorem, i.e.

‖u− vh‖H1/2(Γ) ≤ C5‖u− vh‖H1(Ω) with C5 > 0, lead to

〈φ− φh, vh − uh〉Γ = 〈φ− φh, vh − u〉Γ + 〈φ− φh, u− uh〉Γ

≤ C5‖φ− φh‖H−1/2(Γ)‖u− vh‖H1(Ω) + 〈φ− φh, u− uh〉Γ .


From the definition of p0 and p1, the orthogonal relations (2.28) and (2.29) and because

K is adjoint to K∗ we write with δ = ξ − ξh and ǫ = φ− φh

(φ− φh, u− uh)Γ = (ǫ, (1/2 + K)δ)Γ − (ǫ,Vǫ)Γ − (ǫ, p0)Γ (2.38)

≤ −CV‖ǫ‖2H−1/2(Γ) − CW‖δ‖2

H1/2(Γ) − 〈p0, φ− φ〉Γ + 〈p1, ξ − ξh〉Γ,

where φ is the EΓ-piecewise integral mean of φ and ξh ∈ S1∗ (EΓ) is chosen arbitrary. Note

that CV > 0 and CW > 0 are the ellipticity constants from the operators V and W,

respectively. Next we estimate

−(p0, φ− φ

)Γ

=(e− eE , φ− φ

)Γ

+ ‖Vǫ− (1/2 + K)δ‖H1/2(Γ)‖φ− φ‖H−1/2(Γ),

(2.39)

where eE is the EΓ-piecewise integral mean of e on Γ. We calculate with the trace inequal-

ity (1.27) and continuity of V and K

−(p0, φ− φ

)Γ

≤ C6

(‖∇e‖L2(Ω)‖h1/2

E (φ− φ)‖L2(Γ)

+(‖ǫ‖H−1/2(Γ) + ‖δ‖H1/2(Γ)

)‖φ− φ‖H−1/2(Γ)

)

≤ C7|||e|||2Ω + C−17 ‖h1/2

E (φ− φ)‖2L2(Γ)

+ C8‖ǫ‖2H−1/2(Γ) + C8‖δ‖2

H1/2(Γ) + C−18 ‖h1/2

E (φ− φ)‖2L2(Γ),

where we have used Young’s inequality in the last estimation and [23, Lemma 4.3], i.e.

‖φ−φ‖H−1/2(Γ) ≤ C8′‖h1/2E (φ−φ)‖L2(Γ) with C8′ > 0, for the last term. Note that we can

fix C7, C8 > 0. Similarly as above we estimate

(p1, ξ − ξh

)Γ

≤(‖(1/2 + K∗)ǫ‖H−1/2(Γ) + ‖Wδ‖H1/2(Γ)

)‖ξ − ξh‖H−1/2(Γ)

≤ C9‖ǫ‖2H−1/2(Γ) + C9‖δ‖2

H1/2(Γ) + C−19 ‖ξ − ξh‖2

H1/2(Γ),

which follows from the continuity of K∗ and W, C9 > 0. Now we fix the constants C8 and

C9 such that C10 := CV − C8 − C9 > 0 and C11 := CW − C8 − C9 > 0. Thus, we get

for (2.38)

(φ− φh, u− uh)Γ ≤ −C10‖ǫ‖2H−1/2(Γ) − C11‖δ‖2

H1/2(Γ) + C7|||e|||2Ω+ max

C−1

7 , C−18

‖h1/2E (φ− φ)‖2

L2(Γ) + C−19 ‖ξ − ξh‖2

H1/2(Γ).

For the last term in (2.37) we may apply (1.18)

(φh, vh − uh − I∗h(vh − uh))Γ = 0

since φh ∈ P0(EΓ). If we choose C1, C3, C4, C7 < 1 we conclude the proof for hT small

enough (because of C−13 ). The second assertion follows directly from the first by applying

the approximation theorem, e.g. [10, II-§2-Theorem 6.4].


Remark 2.3.11. Note that we need the additional regularity φ ∈ L2(Γ) to estimate(e, φ− φ

)Γ

in (2.39). If we only allow Cbc,1 > 0, then φ ∈ H−1/2(Γ) is sufficient to prove

Theorem 2.3.10. This can be seen by the following estimations. Let us choose φh ∈ P0(EΓ)

arbitrary. Then we get(e, φ− φh

)Γ

instead of(e, φ− φ

)Γ, which follows easily by (2.28).

We estimate

(e, φ− φh

)Γ

≤ ‖e‖H1/2(Γ)‖φ− φh‖H−1/2(Γ) ≤ C ′‖e‖H1(Ω)‖φ− φh‖H−1/2(Γ)

≤ C ′′|||e|||Ω‖φ− φh‖H−1/2(Γ),

where we have used the trace theorem [71, Theorem 2.21], i.e. ‖v‖H1/2(Γ) ≤ C ′‖v‖H1(Ω)

with C ′ > 0, and the fact that in this case we can estimate the H1-norm by the energy

norm, i.e. C ′′ depends on the model data. Thus we are finished if we replace

‖h1/2E (φ− φ)‖L2(Γ) by inf

φh∈P0(EΓ)

‖φ− φh‖H−1/2(Γ)

in the rest of the proof.

With the help of Theorem 2.3.10 we prove the existence and uniqueness of the discrete

problem in Definition 2.3.2.

Corollary 2.3.12. Let b · n be piecewise constant on Γin, e.g. b · n|Γin ∈ P0(E inΓ ). For

Cbc,1 = 0 we also require div b + c = 0 on Ω and b · n = 0 on Γin. The discrete solution

uh ∈ S1(T ), ξh ∈ S1∗ (EΓ), φh ∈ P0(EΓ) of the discrete problem in Definition 2.3.2 to the

solution u ∈ H1(Ω), ξ ∈ H1/2∗ (Γ), φ ∈ H−1/2(Γ) of our model problem in Definition 2.0.2

exists and is unique.

Proof. It is easy to see that the discrete problem in Definition 2.3.2 leads to a linear

system of Ns = #N + #EΓ + #NΓ equations with Ns unknowns. Let us assume that uh,

ξh and φh is a solution of the system with the right-hand side 0. We can write this system

as As xs = 0 with the system matrix As ∈ RNs×Ns and the unknown vector xs ∈ R

Ns×1,

which consists the unknowns of uh, ξh and φh. For the data f = 0, u0 = 0 and t0 = 0

we observe by the equivalent weak formulation in Definition 2.2.2 that u = 0, ξ = 0 and

φ = 0 hold and this continuous solution is unique because of Theorem 2.2.7. Obviously

uh, ξh and φh is a discrete solution of this continuous system and Theorem 2.3.10 holds.

Therefore, we get with the chosen data

|||uh|||Ω + ‖ξh‖H1/2(Γ) + ‖φh‖H−1/2(Γ) ≤ 0. (2.40)

Note that we can estimate |||uh|||Ω ≥ C1/2A,1‖∇uh‖L2(Ω). Thus, we get from (2.40) that

ξh = 0, φh = 0 and ∇uh = 0, which implies that uh is constant in Ω. From (2.17b) we

observe with ξh = 0 and φh = 0 that 〈uh, ψh〉Γ = 0 for all ψh ∈ P0(EΓ), in particular

the integral mean of uh on the boundary Γ is zero and thus uh = 0 on Ω. That means

As is injective and thus bijective (since As is square), which proves the existence and

uniqueness of the discrete solution.


T

aj

τTij

ai

Vi

ni

Figure 2.3. In the proof of Lemma 2.3.14 we need the line τTij and the normal vector ni,

which is defined as the outer normal vector with respect to the element Vi associated with ai.

Remark 2.3.13. Note that for Corollary 2.3.12 we do not need additionally regularity

for φ, neither for Cbc,1 = 0.

The following lemma gives us an estimate between the common and the upwind finite

volume element bilinear form. We will need this to prove an a priori result for the dis-

cretization problem with upwinding in Definition 2.3.7.

Lemma 2.3.14. For all vh, wh ∈ S1(T ) there holds

|AV (vh, I∗hwh) − Aup

V (vh, I∗hwh)| ≤ C

∑

T ∈T

(hT |||vh|||T ‖∇wh‖L2(T )

)

with a constant C > 0, which depends on the model data on A, b, c and the shape regularity

constant.

Proof. Let us define w∗h := I∗

hwh. With the bilinear forms (2.15) and (2.26) we get

AV (vh, w∗h) − Aup

V (vh, w∗h)

=∑

ai∈Nw∗

i

∑

j∈Ni

( ∫

τij

b · ni vh ds−∑

τTij ⊂τij

∫

τTij

b · ni vTh,ij ds

).

For the notation we recall Figure 1.3(a) and Figure 2.3. We can express this sum over the

elements of T .

AV (vh, w∗h) − Aup

V (vh, w∗h)

=∑

T ∈T

∑

τTij ∈DT

(w∗i − w∗

j )

∫

τTij

b · ni(vh − vTh,ij) ds.

Note that vTh,ij = λT

ijvh(ai) + (1 − λTij)vh(aj) with λT

ij ∈ [0, 1]. With the Cauchy-Schwarz

inequality and a similar argument as in the proof of Lemma 1.4.2 we get

(w∗i − w∗

j )

∫

τTij

b · ni(vh − vTh,ij) ds ≤ ‖b · ni‖L∞(τT

ij )‖w∗i − w∗

j ‖L2(τTij )‖vh − vT

h,ij‖L2(τTij )

≤ ChT ‖∇wh‖L2(τTij )hT ‖∇vh‖L2(τT

ij ).


Here, the constant C > 0 depends on the weighting factor λTij and b. We easily calculate

‖∇wh‖L2(τTij ) ≤ h

−1/2T ‖∇wh‖L2(T ) and ‖∇vh‖L2(τT

ij ) ≤ h−1/2T ‖∇vh‖L2(T )

and conclude the proof.

Similar as in Theorem 2.3.10 we state an a priori result for the coupling with upwinding.

Theorem 2.3.15 (A Priori Convergence Estimation for Upwinding). Let us as-

sume that b · n is piecewise constant on Γin, e.g. b · n|Γin ∈ P0(E intΓ ). If Cbc,1 = 0 we also

require div b + c = 0 on Ω and b · n = 0 on Γin. For the solution u ∈ H1(Ω), ξ ∈ H1/2∗ (Γ),

φ ∈ L2(Γ) of our model problem in Definition 2.0.2 there holds with a discrete solution

uh ∈ S1(T ), ξh ∈ S1∗ (EΓ), φh ∈ P0(EΓ) of the discrete problem in Definition 2.3.7 and hT

small enough

|||u− uh|||Ω + ‖ξ − ξh‖H1/2(Γ) + ‖φ− φh‖H−1/2(Γ)

≤ C(‖hT f‖L2(Ω) + |||hT u|||Ω + ‖h1/2

E (t0 − t0)‖L2(Γ) + ‖h1/2E (φ− φ)‖L2(Γ)

+ infvh∈S1(T )

‖u− vh‖H1(Ω) + infξh∈S1

∗(EΓ)

‖ξ − ξh‖H1/2(Γ)

),

where φ and t0 is the EΓ-piecewise integral mean of φ and t0, respectively, and C > 0 is a

constant, which depends on the model data and the shape regularity constant.

Furthermore, for u ∈ H2(Ω), ξ ∈ H1∗ (Γ) ∩ H2(EΓ), φ ∈ L2(Γ) ∩ H1/2(EΓ) and t0 ∈

L2(Γ) ∩H1/2(EΓ) we get

|||u− uh|||Ω + ‖ξ − ξh‖H1/2(Γ) + ‖φ− φh‖H−1/2(Γ) = O(h)

with h := maxT ∈T hT .

Proof. The proof follows exactly the lines of the proof of Theorem 2.3.10. With

vh ∈ S1(T ) we get in the same way (2.37), now with AupV (uh, I∗

h(vh − uh) instead of

AV (uh, I∗h(vh − uh). Thus, we plug in AV (uh, I∗

h(vh − uh) − AV (uh, I∗h(vh − uh) in (2.37).

Note that we can estimate −A(uh, vh − uh) + AV (uh, I∗h(vh − uh) by Lemma 2.3.9 and

AupV (uh, I∗

h(vh − uh) − AV (uh, I∗h(vh − uh) with Lemma 2.3.14 and the other terms as in

the proof of Theorem 2.3.10.

Remark 2.3.16. We refer to Remark 2.3.11 for a discussion on φ ∈ H−1/2(Γ).

With the help of Theorem 2.3.15 we prove the existence and uniqueness of the discrete

problem in Definition 2.3.2.

Corollary 2.3.17. Let b · n be piecewise constant on Γin, e.g. b · n|Γin ∈ P0(E inΓ ). For

Cbc,1 = 0 we also require div b + c = 0 on Ω and b · n = 0 on Γin. The discrete solution

uh ∈ S1(T ), ξh ∈ S1∗ (EΓ), φh ∈ P0(EΓ) of the discrete problem in Definition 2.3.7 to the

solution u ∈ H1(Ω), ξ ∈ H1/2∗ (Γ), φ ∈ H−1/2(Γ) of our model problem in Definition 2.0.2

exists and is unique.

Proof. The proof is the same as for Corollary 2.3.12.


2.4 Coupling with the Cell-Centered Finite Volume Method

In this section we provide a discretization of the coupling in a cell-centered finite volume

sense. First, we define a system in a general sense. The second subsection discusses the

approximation problems on the interface Γ and two methods for discretizing the diffusion

flux, namely the first order difference scheme for admissible meshes in the sense of [45]

and the diamond-path method for arbitrary meshes. An upwind scheme for the convection

flux follows the idea of Subsection 2.3.2. Throughout, we assume A = αI with α ∈ R+

for the diffusion matrix.

2.4.1 Discretization in a Cell-Centered Finite Volume Sense

As in Subsection 2.3.1 we assume that the conormal derivative γ1u ∈ L2(Γ), the Cauchy

data γ1uc = φ ∈ L2(Γ) and the jump term t0 ∈ L2(Γ). The control volumes are now the

elements of the primal mesh T . Thus, we integrate (2.1a) over all primal elements T ∈ Tand apply the divergence theorem in order to get

∫

Tdiv(−α∇u+ bu) dx+

∫

Tcu dx =

∫

∂T(−α∇u+ bu) · n ds+

∫

Tcu dx =

∫

Tf dx.

Then we insert (2.1e) and (2.1f), respectively, and write for all T ∈ T∫

∂T \Γ(−α∇u+ bu) · n ds+

∫

Tcu dx+

∫

∂T ∩Γoutb · nu ds

−∫

∂T ∩Γout(φ+ t0) ds =

∫

Tf dx.

We define the diffusion flux ΦDT,E(u) :=

∫E −α∇u · n ds, the convective flux ΦC

T,E(u) :=∫

E b · nu ds over an edge E ∈ ET and the reaction term ΦRT (u) :=

∫T cu dx. Here, the

normal vector n always points outward with respect to T . Then we get for all T ∈ T∑

E∈ET \Γ

ΦDT,E(u) +

∑

E∈ET \Γin

ΦCT,E(u) + ΦR

T (u)

−∫

∂T ∩Γφds =

∫

Tf dx+

∫

∂T ∩Γt0 ds,

(2.41)

which is also known as the balance equation. For the cell-centered finite volume method,

one replaces the continuous diffusion flux ΦDT,E(u) by a discrete diffusion flux FD

T,E(uh),

the continuous convective flux ΦCT,E(u) by a discrete convective flux FC

T,E(uh) and the

reaction term ΦRT (u) by an approximation denoted by FR

T (uh), which we will discuss in

Subsection 2.4.2. Here, uh ∈ P0(T ) is a piecewise constant approximation of u, namely

uT := uh|T ≈ u(xT ), where xT denotes an interior point of an element T ∈ T . The Cauchy

data φ = ∂uc/∂n is approximated through a piecewise constant function φh ∈ P0(EΓ).

Thus, we get an approximated equation for (2.41), which reads for all T ∈ T∑

E∈ET \Γ

FDT,E(uh) +

∑

E∈ET \Γin

FCT,E(uh) + FR

T (uh) −∫

∂T ∩Γφh ds =

∫

Tf dx+

∫

∂T ∩Γt0 ds.

2.4. Coupling with the Cell-Centered Finite Volume Method 45

The cell-centered finite volume method provides a piecewise constant function uh ∈ P0(T )

in the interior domain Ω. The first idea of a coupling is to use

〈uh, ψh〉Γ = 〈ξh, ψh〉Γ + 〈u0, ψh〉Γ for all ψh ∈ P0(EΓ),

which is motivated by (2.1d) and we write with the Calderón system (1.13) as in Subsec-

tion 2.3.1

− 〈uh, ψh〉Γ − 〈Vφh, ψh〉Γ + 〈(1/2 + K)ξh, ψh〉Γ = − 〈u0, ψh〉Γ for all ψh ∈ P0(EΓ),

〈(1/2 + K∗)φh, θh〉Γ + 〈Wξh, θh〉Γ = 0 for all θh ∈ S1∗ (EΓ).

Here, ξh ∈ S1∗ (EΓ) is the piecewise affine approximation of ξ ∈ H

1/2∗ (Γ) on Γ and φh ∈

P0(EΓ) the piecewise constant approximation of φ ∈ H−1/2(Γ) on Γ, respectively.

Definition 2.4.1 (Discrete Problem). Additionally to the data Assumption 2.0.1,

we demand A = αI with α ∈ R+ and t0 ∈ L2(Γ). Find uh ∈ P0(T ), ξh ∈ S1

∗ (EΓ) and

φh ∈ P0(EΓ) such that

∑

E∈ET \Γ

FDT,E(uh) +

∑

E∈ET \Γin

FCT,E(uh) + FR

T (uh)

−∫

∂T ∩Γφh ds =

∫

Tf dx+

∫

∂T ∩Γt0 ds, (2.42a)

− 〈uh, ψh〉Γ − 〈Vφh, ψh〉Γ + 〈(1/2 + K)ξh, ψh〉Γ = − 〈u0, ψh〉Γ , (2.42b)

〈(1/2 + K∗)φh, θh〉Γ + 〈Wξh, θh〉Γ = 0 (2.42c)

for all T ∈ T , θh ∈ S1∗(EΓ), ψh ∈ P0(EΓ).

Remark 2.4.2. There is neither an existence proof nor an a priori result available for

this coupling.

We do not have sufficient numerical tests, which show the robustness and convergence rate

at least in an experimental way for the discrete problem in Definition 2.4.1, especially for

local refined meshes, see Remark 2.4.3. Note that there are a piecewise constant function

uh ∈ P0(T ) from the cell-centered finite volume method and a piecewise affine function

ξh ∈ S1∗ (EΓ) from the boundary element method, which cross each other on the boundary.

This could influence the convergence rate. Therefore, the second approach is to use an

extended piecewise affine discrete solution uh,Γ ∈ S1(EΓ) on the coupling boundary Γ,

which is calculated from the piecewise constant finite volume solution uh of the interior

problem. To get uh,Γ ∈ S1(EΓ) on the coupling boundary Γ, we have to project our

piecewise constant function uh onto the boundary Γ. Therefore, we write

uh,Γ(x) :=∑

a∈NΓ

uaηa(x), (2.43)

where ηa is the standard nodal linear basis function on EΓ associated with the node a,

which defines the nodal basis for S1(EΓ). The approximation of ua is done by an inter-

polation value ua of certain values uT of T ∈ T and a mean value ςa = ςa,h + ςa,t0 of


the approximated conormal uc on Γ, which is given by the solution φh of the boundary

element method for the exterior problem and the jump term t0. For details we refer to

Subsection 2.4.2. This leads us to a second equation block, namely

ua = ua + ςa,h + ςa,t0 for all a ∈ NΓ. (2.44)

For the exterior problem we use the Calderón system (1.13) as in Subsection 2.3.1 and

〈uh,Γ, ψh〉Γ = 〈ξh, ψh〉Γ + 〈u0, ψh〉Γ for all ψh ∈ P0(EΓ),

which is motivated by (2.1d). We get

− 〈uh,Γ, ψh〉Γ − 〈Vφh, ψh〉Γ + 〈(1/2 + K)ξh, ψh〉Γ = − 〈u0, ψh〉Γ for all ψh ∈ P0(EΓ),

〈(1/2 + K∗)φh, θh〉Γ + 〈Wξh, θh〉Γ = 0 for all θh ∈ S1(EΓ).

Here again, ξh ∈ S1∗ (EΓ) is the piecewise affine approximation of ξ ∈ H

1/2∗ (Γ) on Γ and

φh ∈ P0(EΓ) the piecewise constant approximation of φ ∈ H−1/2(Γ) on Γ, respectively.

Remark 2.4.3. Note that this approach gives us additional unknowns on the nodes of

the coupling boundary Γ. Since one of our main focus in this thesis is on local mesh-

refinement, see Chapter 3, we will see in Subsection 2.4.2 and Subsection 3.2.1 that this

approach is more convenient for our purpose, since we will need approximated values also

on the nodes.

The extended discrete problem for the coupling of the cell-centered finite volume method

and boundary element method reads:

Definition 2.4.4 (Extended Discrete Problem). Additionally to the data As-

sumption 2.0.1, we demand A = αI with α ∈ R+ and t0 ∈ L2(Γ). Find uh ∈ P0(T ),

uh,Γ ∈ S1(EΓ), ξh ∈ S1∗ (EΓ) and φh ∈ S0(EΓ) such that

∑

E∈ET \Γ

FDT,E(uh) +

∑

E∈ET \Γin

FCT,E(uh) + FR

T (uh)

−∫

∂T ∩Γφh ds =

∫

Tf dx+

∫

∂T ∩Γt0 ds, (2.45a)

−ua + ua + ςa,h = −ςa,t0 , (2.45b)

− 〈uh,Γ, ψh〉Γ − 〈Vφh, ψh〉Γ + 〈(1/2 + K)ξh, ψh〉Γ = − 〈u0, ψh〉Γ , (2.45c)

〈(1/2 + K∗)φh, θh〉Γ + 〈Wξh, θh〉Γ = 0 (2.45d)

for all T ∈ T , a ∈ NΓ, θh ∈ S1∗(EΓ), ψh ∈ P0(EΓ) with uh,Γ =

∑a∈NΓ

uaηa(x). The values

ua, ςa,h and ςa,t0 are defined in (2.46)–(2.51) in Subsection 2.4.2.

This leads to a system of linear equations.

Remark 2.4.5. Note that in the second equation ςa,t0 is on the right-hand side of the

resulting linear system of equations, since t0 is known. We want to point out that there is

neither an existence proof nor an a priori result available for this type of coupling. Thus,

we assume that this systems is well-defined and gives a unique solution.


T1 T2

T3

E1 E2

xT1

xT2

xT3

xa

anE1

nE2

(a) nE1= nE2

.

T1

T2

E1

E2

xT1

xT2

xa

a

nE1

nE2

(b) nE16= nE2

and #ωa > 1.

E1

E2a

nE1

nE2

xa = xT

T

(c) nE16= nE2

and #ωa = 1.

E1

E2xT1

xT2xT3

nE1

nE2

xa = a

(d) nE16= nE2

, #ωa > 1 and xa = a.

Figure 2.4. The different cases for calculating ua with a ∈ NN and E1, E2 ∈ EN or a ∈ NΓ

and E1, E2 ∈ EΓ. The value ua is an approximation of u in the node a.

2.4.2 Approximation of the Boundary Values and the Fluxes

In this subsection we want to show how we calculate the discrete numerical fluxes FDT,E(uh)

and FCT,E(uh) and the reaction term FR

T,E(uh) of (2.45a). But first, we introduce a method

to approximate u in a point a ∈ N with the values uh ∈ P0(T ) of the cell-centered finite

volume scheme. Later we need these values to approximate the diffusion flux and to define

an a posteriori error estimator. Additionally, this method defines (2.45b). Here, we will

also consider the case of Dirichlet and Neumann boundaries in the interior domain Ω. We

refer to Remark 2.0.3 for the notation.

Approximation of u on a Node a ∈ N . For each node a ∈ N , we define

ua =

∑

T ∈ωa

ΥT (a)uT for all a ∈ NI ,

uD(a) for all a ∈ ND

ua + ςa for all a ∈ NN ∪ NΓ

(2.46)

for certain weightsΥT (a)

∣∣T ∈ T , a ∈ NT

. We use a least square interpolation to cal-


culate the weights. For details on the computation of the weights, the reader is referred

to [30, 34, 35, 39]. We stress that the computation can be done in linear complexity with

respect to the number #T of elements and thus ua for a ∈ NI is interpolated linearly from

the values uT of the node patch ωa. The computation of ua and ςa in case of a Neumann

node a ∈ NN or coupling node a ∈ NΓ is more complicated, see Figure 2.4. First, we give

here the construction for a ∈ NN , which was successfully applied by the author in [42].

To a ∈ NN correspond two edges E1, E2 ∈ EN such that a = E1 ∩ E2. Let nEj denote

the normal vector of Ej . In case of #ωa > 1, let T1, T2 ∈ ωa with T1 6= T2. We define xa

as the intersection of the line γ1(s) = a+ s(n1 + n2)/2 and the line γ2(t) = t(xT1 − xT2).

Moreover, provided that #ωa > 2, we assume that |xa − a| is minimized over all pairs

T1, T2 ∈ ωa. Then, ua ≈ u(xa) is interpolated linearly from uT1 and uT2 ,

ua =uT2 − uT1

|xT2 − xT1 | |xa − xT1 | + uT1 .

For nE1 = nE2 , we choose

ςa = |xa − a|[α−1 1

|E1|( ∫

E1

gN ds+

∫

E1∩Γinb · nE1 uh,ΓN

ds)

+ α−1 1

|E2|( ∫

E2

gN ds+

∫


ds)]/

2

and finally for nE1 6= nE2 , we choose

ςa = λα−1 1

|E1|( ∫

E1

gN ds+

∫


ds)

+ µα−1 1

|E2|( ∫

E2

gN ds+

∫


ds),

where λ, µ ∈ R are calculated from the linear equation a − xa = λnE1 + µnE2 . In case

ωa = T, i.e. a is the node of only one element T ∈ T , we choose xa = xT and ua = uT ,

whereas ςa is computed as before.

Remark 2.4.6. Provided xa = a, we obtain a− xa = 0, λ = µ = 0, and ςa = 0.

The computation of ua in case of a coupling node a ∈ NΓ is the same as before, for ςa

we simply replace g through the coupling conditions (2.1e) and (2.1f), respectively. Note

that t0 is known and φ ∈ H−1/2(Γ), the conormal of uc, is replaced by its discrete value

φh ∈ P0(EΓ) and thus we get for nE1 = nE2

ςa = |xa − a|[α−1 1

|E1|( ∫

E1

(φh + t0) ds+

∫

E1∩Γinb · nuh,Γ ds

)(2.47)

+ α−1 1

|E2|( ∫

E2

(φh + t0) ds+

∫


)]/2 (2.48)

and for nE1 6= nE2

ςa = λα−1 1

|E1|( ∫

E1

(φh + t0) ds+

∫


)(2.49)

+ µα−1 1

|E2|( ∫

E2

(φh + t0) ds+

∫


). (2.50)


T ′T

E

xT ′xTn

(a) For E ∈ EI .

T

E

xT

xEm

n

(b) For E ∈ ED.

Figure 2.5. The orthogonality condition for an interior edge E ∈ EI (a) and a Dirichlet edge

E ∈ ED (b), respectively, for an admissible mesh in the sense of [45] for Definition 2.4.7.

There appear the unknowns φh and uh,Γ in ςa. Thus, we split

ςa = ςa,h + ςa,t0 (2.51)

in the unknown part ςa,h and the known part ςa,t0 .

Discretization of the Diffusion Flux. Note that ΦDE (u) = − ∫E gN ds is known for a

Neumann edge E ∈ EN . Therefore, we define

FDT,E(uh) := ΦD

T,E(u) = −∫

EgN ds for E ∈ EN . (2.52)

Here, there is a slight impreciseness of notation, since there holds ΦDT,E(u) =

∫E(−α∇u+

bu)·n ds for E ∈ E inN , which is not only the pure diffusion flux. The simplest approximation

for the diffusive fluxes on the other edges is a first order difference scheme as it is used in

several computer codes [45]. But first, we have to define an admissible mesh, since this

approximation is limited to this kind of meshes.

Definition 2.4.7 (Admissible Mesh). We say that the triangulation T is admissible

in the sense of [45, Definition 9.1] if the following additional conditions to T hold:

• For E ∈ EI we find T, T ′ ∈ T such that E = T ∩T ′, i.e. T ′ is the neighbor of T . For

the centers xT ⊂ T and xT ′ ⊂ T ′ we assume xT 6= xT ′ . Then the straight line going

through xT and xT ′ is orthogonal to E, see Figure 2.5(a).

• For E ∈ ED we find a T ∈ T such that E = T ∩ ΓD. For a center xT ⊂ T , which is

not on E, let s be the straight line going through xT and orthogonal to E, then we

assume s ∩ E 6= ∅, see Figure 2.5(b).

Remark 2.4.8. Definition 2.4.7 can be extended for E ∈ EΓ and E ∈ EN , where we would

demand the same condition as for E ∈ ED. But since we apply the following first order

difference scheme only on E ∈ EI and E ∈ ED, this definition is sufficient.


xT

xEp

xT ′

xEq

|(xT ′ − xT ) · t|dE = |(xT ′ − xT ) · n|

hE

T ′

T

E

tn

Figure 2.6. Diamond-path, the dotted lines are the control volumes T and T ′.

For an admissible mesh, a first order difference scheme leads to

FDT,E(uh) :=

−hEαuT ′ − uT

|xT ′ − xT | if E ∈ EI and E = T ∩ T ′,

−hEαuEm − uT

|xEm − xT | if E ∈ ED and E = T ∩ ΓD

(2.53)

with uT = uh|T ≈ u(xT ) and uT ′ ≈ u(xT ′) as well as, for E ∈ ED, uEm ≈ uD(xEm) with

a point xEm on E. Another interpretation is that uT represents the integral mean of u

on T . We stress that we do not have a diffusion flux on a coupling edge E ∈ EΓ and a

Neumann edge E ∈ EN in the sense of (2.53), see also (2.45a). The admissibility of the

mesh T allows to choose the centers xT for T ∈ T in a way that the edges E = T ∩ T ′

for any T, T ′ ∈ T are orthogonal to the directions xT −xT ′ , see Figure 2.5(a). For general

meshes, it is not possible to choose the centers xT appropriately, and the approximation

(2.53) is not consistent [45].

Remark 2.4.9. Even if a triangular mesh is admissible in the sense of [45, Definition 9.1],

local mesh-refinement is nontrivial: One has to guarantee that all angles are strictly less

than π/2, i.e. one cannot avoid re-meshing of the domain.

A possible choice of FDT,E(uh) for general meshes is the so-called diamond-path method,

which has been mathematically analyzed in [34, 35] for rectangular meshes with maximum

one hanging node per edge. With the notations from Figure 2.6, where xEp and xEq are

the starting and end point of E ∈ EI ∪ ED, we compute FDT,E(uh). For an interior edge

E ∈ EI we have

FDT,E(uh) := −hEα

(uT ′ − uT

dE− ϕE

uEq − uEp

hE

)

with ϕE =(xT ′ − xT ) · t

(xT ′ − xT ) · n, dE = (xT ′ − xT ) · n.

(2.54)

Here, the normal vector n for FDT,E always points outward with respect to T . The additional

unknowns uEq and uEp are located at the nodes xEq and xEp and are computed by (2.46).


For a boundary edge E ∈ ED, we compute FDT,E(uh) by (2.54), where xT ′ is now replaced

by the midpoint xEm of E and uT ′ becomes uD(xEm).

Remark 2.4.10. We stress that xEq and xEp may occur on the coupling boundary, and

therefore we need the approximation of u on Γ too. We use (2.46) and this leads to an

additional block (2.45b) in the discrete coupling system.

Discretization of the Convective Flux. For the approximation of the convective flux

we adopt the upwind scheme of Subsection 2.3.2 for the cell-centered finite volume method.

In order to avoid technicalities we assume that b · n does not change sign over E ∈ E . For

T ∈ T and E ∈ ET we define

βT,E :=1

|E|

∫

Eb · n ds.

For E ∈ EI let us denote by T ′ the neighbor of T with E = T ∩ T ′. Then we write the

general approximation by

uTh,E = λT

EuT + (1 − λTE)uT ′ ,

where we define λTE := Φ(βT,EhE/α). Here, Φ : R → [0, 1] is a weighting function, see

Subsection 2.3.2 for a detailed discussion. If E ∈ ED with E ∈ ET we choose

uTh,E := λT

EuT + (1 − λTE)uD(xEm).

Note that on E ∈ ED uT always exists, but not uT ′ , thus we use uD(xEm) with the

midpoint xEm of E. For E ∈ EoutΓ ∪ Eout

N with E ∈ ET we take

uTh,E := λT

EuT + (1 − λTE)uh,Γ(xEm)

with the midpoint xEm on E. We remind that uh,Γ is the extended approximation on the

coupling boundary Γ.

Remark 2.4.11. For E ∈ E inΓ ∪E in

N we only have an implicit convective flux in t0 and gN ,

respectively.

For E ∈ ET \(E inΓ ∪ E in

N ) the numerical convection flux reads

FCT,E(uh) := hEβT,Eu

Th,E . (2.55)

Remark 2.4.12. If we use the full upwind scheme, i.e. the weighting function Φ in (2.23),

the convection flux for the cell-centered finite volume method reads as follows. For all

interior edges E ∈ EI we get

FCT,E(uh) :=

hEβT,EuT if βT,E ≥ 0,

hEβT,EuT ′ otherwise.


For E ∈ ED

FCT,E(uh) :=

hEβT,EuT if βT,E ≥ 0,

hEβT,EuD(xEm) otherwise.

For E ∈ EoutΓ ∪ Eout

N

FCT,E(uh) := hEβT,EuT .

Discretization of the Reaction Term. Since uh ∈ P0(T ) the approximation of the

reaction term reads

FRT (uh) := uT

∫

Tc dx for all T ∈ T .

Chapter 3

A Posteriori Error Estimates

In a priori estimates we usually need additional knowledge of the unknown solution, e.g.

regularity, to characterize the error of a discrete solution. On the other hand, a posteriori

estimates work directly with the discrete solution. In context of the finite element method

these estimates are used to do adaptive mesh-refinement, which often leads to an improved

discrete solution, even if the exact solution is not smooth enough. The critical point is if we

can write the a posteriori error estimator in local terms. In recent years a posteriori error

estimates for finite volume methods have been developed, see e.g. [22] for the finite volume

element method and [60, 42, 61, 76] for the cell-centered finite volume method. A posteriori

error estimation for the coupling of finite element method and boundary element method

is well-known, e.g. [25, 14] for the coupling with conforming elements and [19] for non-

conforming elements to mention only a few but not all. Thus, it is a logical consequence to

consider such estimators for our coupling method and we will provide such estimators for

both, the coupling with the finite volume element and with the cell-centered finite volume

method. We have a special focus on estimates, which are robust against the data of the

model problem.

If not otherwise specified, we consider the model data of Assumption 2.0.1. To abbreviate

notation we use the symbol . if an estimate holds up to a multiplicative constant, which

depends only on the shape regularity constant of the elements in T , but neither on the

size nor the number of elements in Ω.

3.1 Estimation for the Coupling with the Finite Volume El-

ement Method

In this section we prove an a posteriori error estimator for T -piecewise constant diffusion

coefficients for the interior problem, which is independent of the variation of the diffusion

coefficients, i.e. the ratio between the maximum and the minimum. This is very important

if we describe a model in Ω with layers of different material. We remark that the proofs

for a diffusion matrix A of Assumption 2.0.1 can be done by an obvious modification.

53

54 Chapter 3. A Posteriori Error Estimates

Theorem 3.1.19Reliability in the energy norms with ap-propriate refinement indicators ηT (3.20):

|||u − uh|||Ω + ‖ξ − ξh‖H1/2(Γ) + ‖φ − φh‖H−1/2(Γ)

≤ Crel

(∑T ∈T

η2T

)1/2.

Definition 3.1.1

Quasi-monotonicity for a T -piecewise

constant diffusions coefficient α.

Lemma 3.1.6–3.1.7

Appropriate interpolant Ih (3.2)

with stability estimates.

Lemma 3.1.11

An orthogonality prop-

erty for the residual R (3.7)

and the jump term J (3.8).

Lemma 3.1.12–3.1.15

Stability estimates of the resid-

ual R (3.7) and the jump J (3.8)

with respect to α, b and c.

Lemma 3.1.16– 3.1.17

L2-Localization of H1/2-

and H−1/2-norm terms.

Theorem 3.1.23

Reliability for upwinding with

Lemma 3.1.21 and 3.1.22 and Theorem 3.1.19

and the additional quantity ηT,up (3.23).

Figure 3.1. The main steps to the proof of Theorem 3.1.19, which shows the reliability of theerror estimator for the coupling with the finite volume element method.

Furthermore, our estimator is robust when we have small diffusion with respect to the

convection field b or the reaction term c. We also mention in some important definitions

what happens, if we have additional boundaries than the coupling boundary, since we

adopt some results from the a posteriori error estimation theory in context with the finite

element method. In particular, these additional boundary conditions are Dirichlet and/or

Neumann boundaries, see also Remark 2.0.3. For technical reasons the prescribed jumps

are slightly more regular, i.e. u0 ∈ H1(Γ) and t0 ∈ L2(Γ). Furthermore, let u ∈ H1(Ω),

ξ ∈ H1/2∗ (Γ), φ ∈ H−1/2(Γ) to be the solution of the weak form in Definition 2.2.2 and uh ∈

S1(T ), ξh ∈ S1∗ (EΓ), φh ∈ P0(EΓ) the solution of the discrete problem in Definition 2.3.2.

Then we define the error e := u − uh ∈ H1(Ω) in the interior domain, the trace error

δ := ξ − ξh ∈ H1/2(Γ) and the conormal error ǫ := φ − φh ∈ H−1/2(Γ) of the exterior

problem. For an overview of the steps to get an appropriate error estimator we refer to

Figure 3.1.

3.1. Estimation for the Coupling with the Finite Volume Element Method 55

Ω1

Ω2

Ω3

Ω4

Ω5

Ω6

Figure 3.2. The partition of Ω into subdomains Ωℓ, 1 ≤ ℓ ≤ L. The different gray colors

show the different α und the light lines the triangulation T , which fits to the partition.

3.1.1 The Piecewise Constant Diffusion Coefficient and Quasi-

Monotonicity

We assume that the diffusion matrix A can be written as A = αI, where I is the identity

matrix and α : Ω → R a given T -piecewise constant function. Additionally, we suppose

that Ω can be partitioned into a finite number of open disjoint subdomains Ωℓ, 1 ≤ ℓ ≤ L

such that the function α is equal to a constant αℓ ∈ R on each Ωℓ and the triangulation

T of Ω fits to Ωℓ that means ∂Ωℓ consists of edges of the underlying triangulation. Thus,

for two subdomains Ωk,Ωℓ with ∂Ωk ∩ ∂Ωℓ 6= ∅ there holds αk 6= αℓ. Otherwise, one can

merge Ωk and Ωℓ with αk = αℓ to a new subdomain, see Figure 3.2. For the piecewise

constant function on α ∈ P0(T ) we write

αT := α|T for all T ∈ T ,

which obviously gives αT = αℓ in Ωℓ, and for all E ∈ E we define

αE :=

maxαT1 , αT2

for E ∈ EI with E = T1 ∩ T2,

αT for E ∈ EΓ with E ∈ ET ,(3.1)

where T1, T2, T ∈ T . Additionally, we define two parameters

αmin = min1≤ℓ≤L

αℓ, αmax = max1≤ℓ≤L

αℓ

and we assume that αmin is positive. Furthermore, we will need the set Qa, which denotes

the union of all simplexes T ∈ ωa for a ∈ N , where αT achieves a maximum in ωa, e.g.

Qa =⋃

T ∈Qa

T, where Qa :=T ∈ ωa

∣∣αT ≥ αT ′ , for all T ′ ∈ ωa.

Note that we allow a large ratio αmax/αmin, and later we want to establish estimates,

which are independent of this ratio, which was first proved for the finite element method

in [9]. There, only meshes T are allowed, where α sufficed a monotone criteria along a path


a

αT1= 0.1

αT2= 1

αT3= 1

αT4= 1

αT5= 100

αT6= 1

(a) Quasi-monotone.

a

αT1= 10

αT2= 1

αT3= 1

αT4= 1

αT5= 100

αT6= 1

(b) Not quasi-monotone.

Figure 3.3. In (a) we see a quasi-monotone distribution of diffusion coefficients α with respect

to a ∈ N . The simplex T is colored dark and the set Qa,T is colored gray. Note that even this

distribution does not fulfill the condition from [9], because there is no monotone path from T2

to T6. In (b) the distribution of α is not quasi-monotone with respect to a.

between two subdomains, which share a least a point. As in [63, 47] for finite elements,

we can prove estimates for a larger class of α, namely of quasi-monotone type. This leads

to the definition of quasi-monotonicity, which was introduced in [37], see Figure 3.3.

Definition 3.1.1 (Quasi-Monotonicity). Let a ∈ N . We say α is quasi-monotone in

ωa with respect to a, if for all triangles T ∈ ωa there exists an open and simply connected

set Qa,T with T ∪Qa ⊂ Qa,T ⊂ ωa such that

αT ≤ αT ′ for all T ′ ⊂ Qa,T , T ′ ∈ ωa.

We call α quasi-monotone, if α is quasi-monotone for all a ∈ N .

Remark 3.1.2. For a ∈ ND we also have the condition |∂Ta,qm ∩ ΓD| > 0 in Defini-

tion 3.1.1.

Remark 3.1.3. A sufficient condition for α to be quasi-monotone is the following: For a

point a ∈ N we denote by n the number of subdomains Ωℓ (1 ≤ ℓ ≤ L) to whose closure

a belongs to. The function α is quasi-monotone with respect to a ∈ NI , if n ≤ 3 and for

a ∈ NΓ it is quasi-monotone, if n ≤ 2. The same is valid for pure Dirichlet or Neumann

boundary condition. If we have mixed boundary condition (Dirichlet and Neumann), we

have a point a, where the boundary condition change, n ≤ 1. If a ∈ N lies in a subdomain

Ωℓ, α is always quasi-monotone with respect to a.

Interpolant. For the analysis of an a posteriori error estimator we need an interpolant Ih :

H1(Ω) → S1(T ), which is also an essential tool in the proof of residual-based a posteriori

estimates for the finite element method. A well-known approximation of H1 functions into


the finite element space was established in [29]. The general form of Ih is

Ihv :=∑

a∈NΠav ηa,

where the operator Πa : H1(ω) → R on a domain ω ⊂ Ω is linear and continuous and

ηa is the standard nodal linear basis function associated with the node a. For Πa there

exists several possibilities, e.g. [29, 69, 9, 64, 16, 47] to mention a few but not all and

applied in context of the finite element method. For our purpose the interpolants and

estimates in [9, 63, 64] and [47] are appropriate, because the multiplicative constants of

the estimates do not depend on the ratio αmax/αmin. In [9] they consider a posteriori

estimates for non-smooth diffusion coefficients α with this desirable property, where the

projection Πa is the integral mean on a certain domain ω ⊂ Ω. The extension of this

operator type to quasi-monotone α can be found in [63, 64]. The operator in [16] on

the other hand modifies the Clément operator [29] in the setting of a partition of unity

with the effect that the approximation error has a local average of zero. This results in

a residual-based a posteriori error estimate with a volume contribution, which is smaller

than in the standard estimate. A modification to handle quasi-monotone α can be found

in [47]. Thus, we want to mention the operators of [63, 64] and [47] and its useful estimates

in more detail. The interpolation operator IPh of [63, 64] for v ∈ H1(Ω) is defined by

IPh v :=

∑

a∈NΠav ηa, where Πav :=

1

|Qa|

∫

Qa

v dx

and we state the following lemma:

Lemma 3.1.4. Let v ∈ H1(Ω), T ∈ T , E ∈ E and α is quasi-monotone. Then the

following inequalities hold:

‖IPh v‖L2(T ) . ‖v‖L2(ω

T ),

‖v − IPh v‖L2(T ) . hT ‖∇v‖L2(ω

T ) . α−1/2T hT ‖α∇v‖L2(ω

T ),

‖v − IPh v‖H1(T ) . ‖∇v‖L2(ω

T ) . α−1/2T ‖∇v‖L2(ω

T ),

α1/2E ‖v − IP

h v‖L2(E) . h1/2E ‖α∇v‖L2(ω

E).

Proof. For a proof see e.g. [63], where the statements are shown for two and three

dimensions and for the interpolant, which satisfies the Dirichlet condition IPh : H1

D(Ω) →S1

D(T ). Here, H1D(Ω) denotes the Sobolev space H1(Ω), where the traces on ΓD are zero,

S1D(T ) = S1(T )∩H1

D(Ω). In the definition and in the proofs of [63] the Dirichlet boundary

ΓD 6= ∅ is not mandatory, thus we skip the proof.

Remark 3.1.5. Note that in the standard analysis of a posteriori error estimates for the

finite element method with Dirichlet boundary conditions one uses the Galerkin orthogo-

nality, which is valid for test functions vh ∈ S1D(T ). Thus, Ih has to be a mapping from

H1D(Ω) to S1

D(T ). In our analysis we do not need the Galerkin orthogonality, thus for us

Ih : H1(Ω) → S1(T ) is enough, even if we consider ΓD 6= ∅.


For the interpolation operator of [47] we write Ih : H1(Ω) → S1(T ) and we will refer in our

further analysis to this operator and its estimates. The operator is defined for v ∈ H1(Ω)

by

Ihv :=∑

a∈NΠav ηa, where Πav :=

∫

ωa

αηav dx/ ∫

ωa

αηa dx. (3.2)

Note that in [16] and [47] the numerator has an additional partition of unity, which is not

needed here, since we do not consider Dirichlet boundary conditions. The proofs of the

following estimates can be found in [47] and are still valid for ΓD = ∅. There, one can also

find an explicit upper bound for the constant in the estimates, which depends only on the

shape of the elements of T ∈ T .

Lemma 3.1.6. Let v ∈ H1(Ω). Then there holds

‖Ihv‖L2(T ) . ‖v‖L2(T ). (3.3)

Lemma 3.1.7. Let v ∈ H1(Ω) and α be quasi-monotone. Then the following bounds holds

(∑

T ∈Th−2

T αT ‖v − Ihv‖2L2(T )

)1/2

. ‖α1/2∇v‖L2(Ω). (3.4)

Proof. See [47, Theorem 2.63]. There, the constant consists of the term maxa∈N (ha/hTa)

with ha := diam(ωa) and hTa = maxT ∈ωa, which is bounded since T is assumed to be

regular and does neither depend on the size nor on the number of the elements.

Lemma 3.1.8. Let v ∈ H1(Ω) and α be quasi-monotone. Then the following bound holds

(∑

T ∈TαT ‖∇Ihv‖2

L2(T )

)1/2

. ‖α1/2∇v‖L2(Ω). (3.5)

Proof. See [47, Theorem 2.65]. In the constant we have the term

maxa∈N maxT ∈ωaha/ρa,T where ρa,T for a ∈ N and T ∈ T denotes the distance from

one corner a of T to the opposite side. Since T is assumed to be regular this term is

bounded and does neither depend on the size nor on the number of the elements.

Lemma 3.1.9. Let v ∈ H1(Ω), α be quasi-monotone and for T ∈ T there is at least one

node in Ω. Then we have

(∑

E∈Eh−1

E αE‖v − Ihv‖2L2(E)

)1/2

. ‖α1/2∇v‖L2(Ω). (3.6)

Proof. See [47, Theorem 2.68].

Remark 3.1.10. The condition that at least one node of T has to be in Ω can be easily

achieved through local mesh-refinement and is not a strong restriction.


3.1.2 Residual-Based Error Estimation

One of the main tool to prove a residual-based error estimator for the finite element method

is to use the Galerkin orthogonality of the error. This allows us to insert functions of the

type of the test space. More precisely we can insert the interpolant Ih and with the above

inequalities (3.3)–(3.6) for Ih, we naturally get terms with the local mesh size hT , which

are appropriate for adaptive mesh-refining. In the finite volume element method we only

have an adapted orthogonality, i.e. there does not hold a orthogonality property of the

error with respect to the test space. The main idea for the proof of our a posteriori estimate

is to insert the interpolant Ih to get a similar part to the finite element analysis and a

second (discrete) part, where we can prove estimates with respect to the test functions

P0(T ∗) as well. But first, we have to introduce some important terms, namely the residual

R := R(uh) = f − div(−α∇uh + buh) − cuh on T ∈ T (3.7)

and an edge-residual or jump J : L2(E) → R by

J |E := J(uh)|E =

[[−α∇uh]] · n for all E ∈ EI ,

(−α∇uh + buh) · n + φh + t0 for all E ∈ E inΓ ,

−α∇uh · n + φh + t0 for all E ∈ EoutΓ .

(3.8)

Although for T -piecewise constant α the term div(−α∇uh) = 0 on T ∈ T , we do not

neglect it. Thus, we can see what happens if α would not be constant on T . We need

additional quantities to develop a robust a posteriori estimator with respect to the model

data. Additionally to

αE = maxαT1 , αT2

for EI with E ∈ T1 ∩ T2,

αE = αT for EΓ with E ∈ ET ,

we define

βT := minx∈T

1

2div b(x) + c(x)

for all T ∈ T .

Furthermore, we define

βE := minβT1 , βT2

for E ∈ EI with E ∈ T1 ∩ T2,

βE := βT for E ∈ EΓ with E ∈ ET .

Later we will need minβ−1

T , h2Tα

−1T

and min

β

−1/2E , hEα

−1/2T

. If the first argument

is 0, we take the second argument as minimum. We refer also to Figure A in the In-

dex of Notation. For the error e := u − uh ∈ H1(Ω) we define its discrete error by

eh := Ihe ∈ S1(T ) and e∗h := I∗

heh ∈ P0(T ∗). As mentioned above the natural Galerkin

orthogonality of the error is a powerful tool for the proof of a posteriori estimates for the


finite element method. This property occurs in the coupling of the finite element method

and the boundary element method as well, since the discrete ansatz and test space are sub-

spaces of the continuous spaces. For finite volume schemes we need a different approach.

Here, the balance equation (2.17a) plays a key role to prove the following L2-orthogonality

property.

Lemma 3.1.11. With the notation from above we get for e∗h ∈ P0(T ∗)

∑

T ∈T

∫

TRe∗

h dx+∑

T ∈E

∫

EJe∗

h ds = 0. (3.9)

Proof. The proof follows the ideas of [22]. Here, we also have to consider the coupling

boundary Γ. For each control volume V ∈ T ∗ we have from (2.17a)

∫

∂V \Γ(−α∇uh + buh) · n ds (3.10)

=

∫

Vf dx−

∫

Vcuh dx−

∫

∂V ∩Γout(b · n)uh ds+

∫

∂V ∩Γt0 ds+

∫

∂V ∩Γφh ds.

On the other hand we get by use of the Gauss divergence theorem for all V ∈ T ∗

∑

T ∈T

∫

T ∩Vdiv(−α∇uh + buh) dx

=∑

ζ∈EV \Γ

∫

ζ[[−α∇uh]] · n ds+

∫

∂V(−α∇uh + buh) · n ds

and write

∫

∂V \Γ(−α∇uh + buh) · n ds =

∑

T ∈T

∫

T ∩Vdiv(−α∇uh + buh) dx

−∑

ζ∈EV \Γ

∫

ζ[[−α∇uh]] · n ds−

∫

∂V ∩Γ(−α∇uh + buh) · n ds.

(3.11)

Next we subtract (3.11) from (3.10) and multiply this difference by e∗h ∈ P0(T ∗). This

yields to

∫

V(f − div(−α∇uh + buh) − cuh)e∗

h dx+∑

ζ∈EV \Γ

∫

ζ[[−α∇uh]] · ne∗

h ds+

∫

∂V ∩Γt0e

∗h ds

+

∫

∂V ∩Γ(−α∇uh + buh) · n e∗

h ds−∫

∂V ∩Γoutb · nuhe

∗h ds+

∫

∂V ∩Γφhe

∗h ds = 0

for all V ∈ T ∗. Summing over all V ∈ T ∗ and merging terms with (3.7) and (3.8) proves

the lemma.

For the proof of the reliability of an error estimator we need robust estimates of e − Ihe

in the energy norm, which are provided in Lemma 3.1.12 and Lemma 3.1.13. Besides the

Cauchy-Schwarz inequality we use mainly Lemma 3.1.6–3.1.9 for the proof.


Lemma 3.1.12. For the residual R, the error e ∈ H1(Ω) and eh = Ihe ∈ S1(T ) we have

∑

T ∈T

∫

TR(e− eh) dx .

(∑

T ∈Tmin

β−1

T , h2Tα

−1T

‖R‖2

L2(T )

)1/2

|||e|||Ω. (3.12)

Proof. The Cauchy-Schwarz inequality yields

∑

T ∈T

∫

TR(e− eh) dx

≤(∑

T ∈Tmin

β−1

T , h2Tα

−1T

‖R‖2

L2(T )

)1/2

(∑

T ∈T

(min

β−1

T , h2Tα

−1T

)−1‖e− eh‖2

L2(T )

)1/2

.

The second sum on the right-hand side can be split into two parts, namely

∑

T ∈T

h−2T

αT ≤βT

βT ‖e− eh‖2L2(T ) ≤

∑

T ∈TβT ‖e− eh‖2

L2(T ) .∑

T ∈TβT ‖e‖2

L2(T )

≤∑

T ∈T|||e|||2T = |||e|||2Ω,

where we have used (3.3) to estimate ‖e− eh‖2L2(T ) ≤ ‖e‖2

L2(T ) + ‖eh‖2L2(T ) . ‖e‖2

L2(T ) and

the property βT ≤ 12 div b(x) + c(x) for x ∈ T . For the second term we use (3.4) to get

∑

T ∈T

h−2T

αT >βT

h−2T αT ‖e− eh‖2

L2(T ) ≤∑

T ∈Th−2

T αT ‖e− eh‖2L2(T ) . ‖α1/2∇e‖2

L2(Ω) = |||e|||2Ω,


Additionally to the above techniques, we need Young’s inequality and the trace inequal-

ity (1.26) of Section 1.5 for the following proof.

Lemma 3.1.13. Suppose that the jump J ∈ L2(E) for E ∈ E. Then with the error

e ∈ H1(Ω) and eh = Ihe ∈ S1(T ) we have

∑

E∈E

∫

EJ(e− eh) ds .

(∑

E∈Eα

−1/2E min

β

−1/2E , hEα

−1/2E

‖J‖2

L2(E)

)1/2

|||e|||Ω. (3.13)

Proof. If we use the Cauchy-Schwarz inequality we obtain

∑

E∈E

∫

EJ(e− eh) ds ≤

(∑

E∈Eα

−1/2E min

β

−1/2E , hEα

−1/2E

‖J‖2

L2(E)

)1/2

(3.14)

(∑

E∈E

(α

1/2E min

β

−1/2E , hEα

−1/2E

)−1‖e− eh‖2

L2(E)

)1/2

.


As above we consider two cases for the last sum in (3.14). For the first case we estimate

∑

E∈E

h−2E

αE≤βE

α1/2E β

1/2E ‖e− eh‖2

L2(E)

.∑

E∈E

h−2E

αE≤βE

α1/2E β

1/2E

(h−1

E ‖e− eh‖2L2(TE) + ‖e− eh‖L2(TE)‖∇(e− eh)‖L2(TE)

),

where we have used the trace inequality (1.26). For the latter, the element TE adjacent

to E must be chosen such that αTEis maximal, e.g. αTE

= αE . If we use the property

h−1E α

1/2E ≤ β

1/2E and Young’s inequality with a parameter γE > 0 we may write

∑

E∈E

h−2E

αE≤βE

α1/2E β

1/2E ‖e− eh‖2

L2(E)

.∑

E∈E

h−2E

αE≤βE

βE‖e− eh‖2L2(TE) +

α1/2E β

1/2E

2γE‖e− eh‖2

L2(TE)

+α

1/2E β

1/2E γE

2‖∇(e− eh)‖2

L2(TE)

.∑

E∈EβE‖e‖2

L2(TE) +∑

E∈EαE‖∇e‖2

L2(TE) +∑

E∈EαE‖∇eh‖2

L2(TE)

.∑

E∈E|||e|||2TE

+∑

E∈E‖α1/2∇e‖2

L2(TE) +∑

E∈E‖α1/2∇eh‖2

L2(TE),

where we have used γE = α1/2E /β

1/2E and (3.3) to estimate ‖e − eh‖2

L2(T ) . ‖e‖2L2(T ),

βE ≤ βTEand αE = αTE

. If we use (3.5) we get

∑

E∈E

h−2E

αE≤βE

α1/2E β

1/2E ‖e− eh‖2

L2(E) . |||e|||2Ω + ‖α∇e‖2L2(Ω) + ‖α∇eh‖2

L2(Ω) . |||e|||2Ω.

The second part of the right sum of (3.14) can be estimated by

∑

E∈E

h−2E

αE>βE

α1/2E h−1

E α1/2E ‖e− eh‖2

L2(E) . ‖α1/2∇e‖2L2(Ω),

where we can apply (3.6), and this completes the proof.

Additionally to the above tools, the property of the T ∗-piecewise interpolation operator,

see Lemma 1.4.2, is used to prove the following lemmas.

Lemma 3.1.14. For the residual R, eh = Ihe ∈ S1(T ) and e∗h = I∗

heh ∈ P0(T ∗) we have

∑

T ∈T

∫

TR(eh − e∗

h) dx .

(∑

T ∈Tmin

β−1

T , h2Tα

−1T

‖R‖2

L2(T )

)1/2

|||e|||Ω. (3.15)


Proof. The Cauchy-Schwarz inequality yields to

∑

T ∈T

∫

TR(eh − e∗

h) dx ≤(∑

T ∈Tmin

β−1

T , h2Tα

−1T

‖R‖2

L2(T )

)1/2

(∑

T ∈T

(min

β−1

T , h2Tα

−1T

)−1‖eh − e∗

h‖2L2(T )

)1/2

.

We may estimate

∑

T ∈T

h−2T

αT ≤βT

βT ‖eh − e∗h‖2

L2(T ) ≤∑

T ∈TβT ‖eh − e∗

h‖2L2(T ) .

∑

T ∈TβT ‖eh‖2

L2(T )

.∑

T ∈TβT ‖e‖2

L2(T ) ≤ |||e|||2Ω,

where we have used (1.22), since each T contains 6 triangles from⋃

V ∈T ∗ ZV , and (3.3).

The remaining terms of the sum are estimated by

∑

T ∈T

h−2T

αT >βT

h−2T αT ‖eh − I∗

heh‖2L2(T ) ≤

∑

T ∈Th−2

T αT ‖eh − I∗heh‖2

L2(T ) .∑

T ∈TαT ‖∇eh‖2

L2(T )

= ‖α1/2∇eh‖2L2(Ω) . ‖α1/2∇e‖2

L2(Ω) ≤ |||e|||2Ω,

where we have used (1.21), since each T contains 6 triangles from⋃

V ∈T ∗ ZV and (3.5).

Lemma 3.1.15. Suppose that the jump J ∈ L2(E) for E ∈ E. Then with eh = Ihe ∈S1(T ) and e∗

h = I∗heh ∈ P0(T ∗) we have

∑

E∈E

∫

EJ(eh − e∗

h) ds .

(∑

E∈Eα

−1/2E min

β

−1/2E , hEα

−1/2E

‖J‖2

L2(E)

)1/2

|||e|||Ω. (3.16)

Proof. If we use the Cauchy-Schwarz inequality we obtain

∑

E∈E

∫

EJ(eh − e∗

h) ds ≤(∑

E∈Eα

−1/2E min

β

−1/2E , hEα

−1/2E

‖J‖2

L2(E)

)1/2

(3.17)

(∑

E∈E

(α

−1/2E min

β

−1/2E , hEα

−1/2E

)−1‖eh − e∗

h‖2L2(E)

)1/2

.

For the first part of the right sum we find for every ζ ∈ EV a unique E ∈ E with ζ ⊂ E

and because of the construction of T ∗ here holds 2hζ = hE . Thus, we define αζ := αE

and βζ := βE , where TE is the element adjacent to E such that αTEis maximal, e.g.

αTE= αE = αζ . Additionally, Zζ ∈ ZV is the unique triangle with Zζ ⊂ TE and ζ

is a side of Zζ . Note that in general there does not hold βE = βTE. This leads to the


estimation

∑

E∈E

h−2E

αE≤βE

α1/2E β

1/2E ‖eh − e∗

h‖2L2(E) =

∑

V ∈T ∗

∑

ζ∈EV(2hζ )−2αζ ≤βζ

α1/2ζ β

1/2ζ ‖eh − e∗

h‖2L2(ζ)

.∑

V ∈T ∗

∑


α1/2ζ β

1/2ζ

(h−1

ζ ‖eh − e∗h‖2

L2(Zζ)

+ ‖eh − e∗h‖L2(Zζ)‖∇(eh − e∗

h)‖L2(Zζ)

),

where we have used the trace inequality (1.26). If we use the property (2hζ)−1α1/2ζ ≤ β

1/2ζ

and Young’s inequality with a parameter γζ > 0 we may write

∑

E∈E

h−2E

αE≤βE

α1/2E β

1/2E ‖eh − e∗

h‖2L2(E)

.∑

V ∈T ∗

∑


βζ‖eh − e∗h‖2

L2(Zζ) +α

1/2ζ β

1/2ζ

2γζ‖eh − e∗

h‖2L2(Zζ)

+α

1/2ζ β

1/2ζ γζ

2‖∇(eh − e∗

h)‖2L2(Zζ)

.∑

V ∈T ∗

∑

ζ∈EV

βζ‖eh‖2L2(Zζ) + αζ‖∇eh‖2

L2(Zζ),

where we have used γζ = α1/2ζ /β

1/2ζ , (1.22) and the fact that ∇e∗

h = 0 on Zζ . Finally, we

get with βE ≤ βTE, (3.3) and (3.5)

∑

E∈E

h−2E

αE≤βE

α1/2E β

1/2E ‖eh − e∗

h‖2L2(E) .

∑

E∈EβTE

‖eh‖2L2(TE) + αTE

‖∇eh‖2L2(TE)

≤∑

T ∈TβT ‖eh‖2

L2(T ) +∑


L2(T ) . |||e|||2Ω.

The last step in the proof is the estimate

∑

E∈E

h−2E

αE>βE

α1/2E h−1

E α1/2E ‖eh − e∗

h‖2L2(E) =

∑

V ∈T ∗

∑

ζ∈EV

(2hζ)−1αζ‖eh − e∗h‖2

L2(ζ)

.∑

V ∈T ∗

∑

ζ∈EV

(2hζ)−1αζ

(h−1

ζ ‖eh − e∗h‖2

L2(Zζ) + hζ‖∇eh‖2L2(Zζ)

),

where we have used the trace inequality (1.25) and ∇e∗ = 0 on Zζ . If we use (1.21) we

∑

E∈E

h−2E

αE>βE

α1/2E h−1

E α1/2E ‖eh − e∗

h‖2L2(E) .

∑

V ∈T ∗

∑

ζ∈EV

αζ‖∇eh‖2L2(Zζ) .

∑

E∈EαTE

‖∇eh‖2L2(TE)

.∑


L2(T ) . ‖α1/2∇e‖2L2(Ω) ≤ |||e|||2Ω

and conclude with the estimate of (3.5).


The next two lemmas are well-known in the context of a posteriori estimates for boundary

element methods, e.g. [15]. In [25, 14] the first lemma was also used for the analysis of the

coupling of the finite element method and the boundary element method. In the following,

∂/∂s denotes the arc length derivative. The constant C(EΓ) depends on the (boundary-)

mesh EΓ. The mesh-dependence is very weak,

C(EΓ) := C (log(1 +))1/2 ,

where C > 0 and := maxhEi/hEj : Ei ∈ EΓ is a neighbor of Ej ∈ EΓ.

Lemma 3.1.16. Assume v ∈ H1(Γ) has at least one root in each element of EΓ. Then

there holds

‖v‖H1/2(Γ) ≤ C(EΓ)

∑

E∈EΓ

hE‖∂v/∂s‖2L2(E)

1/2

. (3.18)

Proof. See e.g. [15, Theorem 1] for a proof.

Lemma 3.1.17. If v ∈ L2(Γ) is L2(Γ)−orthogonal to S1(EΓ), then there holds

‖v‖H−1/2(Γ) ≤ C(EΓ)

∑

E∈EΓ

hE‖v‖2L2(E)

1/2

. (3.19)

Proof. See e.g. [15, Theorem 2] for a proof.

Remark 3.1.18. The constant C(EΓ) is bounded because we can bound to a certain

value, e.g. ≤ 10. This can easily be achieved if we refine the element with the edge

on the boundary Γ, where the ratio with the neighbor boundary edge exceeds. Thus, our

notation . is valid for (3.18)–(3.19) as well.

3.1.3 Reliability of the Error Estimator

In this subsection we prove reliability of an a posteriori error estimator. First, we want to

give some important notations. For each element T ∈ T , we define

µT := minβ

−1/2T , hTα

−1/2T

and for each edge E ∈ E

µE := minβ

−1/2E , hEα

−1/2E

and we remember the definitions αE = maxαT1 , αT2 and βE = minβT1 , βT2 for EI with

E ∈ T1 ∩ T2 and αE = αT and βE = βT for E ∈ EΓ with E ∈ ET . We refer to Figure A in

the Index of Notation to get an overview of the introduced quantities.


Error Estimator. This leads us to the definition of the refinement indicator

η2T := µ2

T ‖R‖2L2(T ) +

1

2

∑

E∈EI∩ET

α−1/2E µE‖J‖2

L2(E) +∑

E∈EΓ∩ET

α−1/2E µE‖J‖2

L2(E)

+∑

E∈EΓ∩ET

hE‖ ∂∂suh − ∂

∂s

(u0 − Vφh + (1/2 + K)ξh

)‖2L2(E)

+∑

E∈EΓ∩ET

hE‖Wξh + (1/2 + K∗)φh‖2L2(E)

(3.20)

with R and J from (3.7) and (3.8), respectively. We are now able to formulate and prove

an upper bound of the coupling errors expressed by the refinement indicators.

Theorem 3.1.19 (Reliability without Upwinding). Let u ∈ H1(Ω), ξ ∈ H1/2∗ (Γ),

φ ∈ H−1/2(Γ) be the solution of our model problem in Definition 2.0.2 and uh ∈ S1(T ),

ξh ∈ S1∗ (EΓ), φh ∈ P0(EΓ) the discrete solution of the discrete problem in Definition 2.3.2.

There is a constant Crel > 0, which depends only on the shape of the elements in T but

neither on the size nor on the number of elements such that

|||u− uh|||Ω + ‖ξ − ξh‖H1/2(Γ) + ‖φ− φh‖H−1/2(Γ) ≤ Crel

(∑

T ∈Tη2

T

)1/2

.

Proof. By the definition of the energy norm we start with

|||e|||2Ω ≤ (α∇e− be,∇e)Ω + (ce, e)Ω + 〈b · n e, e〉Γout

= (α∇u− bu,∇e)Ω + (cu, e)Ω + 〈b · nu, e〉Γout

+ (−α∇uh + buh,∇e)Ω − (cuh, e)Ω − 〈b · nuh, e〉Γout

=∑

T ∈T

∫

T(f − cuh)e dx+

∑

E∈EΓ

∫

Et0e ds+ 〈φ, e〉Γ

+∑

T ∈T

∫

T(−α∇uh + buh) · ∇e dx−

∑

E∈EoutΓ

∫

Eb · nuhe ds

=∑

T ∈T

∫

T(f − cuh)e dx+

∑

E∈EΓ

∫

Et0e ds+ 〈φ, e〉Γ −

∑

E∈EoutΓ

∫

Eb · nuhe ds

+∑

T ∈T

∫

∂T(−α∇uh + buh) · n e ds−

∑

T ∈T

∫

Tdiv(−α∇uh + buh)e dx,

where we have used (2.8a) and integration by parts. We insert the residual R = f −div(−α∇uh + buh) − cuh and we write the fourth sum over the inner edge EI in order to

get

|||e|||2Ω ≤∑

T ∈T

∫

TRedx+

∑

E∈EΓ

∫

Et0e ds+ 〈φ, e〉Γ +

∑

E∈EI

∫

E[[−α∇uh]] · n e ds

+∑

E∈EinΓ

∫

E(−α∇uh + buh) · n e ds+

∑

E∈EoutΓ

∫

E−(α∇uh) · n e ds.


Subtracting the identity from (3.9) with eh = Ihe we get

|||e|||2Ω ≤∑

T ∈T

∫

TR (e− eh + eh − e∗

h) dx+∑

E∈EI

∫

E[[−α∇uh]] · n (e− eh + eh − e∗

h) ds

+∑

E∈EinΓ

∫

E((−α∇uh + buh) · n + φh + t0)(e− eh + eh − e∗

h) ds (3.21)

+∑

E∈EoutΓ

∫

E(−(α∇huh) · n + φh + t0) (e− eh + eh − e∗

h) ds+ 〈φ− φh, e〉Γ .

The terms with (e − eh + eh − e∗h) on the right-hand side can be split into two terms,

namely:

Fem =∑

T ∈T

∫

TR (e− eh) dx+

∑

E∈E

∫

EJ (e− eh) ds

and

Fvm =∑

T ∈T

∫

TR (eh − e∗

h) dx+∑

E∈E

∫

EJ (eh − e∗

h) ds,

where we have used (3.8) to merge the sums over an edge. The term Fem can be estimated

by (3.12) and (3.13) and Fvm by (3.15) and (3.16):

Fem + Fvm .

(∑

T ∈Tµ2

T ‖R‖2L2(T )

)1/2

+

(∑

E∈Eα

−1/2E µE‖J‖2

L2(E)

)1/2 |||e|||Ω. (3.22)

Thus, it only remains to estimate the last term of (3.21). The error on the boundary can

be written as e = −p0 − Vǫ+ (1/2 + K)δ. Thus, we estimate

〈φ− φh, e〉Γ = 〈ǫ,−p0〉Γ + 〈ǫ,−Vǫ〉Γ + 〈ǫ, (1/2 + K)δ〉Γ

≤ ‖ǫ‖H−1/2(Γ)‖p0‖H1/2(Γ) − 〈ǫ,Vǫ〉Γ + 〈p1, δ〉Γ − 〈Wδ, δ〉Γ

≤ ‖ǫ‖H−1/2(Γ)‖p0‖H1/2(Γ) + ‖δ‖H1/2(Γ)‖p1‖H−1/2(Γ) − 〈ǫ,Vǫ〉Γ − 〈Wδ, δ〉Γ ,

where we have used the Hölder inequality, the fact that K and K∗ are adjoint and the

definition of p1, see (2.29). To estimate ‖p0‖H1/2(Γ) we remark that u0 ∈ H1(Γ), Vφh :

L2(Γ) 7→ H1(Γ) and Kξh : H1(Γ) 7→ H1(Γ) and thus p0 = uh − u0 + Vφh − (1/2 + K)ξh ∈H1(Γ). Since p0 is L2(Γ)-orthogonal to P0(EΓ) there holds

∫E p0 ds = 0 for all E ∈ EΓ and

thus we have at least one zero of the function p0 in the interior of E. Therefore, we can

apply (3.18). For ‖p1‖H−1/2(Γ) we note that K∗φh : L2(Γ) 7→ L2(Γ) and Wξh : H1(Γ) 7→L2(Γ). Thus, p1 = (1/2 + K∗)ǫ + Wδ = −(1/2 + K∗)φh − Wξh ∈ L2(Γ) and since p1 is

L2(Γ)-orthogonal to S1∗ (EΓ) we can apply (3.19). Altogether we get for (3.21)

|||e|||2Ω .

(∑

T ∈Tµ2

T ‖R‖2L2(T )

)1/2

+

(∑

E∈Eα

−1/2E µE‖J‖2

L2(E)

)1/2 |||e|||Ω

+ ‖ǫ‖H−1/2(Γ)

∑

E∈EΓ

hE‖∂p0/∂s‖2L2(E)

1/2

+ ‖δ‖H1/2(Γ)

∑

E∈EΓ

hE‖p1‖2L2(E)

1/2

− 〈ǫ,Vǫ〉Γ − 〈Wδ, δ〉Γ .


Since V and W are positive definite on H−1/2(Γ) and H1/2∗ (Γ), respectively, we can put

the last two terms to the left-hand side and write

|||e|||2Ω + ‖δ‖2H1/2(Γ) + ‖ǫ‖2

H−1/2(Γ)

.

(∑

T ∈Tµ2

T ‖R‖2L2(T )

)1/2

+

(∑

E∈Eα

−1/2E µE‖J‖2

L2(E)

)1/2 |||e|||Ω

+

∑

E∈EΓ

hE‖∂/∂s (uh − u0 + Vφh − (1/2 + K)ξh) ‖2L2(E)

1/2

‖ǫ‖H−1/2(Γ)

+

∑

E∈EΓ

hE‖Wξh + (1/2 + K∗)φh‖2L2(E)

1/2

‖δ‖H1/2(Γ).

If we use again Cauchy-Schwarz inequality we arrive at

|||e|||2Ω + ‖δ‖2H1/2(Γ) + ‖ǫ‖2

H−1/2(Γ)

.∑

T ∈Tµ2

T ‖R‖2L2(T ) +

∑

E∈Eα

−1/2E µE‖J‖2

L2(E)

+∑

E∈EΓ

hE‖∂/∂s (uh − u0 + Vφh − (1/2 + K)ξh) ‖2L2(E)

+∑

E∈EΓ

hE‖Wξh + (1/2 + K∗)φh‖2L2(E),


Error Estimator with Upwind Approximation. In this paragraph uh ∈ S1(T ),

ξh ∈ S1∗ (EΓ) and φh ∈ P0(EΓ) is the discrete solution of problem in Definition 2.3.7. We

stress that in general the estimator of Theorem 3.1.19 is not valid for this solution. We

emphasize that we will get an additional quantity, which considers the upwinding behavior

of the solution.

Remark 3.1.20. Note that there does not hold AV (uh, v∗) − (φh, v

∗)Γ = (f, v∗)Ω +

(t0, v∗)Γ. We also do not use AV (u, φ, v∗) − (φ, v∗)Γ = (f, v∗)Ω + (t0, v

∗)Γ. Thus our

analysis for the a posteriori estimator for our upwind estimator differs from [22].

The notation in this paragraph is as usual, e := u − uh ∈ H1(Ω) defines the error in the

interior domain, δ := ξ − ξh ∈ H1/2∗ (Γ) and ǫ := φ − φh ∈ H−1/2(Γ) are the trace and

conormal error on the boundary Γ, respectively, eh := Ihe ∈ S1(T ) and e∗h := I∗

heh ∈P0(T ∗). The same is valid for the residual R := R(uh) defined in (3.7) and the edge-

residual J |E := J(uh)|E defined in (3.8), now always defined with the upwind solution uh

of the system in Definition 2.3.7. We define an additional quantity for the error estimator

for T ∈ T , namely

η2T,up := α

−1/2T µT

∑

τTij ∈DT

‖b · ni(uh − uTh,ij)‖2

L2(τTij ), (3.23)


where uTh,ij is defined in (2.22). We refer to Figure 1.3(a) and Figure 2.3 for the notation

of τTij , DT and the normal vector ni. Before we can prove the reliability we provide two

useful lemmas, first an orthogonality property, which is similar to Lemma 3.1.11.

Lemma 3.1.21. With the residual R and the jump J there holds

∑

T ∈T

∫

TRe∗

h dx+∑

E∈E

∫

EJe∗

h ds+ AV (uh, e∗h) − Aup

V (uh, e∗h) = 0.

Proof. Because of (2.27a) we get

(f, e∗h)Ω + (t0, e

∗h)Γ − Aup

V (uh, e∗h) + (φh, e

∗h)Γ + AV (uh, e

∗h) − AV (uh, e

∗) = 0.

Furthermore, we get as in the proof of Lemma 2.3.9 (equation (2.36))

−AV (uh, e∗h) =

∑

T ∈T

(∫

T−( div(−α∇uh + buh) + c

)e∗

h dx

+∑

E∈ET ∩EI

∫

E[[−α∇uh]] · n e∗

h ds+∑

E∈ET ∩EoutΓ

∫

E−α∇uh · n e∗

h ds

+∑

E∈ET ∩EinΓ

∫

E(−α∇uh + buh) · n e∗

h ds

).

This concludes the proof.

A similar technique as in the a priori result in Lemma 2.3.14 leads to the following esti-

mation.

Lemma 3.1.22. There is a constant C > 0, which depends only on the shape of the

elements in T but neither on the size nor on the number of elements such that

AV (uh, e∗h) − Aup

V (uh, e∗h) ≤ C

(∑

T ∈Tη2

T,up

)1/2

|||e|||Ω.

Proof. As in the proof of Lemma 2.3.14 we get together with the Cauchy-Schwarz

inequality

AV (uh, e∗h) − Aup

V (uh, e∗h)

=∑

T ∈T

∑

τTij ∈DT

(e∗i − e∗

j )

∫

τTij

b · ni(uh − uTh,ij) ds

=

∑

T ∈Tα

−1/2T min

β

−1/2T , hTα

−1/2 ∑

τTij ∈DT

‖b · ni(uh − uTh,ij)‖2

L2(τTij )

1/2

∑

T ∈T

(α

−1/2T min

β

−1/2T , hTα

−1/2)−1 ∑

τTij ∈DT

‖e∗i − e∗

j‖2L2(τT

ij )

1/2

.


We estimate the second product term of the right-hand side. First, we use the triangle

inequality to get

‖e∗i − e∗

j‖2L2(τT

ij ) ≤ ‖eh − e∗i ‖2

L2(τTij ) + ‖eh − e∗

j‖2L2(τT

ij ).

Note that τTij = Zi ∩ Zj with Zi ∈ ZVi , Zi ⊂ T and Zj ∈ ZVj , Zj ⊂ T , respectively.

Therefore, we may apply the trace inequalities (1.25) and (1.26) for ‖eh − e∗i ‖L2(τT

ij ) in Zi

and ‖eh − e∗j‖L2(τT

ij ) in Zj with ∇e∗h|Zi = 0 and ∇e∗

h|Zj = 0. This leads us for the case

h−2T αT > βT to

∑

τTij ∈DT

‖e∗i − e∗

j‖2L2(τT

ij ) . h−1T ‖eh − e∗

h‖2L2(T ) + hT ‖∇eh‖2

L2(T )

and for the case h−2T αT ≤ βT to

∑

τTij ∈DT

‖e∗i − e∗

j‖2L2(τT

ij ) . h−1T ‖eh − e∗

h‖2L2(T ) + ‖eh − e∗

h‖L2(T )‖∇eh‖L2(T ).

Note that with these results, the remaining steps are similar to that in the proof of

Lemma 3.1.15, i.e. consider the two different cases of the minimum and choose one of the

estimates above. Thus, we get

∑

T ∈T

∑

τTij ∈DT

(e∗i − e∗

j )

∫

τTij

b · ni(uh − uTh,ij) ds .

(∑

T ∈Tη2

T,up

)1/2

|||e|||Ω,

which proves the lemma.

Collecting all results together we are now able to formulate an a posteriori result for the

error of our discrete system with upwinding and the error indicators defined in (3.20)

and (3.23).

Theorem 3.1.23 (Reliability for Upwinding). Let u ∈ H1(Ω), ξ ∈ H1/2∗ (Γ), φ ∈

H−1/2(Γ) be the solution of our model problem in Definition 2.0.2 and uh ∈ S1(T ), ξh ∈S1

∗ (EΓ), φh ∈ P0(EΓ) the discrete solution of the discrete problem in Definition 2.3.7. There

is a constant Cup > 0 which depends only on the shape of the elements in T but neither

on the size nor the number of elements such that

|||u− uh|||I + ‖ξ − ξh‖H1/2(Γ) + ‖φ− φh‖H−1/2(Γ) ≤ Cup

(∑

T ∈T

(η2

T + η2T,up

))1/2

.

Proof. As in the proof of Theorem 3.1.19 we get

|||e|||2I ≤∑

T ∈T

∫

TRedx+

∑

E∈E

∫

EJ e ds+

∑

E∈EΓ

∫

E(φ− φh)e ds. (3.24)

With Lemma 3.1.21 and Lemma 3.1.22 and the techniques of the proof of Theorem 3.1.19

we conclude the proof.


3.1.4 Efficiency of the Error Estimator

In this subsection we suppose a little bit more regularity for the solution and the jump

terms of the weak form in Definition 2.2.2, namely

u ∈ H1(Ω) with γ0u ∈ H1(Γ),

ξ ∈ H1∗ (Γ), φ ∈ L2(Γ),

u0 ∈ H1(Γ), t0 ∈ L2(Γ).

We consider the discrete solution uh ∈ S1(T ), ξh ∈ S1∗ (EΓ), φh ∈ P0(EΓ) of the discrete

problem in Definition 2.3.2. Additional to Assumption 2.0.1 we demand the following

assumption, which can also be found in [61].

Assumption 3.1.24. Assume that there exists λ1 > 2 and λ2 ∈ [3/2, 2) such that

− div b ≤ λ1c in x ∈ Ω : div b(x) > 0,− div b ≤ λ2c in x ∈ Ω : div b(x) < 0.

Since 12 div b + c ≥ 0 we see that this is equivalent to

| div b + c| ≤ λ(1

2div b + c) almost everywhere in Ω (3.25)

with the constant

λ = 2 max

λ1 − 1

λ1 − 2,λ2 − 1

2 − λ2

> 0.

Remark 3.1.25. Assumption 3.1.24 is not restrictive since it is satisfied in the so called

convection dominated case 12 div b + c ≥ Cbc,1 > 0 in Ω. It also holds if div b + c ≥ 0 and

c ≥ 0 almost everywhere in Ω, see [61].

We refer to Figure 3.4 for an overview of the proof of the efficiency. Before we can present

an efficiency proof of our error estimator, we present some useful inverse inequalities. We

stress that the main idea to prove efficiency is to follow the steps of [73]. Now we may

recall in Lemma 3.1.26–3.1.28 inverse inequalities, which are proved by using classical

scaling techniques and the bubble functions introduced in Section 1.5, see also Figure 1.6.

For the proofs we refer to [73].

Lemma 3.1.26. Let T ∈ T and bT ∈ H1(T ) be the bubble function of (1.28) and p ∈ N0.

For all w ∈ Pp(T ) there holds

C1‖w‖L2(T ) ≤ ‖w bT ‖L2(T ) ≤ ‖w‖L2(T ), (3.26)

‖∇(wbT )‖L2(T ) ≤ C2h−1T ‖∇w‖L2(T ). (3.27)

The constant C1 > 0 depends only on the polynomial degree p and C2 > 0 on p and on the

shape of the elements of T . Moreover, for T ∈ T the bubble function satisfies bT ∈ H10 (T ).


Theorem 3.1.38

Local efficiency gives a local lower bound of the error in

the energy norms with the refinement indicator ηT (3.20).

More regularity on the solution and data and

Assumption 3.1.24.

Lemma 3.1.26–3.1.28

Well known inequalities with the bub-

ble functions bT (1.28) and bE (1.29)

and the lifting operator Fext (1.31).

Lemma 3.1.29

Inequalities for the squeezed

bubble function bE,κE(3.31).

Lemma 3.1.31

Inverse trace inequality to

prove the tangential compo-

nent of the interface solution.

Claim 3.1.33

Estimate residual term.

Claim 3.1.34

Estimate jump

term for E ∈ EI .

Claim 3.1.35

Estimate jump

term for E ∈ E inΓ .

Claim 3.1.36

Estimate jump term

for E ∈ EoutΓ .

Claim 3.1.37

Estimate tangential com-

ponent of the error on Γ.

Figure 3.4. The main steps to the proof of Theorem 3.1.38, which shows the efficiency of theerror estimator for the coupling with the finite volume element method.

Lemma 3.1.27. Let E ∈ E and bE be the edge bubble function of (1.29) and (1.30),

respectively, and p ∈ N0. For all w ∈ Pp(E), there holds

C‖w‖L2(E) ≤ ‖w bE‖L2(E) ≤ ‖w‖L2(E). (3.28)

The constant C > 0 depends only on the shape of the elements of T and the polynomial

degree p.

Lemma 3.1.28. For each edge E ∈ E, p ∈ N0 let bE be the edge bubble function of (1.29)

and (1.30) and Fext : Pp(E) → H1(ωE) the lifting operator (1.31) such that Fext(w)|E = w.

For T ∈ ωE and w ∈ Pp(E) there holds

C1h1/2E ‖w‖L2(E) ≤ ‖Fext(w)bE‖L2(T ) ≤ C2h

1/2E ‖w‖L2(E), (3.29)

‖∇(Fext(w)bE)‖L2(T ) ≤ C3h−1/2E ‖w‖L2(E). (3.30)

The constants C1, C2, C3 > 0 depend only on the shape of the elements in T and the

polynomial degree p.


(a) Classic bubble function bE . (b) Squeezed bubble function bE,κE.

Figure 3.5. The classical bubble function bE on an interior edge E ∈ EI with support on ωE

in (a) and the squeezed bubble function bE,κEin (b), which support lies in ωE .

For the efficiency proof in the energy norm we need modifications of (3.28)–(3.30). The

main idea is to define a squeezed element associated with an element T ∈ T . We follow

the step of [74] and define for the reference element T = conv(0, 0), (1, 0), (0, 1)

the

squeezed reference element Tκ = conv(0, 0), (1, 0), (0, κ)

with 0 < κ ≤ 1. On Tκ with

E = conv(0, 0), (1, 0)

we can define a bubble function b

E,κas in (1.29) introduced in

Section 1.5, where the extension to T is 0, i.e. bE,κ

= 0 on T\Tκ. It is easy to see

that with the affine transformation H, which maps T onto T and E onto E ∈ ET , Tκ is

transformed onto Tκ with Tκ ⊂ T and E ⊂ ∂Tκ. Furthermore, we denote by T1, T2 ∈ Ttwo elements which share the edge E ∈ EI . Then there exist two orientation preserving

affine transformations Hi, i = 1, 2, which maps T onto Ti and E onto E. Therefore, we

define the squeezed edge bubble function for E ∈ EI by

bE,κ :=

bκ H−1i on Ti, i = 1, 2,

0 on Ω\ωE .(3.31)

See also Figure 3.5(b). For E ∈ EΓ the bubble function bE,κ is defined in the same way

with the obvious modifications. We recall now the properties of this squeezed bubble

function,

bE,κ = 0 on ∂Tκ\E with Tκ ⊂ ωE , suppbE,κ = Tκ,

0 ≤ bE,κ ≤ 1, maxx∈E

bE,κ(x) = 1.

The height of T and Tκ differ by the factor κ.

In the following we will replace the factor κ by an edge based factor and define therefore

for all E ∈ E

κE :=

1 for βE = 0,

min

1,

α1/2E

β1/2E hE

else.

The inequalities (3.28)–(3.30) can be modified by the bubble function bE,κE. For the proof

of the following lemma and more details on the squeezed bubble function we refer to [74].


Lemma 3.1.29. For each edge E ∈ E, p ∈ N0 let bE,κEbe the squeeze bubble func-

tion (3.31) and Fext : Pp(E) → H1(ωE) the lifting operator (1.31) such that Fext(w)|E =

w. For T ∈ ωE and w ∈ Pp(E) there holds

C1‖w‖L2(E) ≤ ‖w bE,κE‖L2(E) ≤ ‖w‖L2(E) (3.32)

‖Fext(w)bE,κE‖L2(T ) ≤ C2κ

1/2E h

1/2E ‖w‖L2(E) (3.33)

‖∇(Fext(w)bE,κE)‖L2(T ) ≤ C3κ

−1/2E h

−1/2E ‖w‖L2(E). (3.34)

The constants C1, C2, C3 > 0 depend only on the shape of the elements in T and the

polynomial degree p.

Remark 3.1.30. The squeeze bubble function also provides a useful tool in context of

the a posteriori error estimation for finite element method on anisotropic meshes, see [54].

The next lemma will be very useful to estimate the tangential component on the boundary

of our error estimator.

Lemma 3.1.31. Let v ∈ H1(EΓ)∩C(Γ) and wh ∈ P1(T ) with wh(a) = v(a) for all a ∈ NΓ.

For all T ∈ T with E ∈ ET ∩ EΓ there holds

infw∈H1(T )w|E=v|E

‖∇(wh − w)‖L2(T ) ≤ Ch1/2E ‖∂(wh − v)/∂s‖L2(E) (3.35)

with C > 0, which depends only on the shape of the elements T .

Proof. The proof follows the steps of the proof in [17, Lemma 4.1]. This proof shows

with an harmonic extension, an interpolation estimate, and a one dimensional integration

argument

‖∇w‖L2(T ) . h1/2T ‖∂(wh − v)/∂s‖L2(E)

for a w ∈ H1(T ) with w|E = wh|E − v|E and w|∂T \E = 0. A scaling argument guarantees

that the constant is hT -independent. Defining w ∈ H1(T ) with w|E = v|E and applying

the infimum proofs the lemma.

The following property is useful to write the lower bound in a proper way.

Corollary 3.1.32. For E ∈ E there holds

µE = maxT ∈ωE

µT . (3.36)

Proof. For E ∈ EΓ the proof is clear. Let us assume E ∈ EI , thus there are unique

elements T1, T2 ∈ T with E = T1 ∩ T2. We write

µE = minβ

−1/2E , hEα

−1/2E

= min

min βT1 , βT2−1/2 ,max αT1 , αT2−1/2

.


Since there holds maxαT1 , αT2−1/2 ≤ α−1/2T1

and maxαT1 , αT2−1/2 ≤ α−1/2T2

we get two

cases, namely, βT1 ≤ βT2

µE ≤ minβ

−1/2T1

, hEα−1/2T1

= µT1

and βT2 < βT1

µE ≤ minβ

−1/2T2

, hEα−1/2T2

= µT2 ,

which proves the corollary.

Before we can prove the efficiency of our error estimator, we dominate the different edge

contributions of η2T separately in five claims. Throughout the proofs, we adopt the fore-

going notation for e = u − uh, δ = ξ − ξh, ǫ = φ − φh, R, J and .. We remark that the

constant . may depend on the constant λ of (3.25). We refer to Figure A in the Index of

Notation to get an overview of the introduced quantities.

Claim 3.1.33. For all T ∈ T there holds

µT ‖R‖L2(T ) .(1 + α−1

T ‖b‖L∞(T )hT + µT ‖ div b + c‖1/2L∞(T )

)|||e|||T

+ µT ‖R−RT ‖L2(T ),

where RT ∈ P0(T ) is the piecewise integral mean of the residual R.

Proof. Since

‖R‖L2(T ) ≤ ‖R−RT ‖L2(T ) + ‖RT ‖L2(T )

we estimate by (3.26)

‖RT ‖2L2(T ) .

∫

TRT v dx with v := RT bT ∈ H1

0 (T ), v ∈ H1(Ω).

We rewrite the right-hand side using (2.1a) and integration by parts

∫

TRT v dx =

∫

T(RT −R)v dx+

∫

TRv dx

=

∫

T(RT −R)v dx+

∫

TαT ∇e · ∇v dx+

∫

Tdiv(b e)v dx+

∫

Tcev dx.

The Cauchy-Schwarz inequality and div(b e) = div(b)e+ b∇e yields

∫

TRT v dx ≤

‖RT −R‖L2(T )‖v‖L2(T ) + αT ‖∇e‖L2(T )‖∇v‖L2(T ) + ‖b‖L∞(T )‖∇e‖L2(T )‖v‖L2(T )

+ δ1/2‖ div b + c‖1/2L∞(T )‖

(1

2div b + c

)1/2e‖L2(T )‖v‖L2(T ),


where we have used (3.25) with the constant λ > 0. Applying (3.26) and (3.27) we get

‖RT ‖L2(T ) . ‖RT −R‖L2(T ) + h−1T α

1/2T ‖α1/2

T ∇e‖L2(T ) + α−1/2T ‖b‖L∞(T )‖α1/2

T ∇e‖L2(T )

+ ‖ div b + c‖1/2L∞(T )‖

(1

2div b + c

)1/2e‖L2(T )

and because of µT ≤ hTα−1/2T we get

µT ‖RT ‖L2(T ) . µT ‖RT −R‖L2(T )

+(1 + α−1

T ‖b‖L∞(T )hT + µT ‖ div(b) + c‖1/2L∞(T )

)|||e|||T .

Note that this proof is valid for βT = 0 too.

Claim 3.1.34. For all E ∈ EI there holds

α−1/4E µ

1/2E ‖J‖L2(E) .

[1 + max

T ∈ωE

α−1

T ‖b‖L∞(T )hT

+ max

T ∈ωE

µT maxT ∈ωE

‖ div b + c‖1/2

L∞(T )

]|||e|||ωE

+∑

T ∈ωE

µE‖R‖L2(T ).

Proof. With bE,κE∈ H1

0 (ωE) the corresponding edge bubble function, (3.32) yields

‖J‖2L2(E) .

∫

EJv ds with v := Fext(J)bE,κE

∈ H10 (ωE).

We rewrite the right-hand side and use integration by parts to prove∫

EJv ds =

∑

T ∈ωE

∫

∂T−αT ∇uh · n v dx =

=∑

T ∈ωE

( ∫

Tdiv(−αT ∇uh)v dx+

∫

T−αT ∇uh · ∇v ds

).

Integration by parts with v = 0 on ∂ωE and (2.1a) lead to∫

EJv ds =

∑

T ∈ωE

( ∫


∫

TαT ∇e · ∇v dx

)−∫

Tdiv(−αT ∇u)v dx

=∑

T ∈ωE

(−∫

TRv dx+

∫

TαT ∇e · ∇v dx+

∫

Tdiv(b e)v dx+

∫

Tcev dx

),

where we have used R = f − div(−α∇uh + buh) − cuh. We observe with a similar

calculation as in Claim 3.1.33∫

EJv ds

.∑

T ∈ωE

(‖R‖L2(T )‖v‖L2(T ) + αT ‖∇e‖L2(T )‖∇v‖L2(T ) + ‖b‖L∞(T )‖∇e‖L2(T )‖v‖L2(T )

+ ‖ div b + c‖1/2L∞(T )‖

(1

2div b + c

)1/2e‖L2(T )‖v‖L2(T )

).


With the help of (3.33) and (3.34) we get

‖J‖2L2(E) .

∑

T ∈ωE

[κ

1/2E h

1/2E ‖R‖L2(T ) +

(κ

−1/2E h

−1/2E αT + κ

1/2E h

1/2E ‖b‖L∞(T )

)‖∇e‖L2(T )

+ κ1/2E h

1/2E ‖ div b + c‖1/2

L∞(T )‖(1

2div b + c)1/2e‖L2(T )

]‖J‖L2(E). (3.37)

Now we multiply (3.37) with α−1/4E µ

1/2E and consider the sums separately. We remark

whenever we have a second case in an estimate, this does not appear for βE = 0. The first

sum is estimated by∑

T ∈ωE

α−1/4E µ

1/2E κ

1/2E h

1/2E ‖R‖L2(T ) ≤

∑

T ∈ωE

µE‖R‖L2(T ),

where we have estimated κE ≤ 1 for µE = hEα−1/2E and κE ≤ α

1/2E /(β

1/2E hE) for µE =

β−1/2E .

For the second sum we distinguish two cases: First, for κE = 1 we take µE ≤ hEα−1/2E

and second κE =α

1/2E

β1/2E hE

we take µ1/2E ≤ β

−1/2E . It is easy to verify that this leads in each

case to∑

T ∈ωE

α−1/4E µ

1/2E κ

−1/2E h

−1/2E α

1/2T ‖α1/2

T ∇e‖L2(T ) ≤∑

T ∈ωE

|||e|||T . |||e|||ωE .

Note that αT ≤ αE for T ∈ ωE .

For the third sum of (3.37) we estimate with µE ≤ hEα−1/2E , κE ≤ 1

∑

T ∈ωE

α−1/4E µ

1/2E κ

1/2E h

1/2E α

−1/2T ‖b‖L∞(T )‖α1/2

T ∇e‖L2(T )

≤∑

T ∈ωE

α−1/2E α

−1/2T hE‖b‖L∞(T )|||e|||T ≤ max

T ∈ωE

α−1

T ‖b‖L∞(T )hT

|||e|||ωE .

And finally for the fourth sum of (3.37) we estimate in a similar way as for the first sum

∑

T ∈ωE

α−1/4E µ

1/2E κ

1/2E h

1/2E ‖ div b + c‖1/2

L∞(T )‖(1

2div b + c

)1/2e‖L2(T )

≤ µE maxT ∈ωE

‖ div b + c‖1/2

L∞(T )

|||e|||ωE ,

where we conclude with (3.36).

Claim 3.1.35. Let b be a linear approximation of b and t0 the EΓ-piecewise integral mean

of t0. For all E ∈ E inΓ there holds

α−1/4E µ

1/2E ‖J‖L2(E)

.(1 + α−1


)|||e|||T

+ µE‖R‖L2(T ) + α−1/4E µ

1/2E ‖φ− φh‖L2(E) + α

−1/4E µ

1/2E ‖b · n‖L2(E)‖u− uh‖L2(E)

+ α−1/4E µ

1/2E ‖t0 − t0‖L2(E) + α

−1/4E µ

1/2E ‖(b − b) · nuh‖L2(E),

where T ∈ T is the element with E ∈ ET .


Proof. We start with

‖J‖L2(E) ≤ ‖(−α∇uh + buh) · n + φh + t0‖L2(E)

+ ‖(b − b) · nuh‖L2(E) + ‖t0 − t0‖L2(E).(3.38)

Note that (−α∇uh + buh) · n + φh + t0 ∈ P2(E), thus we define

v := Fext((−α∇uh + buh) · n + φh + t0

)bE,κE

∈ H1(T ).

Hence, thanks to (3.32), integration by parts and (2.8a) we obtain

‖ − α∇uh + buh · n + φh + t0‖2L2(E)

.

∫

E

((−α∇uh + buh) · n + φh + t0

)v ds+

∫

E(b − b) · nuhv ds

=

∫

Tdiv(−αT ∇uh + buh)v dx+

∫

T(−αT ∇uh + buh) · ∇v dx

+

∫

E(φh + t0)v ds+

∫

E(b − b) · nuhv ds

=

∫

Tdiv(−αT ∇uh + buh)v dx+

∫

T(αT ∇e− be)∇v dx+

∫

Tcuv dx

−∫

Tfv dx−

∫

E(φ− φh)v dx−

∫

E(t0 − t0)v ds+

∫

E(b − b) · nuhv ds.

Note that v|E 6= 0 on E, v|T 6= 0 on T , but v|E′ = 0 for E′ ∈ EΓ\E and v = 0 on Ω\T .

Similar as in Claim 3.1.34 we get

‖ − α∇uh + buh · n + φh + t0‖2L2(E)

.‖R‖L2(T )‖v‖L2(T ) + αT ‖∇e‖L2(T )‖∇v‖L2(T ) + ‖b‖L∞(T )‖∇e‖L2(T )‖v‖L2(T )

+ ‖ div b + c‖1/2L∞(T )‖

(1

2div b + c

)1/2e‖L2(T )‖v‖L2(T ) + ‖b · n‖L2(E)‖e‖L2(E)‖v‖L2(E)

+ ‖φ− φh‖L2(E)‖v‖L2(E) + ‖t0 − t0‖L2(E)‖v‖L2(E) + ‖(b − b) · nuh‖L2(E)‖v‖L2(E).

Note that we get an additional term because of

∫

Tb e∇v ds = −

∫

Tdiv(b e)v dx+

∫

Eb · n ev ds.

With the same techniques as in Claim 3.1.34 and together with (3.38) we finally arrive at

α−1/4E µ

1/2E ‖J‖L2(E)

.(1 + α−1


)|||e|||T

+ µE‖R‖L2(T ) + α−1/4E µ

1/2E ‖φ− φE‖L2(E) + α

−1/4E µ

1/2E ‖b · n‖L2(E)‖e‖L2(E)

+ α−1/4E µ

1/2E ‖t0 − t0‖L2(E) + α

−1/4E µ

1/2E ‖(b − b) · nuh‖L2(E),



Claim 3.1.36. For all E ∈ EoutΓ there holds

α−1/4E µ

1/2E ‖J‖L2(E)

.(1 + α−1


)|||e|||T

+ µE‖R‖L2(T ) + α−1/4E µ

1/2E ‖φ− φh‖L2(E) + α−1/4µ

1/2E ‖t0 − t0‖L2(E),

where T ∈ T is the element with E ∈ ET .

Proof. Since −α∇uh · n + φh + t0 ∈ P0(E) we get

‖J‖L2(E) ≤ ‖ − α∇uh · n + φh + t0‖L2(E) + ‖t0 − t0‖L2(E)

and

‖ − α∇uh · n + φh + t0‖2L2(E) .

∫

E(−α∇uh · n + φh + t0)v ds

with v := Fext(−α∇uh · n + φh + t0)bE,κE∈ H1(T ).

For the right-hand side we get∫

E(−α∇uh · n + φh + t0)v ds

=

∫


∫

TαT ∇e∇v dx−

∫

TαT ∇u∇v dx+

∫

E(φh + t0)v ds

=

∫

T−Rv dx+

∫

TαT ∇e∇v dx+

∫

Tdiv(b)ev dx+

∫

Tb · ∇ev dx+

∫

Tcev dx

−∫

E(φ− φh)v ds−

∫

E(t0 − t0)v ds,

where we have used (2.8a) with the test function v ∈ H1(T ). The same calculations as

above leads to the assertion.

The next statement appears similar in [19], but the proof therein is not correct. With the

help of Lemma 3.1.31 we can provide a similar result.

Claim 3.1.37. Let T ∈ T . Let wh ∈ P1(T ) with wh(a) = γ0u(a) for all a ∈ NΓ. Then

for all E ∈ EΓ we have

h1/2E ‖∇uh · t − ∂

∂s

(u0 − Vφh + (1/2 + K)ξh

)‖L2(E)

. α−1/2T |||u− uh|||T + h

1/2E ‖ ∂

∂s

(wh − γ0u

)‖L2(E)

+ h1/2E ‖ ∂

∂s

((1/2 + K)(ξ − ξh)

)‖L2(E) + h1/2E ‖ ∂

∂sV(φ− φh)‖L2(E),

where T is the element with E ∈ ET .

Proof. The Calderón system (1.13) and the jump relation of the traces (2.1d) yield to

‖∇uh · t − ∂

∂s

(u0 − Vφh + (1/2 + K)ξh

)‖L2(E)

= ‖ ∂∂s

(− γ0u+ uh + (1/2 + K)δ − Vǫ)‖L2(E) (3.39)

≤ ‖ ∂∂s

(uh − wh

)‖L2(E) + ‖ ∂∂s

(wh − γ0u

)‖L2(E) + ‖ ∂∂s

((1/2 + K)δ − Vǫ)‖L2(E).


For the first term we set

v = Fext

( ∂∂s

(uh − wh))bE ∈ H1(T )

with the property v = 0 on ∂T\E. Then we observe with (3.28) and integration by parts

that

‖ ∂∂s

(uh − wh)‖2L2(E) .

∫

E∇(uh − wh) · t v ds =

∫

∂Tcurl(uh − wh) · n v ds

=

∫

Tcurl(uh − wh) · ∇v ds = −

∫

T∇(uh − wh) · curl v ds.

For an arbitrary w ∈ H1E(T ) :=

w ∈ H1(T )

∣∣ w|E = 0

we observe with div(curl v) = 0

and v = 0 on ∂T\E∫

T∇w curl v dx = −

∫

Tw div(curl v) dx+

∫

∂Tw curl v · n ds = 0.

Then we assert by the Cauchy-Schwarz inequality

‖ ∂∂s

(uh − wh)‖2L2(E) ≤ ‖∇(uh − wh + w)‖L2(T )‖ curl v‖L2(T )

. ‖∇(uh − wh + w)‖L2(T )h−1/2E ‖ ∂

∂s(uh − wh)‖L2(E),

where we have used (3.30) for ‖ curl v‖L2(T ). This leads us to

‖ ∂∂s

(uh − wh)‖L2(E) . h−1/2E ‖∇(uh − u)‖L2(T ) + h

−1/2E ‖∇(u− wh + w)‖L2(T ),

where we have inserted u and used the triangle inequality. The second sum on the right-

hand side can be estimated with Lemma 3.1.31 by

infw∈H1

E(T )‖∇(u− wh + w)‖L2(T ) = inf

w∈H1(T )w|E=(γ0u)|E

‖∇(wh − w)‖L2(T )

. h1/2E ‖∂(wh − γ0u)/∂s‖L2(E).

Collecting all these results and applying the triangle inequality for the last term in (3.39)

we arrive at

‖∇uh · t − ∂

∂s(u0 − Vφh + (1/2 + K)ξh) ‖L2(E)

. h−1/2E ‖∇(u− uh)‖L2(T ) + ‖ ∂

∂s(wh − γ0u)‖L2(E)

+ ‖ ∂∂s

((1/2 + K)δ

)‖L2(E) + ‖ ∂∂s

Vǫ‖L2(E),

which proves the claim.


Theorem 3.1.38 (Efficiency). With the notation from Claim 3.1.33– 3.1.37 there holds

for all T ∈ T and E ∈ ET

ηT .

[1 + max

T ∈ωT

α−1

T ‖b‖L∞(T )hT

+ max

T ∈ωT

µT maxT ∈ωT

‖ div b + c‖1/2

L∞(T )

]|||u− uh|||ωT

+α

−1/2T |||u− uh|||T

∂T ∩Γ

+ maxT ∈ωT

µT ∑

T ∈ωT

‖R−RT ‖L2(T )

+ h1/2E ‖ ∂

∂s(wh − γ0u)‖L2(E∩Γ) + α

−1/4T µ

1/2T ‖b · n‖L2(E∩Γin)‖u− uh‖L2(E∩Γin)

+ α−1/4T µ

1/2T ‖(b − b) · nuh‖L2(E∩Γin) + α

−1/4T µ

1/2T ‖φ− φh‖L2(E∩Γ)

+ α−1/4T µ

1/2T ‖t0 − t0‖L2(E∩Γ)

+ h1/2E ‖ ∂

∂s

((1/2 + K)(ξ − ξh)

)‖L2(E∩Γ) + h1/2E ‖ ∂

∂sV(φ− φh)‖L2(E∩Γ)

+ h1/2E ‖((1/2 + K∗)(φ− φh)

)‖L2(E∩Γ) + h1/2E ‖W(ξ − ξh)‖L2(E∩Γ).

The notation ·E∩Γ indicates that this term only appears for elements T on the boundary,

the same is valid for norms over E ∩ Γ and E ∩ Γin.

Proof. Apply Claim 3.1.33–3.1.37 and the Cauchy-Schwarz inequality.

This estimate shows that the error indicator ηT is a local estimator, even for T at the

boundary Γ. The termα

−1/2T |||u − uh|||T

E∩Γ

only appears on boundary elements and

comes from the exterior problem.

Discussion of Higher Order Terms. We stress thatα−1

T ‖b‖L∞(T )hT

is the lo-

cal Péclet number defined in (2.20). This number can be very large in the convec-

tion dominated case, which is a typical behavior of an error estimator. First, we

have to resolve the convection dominated regime, where the energy norm error is high,

and then the overall efficiency constant starts to decrease. The elementwise quantity

maxT ∈ωEµT maxT ∈ωE

‖ div b + c‖1/2L∞(T )

is not a dominated factor, even in the con-

vection dominated or singularly perturbed diffusion reaction case, and it decrease for hT

small enough. The other local error terms and local approximation errors of the residual

R, the data b and t0 are generically of higher order.

Corollary 3.1.39. Let E ∈ EΓ. Additionally, we demand γ0u ∈ H1(Γ) ∩H2(EΓ) we may

estimate

‖ ∂∂s

(wh − γ0u)‖L2(E) ≤ hE‖ ∂2

∂s2(γ0u)‖L2(E).

Proof. Note that since wh(a) = γ0u(a) for a ∈ NΓ and wh ∈ P1(T ) there holds

∫

E∂(wh − γ0u)/∂s = 0 and ∂wh/∂s =

1

|E|

∫

E∂γ0u/∂s ds.


The L2-orthogonality of ∂wh/∂s shows

‖ ∂∂s

(wh − γ0u)‖L2(E) ≤ hE‖ ∂2

∂s2(γ0u)‖L2(E),

where the last estimate results from a one dimensional integration argument, since

∂/∂s(wh − γ0u) has at least one root on E.

Let us now consider the terms with the boundary integral operators V, K, K∗ and W and

the boundary mesh quantities

hΓ,max := maxE∈EΓ

hE , hΓ,min := minE∈EΓ

hE .

If we consider the sum of ηT over all T ∈ T there holds the following estimate.

Corollary 3.1.40. Let ξh ∈ S1(EΓ) be the nodal interpolant of ξ ∈ H1∗ (Γ) with respect

to Γ and φ ∈ P0(EΓ) the EΓ-piecewise integral mean of φ. Then there holds the global

estimate

‖h1/2EΓ

∂

∂sEΓ

((1/2 + K)(ξ − ξh)

)‖L2(Γ) + ‖h1/2EΓ

∂

∂sEΓ

V(φ− φh)‖L2(Γ)

+ ‖h1/2EΓ

((1/2 + K∗)(φ− φh)

)‖L2(Γ) + ‖h1/2EΓ

W(ξ − ξh)‖L2(Γ)

.h

1/2Γ,max

h1/2Γ,min

(‖ξ − ξh‖H1/2(Γ) + ‖φ− φh‖H−1/2(Γ)

+ h1/2Γ,max

(‖ξ − ξh‖H1(Γ) + ‖φ− φ‖L2(Γ)

)),

where hEΓ|E := hE is the EΓ-piecewise length and ∂/∂sEΓ

the EΓ-piecewise derivative with

respect to the arc length.

Proof. We can estimate these terms exactly as in the coupling of the finite element

method and the boundary element method in [14, 19]. Thus, we will illustrate it just for

two terms. First, we get according to the boundedness of K

‖h1/2EΓ

∂

∂sEΓ

((1/2 + K)(ξ − ξh)

)‖2L2(Γ) ≤ hΓ,max‖((1/2 + K)(ξ − ξh)

)‖2H1(Γ) (3.40)

. hΓ,max

(‖ξ − ξh‖2H1(Γ) + ‖ξh − ξh‖2

H1(Γ)

).

We obtain from the well-known inverse inequality [13]

‖ξh − ξh‖2H1(Γ) . h−1

Γ,min‖ξh − ξh‖2H1/2(Γ)

≤ h−1Γ,min

(‖ξ − ξh‖2H1/2(Γ) + ‖ξh − ξ‖2

H1/2(Γ)

).

(3.41)

By the well-known H1/2-interpolation estimate, i.e. for v ∈ H1(Γ) it reads ‖v‖2H1/2 ≤

C1‖v‖L2(Γ)‖v‖H1(Γ) with C1 > 0, we deduce

‖ξh − ξ‖2H1/2Γ . ‖ξh − ξ‖L2(Γ)‖ξh − ξ‖H1(Γ) . hΓ,max‖ξh − ξ‖2

H1(Γ). (3.42)

3.2. Estimation for the Coupling with the Cell-Centered Finite Volume Method 83

And finally we get with (3.40)–(3.42)

‖h1/2EΓ

∂

∂sEΓ

((1/2 + K)(ξ − ξh)

)‖2L2(Γ) .

hΓ,max

hΓ,min

(‖ξ − ξh‖2H1/2(Γ) + hΓ,max‖ξh − ξ‖2

H1(Γ)

).

Similar steps as above shows the estimate for ‖h1/2EΓ

((1/2 + K∗)(φ − φh)

)‖L2(Γ). Here, we

use the inverse inequality

‖φ− φ‖2L2(Γ) . h−1

Γ,min‖φ− φ‖2H−1/2Γ

and

‖φ− φ‖2H−1/2(Γ) . hΓ,max‖φ− φ‖2

L2(Γ),

see e.g. [23, Lemma 4.3]. The other terms are estimated by the same techniques.

Remark 3.1.41. Theorem 3.1.38 and Corollary 3.1.40 state that we get an inverse in-

equality to Theorem 3.1.19 in case of a quasi-uniform mesh on the boundary Γ, i.e. the

a posteriori error estimate is sharp. The term quasi-uniform on the boundary was intro-

duced in Remark 1.3.3.

3.2 Estimation for the Coupling with the Cell-Centered Fi-

nite Volume Method

There are only a few works for a posteriori error estimators with respect to an energy

norm for the cell-centered finite volume method in the literature, which all postprocess the

original piecewise finite volume approximation in a way that the local conservative property

still keeps valid, see [60, 42, 61, 76]. In this section we consider the Morley interpolant

ansatz, which is used in [61]. We extend it in an appropriate way and can provide an

estimator for our coupling problem. We suppose that uh ∈ P0(T ), uh,Γ ∈ S1(EΓ), ξ ∈S1

∗ (EΓ) and φh ∈ P0(EΓ) are the computed discrete solutions of Definition 2.4.4 for the

coupling of the cell-centered finite volume method and the boundary element method. We

remind that we restrict ourself to the diffusion coefficient α ∈ R+ for this coupling method.

We do the analysis only for diffusion or diffusion reaction problems in the interior domain

that means b = (0, 0)T . We refer to Figure 3.6 for an overview of the main steps to prove

an a posteriori estimator.

Remark 3.2.1. We stress that we can apply and extend the analytical ideas of [61] for

convection problems too. Unfortunately, a reproduction of the results in [61] fails for

diffusion convection problems. In other words we do not get experimental convergence in

the H1-seminorm for the Morley error as it is claimed in [61]. We point out that there is

no theoretical convergence result for the Morley interpolant in [61].


Definition 3.2.2

Appropriate finite elements

(T,PT ,ΣT ) of Morley-type.

Equation (3.48)–(3.54)

Define the functionals ΣT for

the Morley interpolant Imuh.

Theorem 3.2.15Reliability in the energy norms with appropriate re-

finement indicators for the Morley interpolant ηT (3.63):|||u − Imuh|||Ω + ‖ξ − ξh‖H1/2(Γ) + ‖φ − φh‖H−1/2(Γ)

≤ Crel

(∑T ∈T

η2T

)1/2.

Lemma 3.2.10

L2-orthogonal property of

the residual RIm(3.55).

Lemma 3.2.11–3.2.12

Orthogonal properties of

the jump term JIm(3.56).

Corollary 3.2.9

Compare Im and uh,Γ ∈S1(EΓ) on the boundary.

Lemma 3.2.13–3.2.14

Stability estimates in the energy

norm with respect to α and c.

Lemma 3.1.16–3.1.17

L2-Localization of H1/2-

and H−1/2-norm terms.

Figure 3.6. The main steps to the proof of Theorem 3.2.15, which shows the reliability of theerror estimator for the coupling with the cell-centered finite volume method.

3.2.1 The Morley Interpolant

In this subsection we define an Morley-type interpolant Imuh for the interior solution uh,

which is appropriate for a posteriori error analysis. The definition, which is an extension

of the definition in [61], satisfies the following properties:

• The interpolant Imuh can be built locally on each T ∈ T .

• The interpolant Imuh is globally continuous.

• The interpolant Imuh is built on a finite element.

• The flux over an edge of Imuh is equal to the numerical flux.

We want to stress that we have to construct the interpolant differently, whether reaction

is present or not. In contrast, the approach in [76] leads to a non-conforming interpolant,

where one can use the same construction for different model equations.

Morley Elements. In [60, 42] exists non-conforming finite elements of Morley type but

for our purpose, more precisely for the analysis, we need a conforming finite element space.

We first define some conforming finite elements, which are suitable for the construction for

different types of equations. For this we need the bubble functions bT and bE introduced

in Section 1.5.


Definition 3.2.2 (Morley Elements). Let T = conva1, a2, a3 ⊂ R2 be a non-

degenerate triangle with edges Ej = convaj , aj+1 (a4 := a1) and the midpoints xEj ∈NM of Ej , j = 1 . . . 3. The dual basis is given by ΣT = (S1, . . . , Si) and the functional Sj

are applied on p ∈ PT .

• The first Morley element (T,PT ,ΣT )1 for diffusion problems is given by

PT =q1 + q2bT

∣∣ q1 ∈ P2(T ), q2 ∈ P1(T ),

Sj(p) = p(aj), Sj+3(p) = p(xEj ),

Sj+6(p) =

∫

Ej

∂p/∂n ds for j = 1, . . . , 3 and p ∈ PT .

(3.43)

• The second Morley element (T,PT ,ΣT )2 for diffusion reaction problems is defined

as

PT =q1 + (q2 + λbT )bT

∣∣ q1 ∈ P1(T ), q2 ∈ P1(T ), λ ∈ R

,

Sj(p) = p(aj), S4(p) =

∫

Tp dx,

Sj+4(p) =

∫

Ej

∂p/∂n ds for j = 1, . . . , 3 and p ∈ PT .

(3.44)

Remark 3.2.3. The elements (3.43)–(3.44) are appropriate for problems with constant

coefficients. For the sake of completeness we notate further elements from [61], which are

used for convection problems (see Remark 3.2.1) and the last is a general finite element

of Morley-type, if the coefficients are not constant.

• The third Morley element (T,PT ,ΣT )3 is suitable for diffusion convection problems

PT =q1 + q2bT

∣∣ q1 ∈ P2(T ), q2 ∈ P1(T ),

Sj(p) = p(aj), Sj+3(p) =

∫

Ej

p ds,

Sj+6(p) =

∫

Ej

∂p/∂n ds for j = 1, . . . , 3 and p ∈ PT .

(3.45)

• The fourth Morley element (T,PT ,ΣT )4 for diffusion convection reaction problems

is

PT =q1 + (q2 + λbT )bT

∣∣ q1 ∈ P2(T ), q2 ∈ P1(T ), λ ∈ R

,

Sj(p) = p(aj), S4(p) =

∫

Tp dx, Sj+4(p) =

∫

Ej

p ds,

Sj+7(p) =

∫

Ej

∂p/∂n ds for j = 1, . . . , 3 and p ∈ PT .

(3.46)


• Finally, the general form of a Morley element (T,PT ,ΣT )5 can be written as

PT =q1 +

∑

E∈ET

λb,EFext(b · n)bE +∑

E∈ET

λα,EbEbT + λcb2T

∣∣∣

q1 ∈ P1(T ), λb,E , λα,E , λc ∈ R

,

Sj(p) = p(aj), S4(p) =

∫

Tp dx, Sj+4(p) =

∫

Ej

p ds,

Sj+7(p) =

∫

Ej

∂p/∂n ds for j = 1, . . . , 3 and p ∈ PT .

(3.47)

Note that in the notation for (3.47) we loose a degree of freedom, if b · n = 0 on an

edge E ∈ ET or c = 0 on T . We assume that there is a typo in [61], because they use

λb,EFext(b · n)bEbT instead of λb,EFext(b · n)bE . Note that this can not lead to a finite

element. Our version is still a conforming finite element.

Lemma 3.2.4 (See [60, 61]). The above triples (T,PT ,ΣT )i, i = 1, . . . , 5, that

means (3.43)–(3.47) are C0-elements.

The Morley Interpolant. On the finite elements (3.43)–(3.44) we can build a Morley

interpolant. This leads mainly to the question of defining the functionals of ΣT . Note that

not all functionals occur in both Morley elements of (3.43)–(3.44). The Morley interpolant

Imuh satisfies elementwise (Imuh)|T ∈ PT for all T ∈ T . We will give the definition of

the interpolant on T in the next equations (3.48)–(3.54) for constant coefficients in the

interior problem, where we provide the construction for possible Dirichlet and Neumann

boundaries in the interior domain Ω as well, see Remark 2.0.3.

For each free node a ∈ NT ∩ NI , the value Imuh(a) satisfies

(Imuh)|T (a) =∑

T ∈ωa

ΥT (a)uh|Ta , (3.48)

where the weights ΥT (a) are the same as for the computation of ua in (2.46). For each

boundary node, the value Imuh|T (a) is prescribed

(Imuh)|T (a) =

uD(a) for all a ∈ ND,

ua + ςa for all a ∈ NN ∪ NΓ,(3.49)

where the calculation of ua and ςa is discussed in Subsection 2.4.2. Additionally, we define

the value for (Imuh)|T (xEm) for each midpoint xEm ∈ NM of an edge E ∈ ET ∩ EI by the

sum

(Imuh)|T (xEm) = ΥETuT + ΥEpuEp + ΥET ′uT ′ + ΥEquEq . (3.50)

Here, uT and uT ′ are the cell-centered finite volume solutions located at xT and xT ′ , respec-

tively, where T and T ′ are the two elements of T , which share the edge E, e.g. E = T ∩T ′,


xT

xT ′

xEp

xEq

xEm

(a) For E ∈ EI .

xEp xEqxEm

(b) For E ∈ EΓ.

Figure 3.7. Notation for the construction of the Morley interpolant Im for an interior edge (a)

and boundary edge (b).

see Figure 3.7. The values uEp and uEq are located at the starting and endpoint of the edge

E, which are calculated in (3.48) and (3.49). The weights ΥET,ΥEp ,ΥET ′ ,ΥEq are calcu-

lated through a least squares ansatz of uT , uEp , uT ′ , uEq by the points xT , xEp , xT ′ , xEq ,

similarly to the weights ΥT for the computation of ua in (2.46). For midpoints xEm ∈ NM

on a boundary edge we compute

(Imuh)|T (xEm) =

uD(xEm) for all xEm on ED,

(uEp + uEq )/2 for all xEm on EN ∪ EΓ,(3.51)

where T is the element which belong to the edge E, i.e. E ∈ ET .

For each edge E ∈ ET \EΓ holds

∫

E−α∇(Imuh)|T · n ds = FD

T,E(uh), (3.52)

where FDT,E(uh) is the numerical diffusion flux from Subsection 2.4.2. Finally, for each

edge E ∈ ET ∩ EΓ holds

∫

Eα∇(Imuh)|T · n ds =

∫

E(φh + t0) ds. (3.53)

Remark 3.2.5. Note that for problems with convection, one would have to distinguish

between edges on Γin and Γout.

For problems with a reaction term c we have for all T ∈ T∫

Tc Imuh dx = FR

T (uh), (3.54)

where FRT (uh) is the numerical approximation of the reaction integral from Subsec-

tion 2.4.2.


Remark 3.2.6. In [60] they use

(Imuh)|T (xEm) = (uT + uT ′)/2

for triangles. We want to point out that we could not get experimental convergence with

this setting, even for a pure Dirichlet problem.

Lemma 3.2.7. The Morley interpolant Imuh is uniquely defined by (3.48)–(3.54). More-

over, Imuh is globally continuous in Ω.

Proof. For an element T ∈ T the interpolant (Imuh)|T is uniquely defined by the

appropriate (3.48)–(3.54) since (T,PT ,ΣT )i, i = 1, 2 are finite elements.

Remark 3.2.8. The Morley interpolant Imuh can be computed locally for each element

T ∈ T by solving a 9 × 9 system in case of a diffusion or by a 7 × 7 system for a diffusion

reaction problem.

Properties of the Morley Interpolant. In this paragraph we will provide some im-

portant properties of our constructed Morley interpolant, which allow us to prove an

a posteriori estimator. First, we define the residual

RIm := f − div(−α∇Imuh) − c Imuh (3.55)

and an edge-residual JIm : L2(E) → R by

JIm |E :=

[[−α∇Imuh]] · n for all E ∈ EI ,

−(α∇hImuh) · n + φh + t0 for all E ∈ EΓ.(3.56)

Our first observation is a property, which gives us a link between the Morley interpolant

and the discrete solution uh,Γ ∈ S1(EΓ) on the coupling boundary.

Corollary 3.2.9. Let us consider the diffusion Morley element (3.43) and the diffusion

reaction Morley element (3.44). On a boundary edge E ∈ EΓ we have

uh,Γ(x) − Imuh(x) = 0 for all x ∈ E, (3.57)

where uh,Γ is given by (2.43). This also holds for the general Morley element (3.47)

(b · n = 0 on E ∈ E).

Proof. Note that uh,Γ ∈ S1(EΓ). Let xEp and xEq the starting and end point of E ∈ EΓ,

see Figure 3.7 (right). Through construction we have the identities uh,Γ(xEp) = Im(xEp)

and uh,Γ(xEq ) = Im(xEq ) and we are finished for diffusion reaction problems since

Imuh|E ∈ P1(E). Note that for diffusion problems we have Imuh|E ∈ P2(E). But through

construction we also gain uh,Γ(xEm) = Im(xEm) for the midpoint xEm of E.

From Definition 2.4.4 of the discrete problem we obtain an additional orthogonality prop-

erty of Imuh.


Lemma 3.2.10. The residual RIm is L2-orthogonal to P0(T ), i.e.

∫

T

(f − div(−α∇Imuh) − c Imuh

)dx = 0 for all T ∈ T . (3.58)

Proof. With the divergence theorem and the properties of the Morley interpolant Imuh

we infer∫

T

(div(−α∇Imuh) + c Imuh)

)dx

=

∫

∂T(−α∇Imuh) · n ds+

∫

Tc Imuh dx

=∑

E∈ET \Γ

FDT,E(uh) + FR

T (uh) −∫

Γ(φh + t0) ds =

∫

Tf dx,

where we have used the balance equation (2.45a) in the last equality.

According to the definition of Imuh on the coupling boundary, we obtain corresponding

orthogonalities. Note that n is as usual the normal vector pointing outward with respect

to T .

Lemma 3.2.11. For all boundary edges E ∈ EΓ hold∫

E

(− α∇Imuh · n + φh + t0)ds = 0 for all E ∈ EΓ. (3.59)

Proof. The lemma follows directly from (3.53).

Finally, we note some orthogonality relations of the normal jump of Imuh.

Lemma 3.2.12. Let n be the normal vector pointing from the element T to T ′ with

E = T ∩ T ′. For the interior edges hold∫

E[[−α∇Imuh]] · n ds = 0 for all E ∈ EI . (3.60)

Proof. The definition of the jump and the discrete flux on interior edges implies∫

E[[−α∇Imuh]] · n ds ds

=

∫

E(−α∇Imuh)|T ′ · n ds−

∫

E(−α∇Imuh)|T · n ds = −FD

T ′,E − FDT,E = 0,

where we have used the conservation property of the diffusive flux.

The next two lemmas prove inequalities against the energy norm. We use the quantities

µT and µE defined in Subsection 3.1.3 and we refer to Figure A in the Index of Notation

for an overview.

Lemma 3.2.13. For v ∈ H1(T ) and the integral mean vT :=∫

T v dx/|T | over T ∈ Tthere holds

µ−2T ‖v − vT ‖2

L2(T ) ≤ |||v|||2T . (3.61)


Proof. Note that µT = minβ

−1/2T , hTα

−1/2

and thus we differ two cases. First, let

h−2T α ≤ βT , then

βT ‖v − vT ‖2L2(T ) ≤ βT ‖v‖2

L2(T ) ≤ |||v|||2T ,

which follows from the fact that vT is the L2-orthogonal projection of v onto a constant

and the definition of the energy norm. For h−2T α > βT we use the Poincaré inequality for

convex domains (1.23) and get

h−2T α‖v − vT ‖2

L2(T ) ≤ h−2T α

h2T

π2‖∇v‖2

L2(T ) ≤ 1

π2|||v|||2T ,

which proves the second case.

Lemma 3.2.14. For v ∈ H1(Ω) and the integral mean vE :=∫

E v dx/|E| over an edge

E ∈ E there holds

∑

E∈Eα1/2µ−1

E ‖v − vE‖2L2(E) . |||v|||2Ω. (3.62)

Proof. Since µE =β

−1/2E , hEα

−1/2E

we split the sum into two parts. We denote by TE

one of the two elements with E ∈ ETEfor an interior edge E ∈ EI . For a boundary edge

this is unique. First, we consider

∑

E∈E

h−2E

α≤βE

α1/2β1/2E ‖v − vE‖2

L2(E) ≤∑

E∈E

h−2E

α≤βE

α1/2β1/2E ‖v‖2

L2(E),

since vE is the L2-orthogonal projection on E. Similar as in the proof of Lemma 3.1.15, we

get with the trace inequality (1.26), the property h−1E α1/2 ≤ β

1/2E and Young’s inequality

∑

E∈E

h−2E

α≤βE

α1/2β1/2E ‖v − vE‖2

L2(E) .∑

E∈E

h−2E

α≤βE

(βE‖v‖2

L2(TE) + α‖∇v‖2L2(TE)

) ≤ |||v|||2Ω.

The second case follows by the trace inequality (1.27)

∑

E∈E

h−2E

α>βE

h−1E α‖v − vE‖2

L2(E) .∑

E∈E

h−2E

α>βE

α‖v‖2L2(TE) ≤ |||v|||2Ω,

which proves the lemma.

3.2.2 Reliability of the Error Estimator

In this subsection we prove reliability of an a posteriori error estimator, which is based

on the Morley interpolant. Because of the properties of this interpolant we adapt the

well-known a posteriori analysis from the context of the finite element method and the


analysis from Theorem 3.1.19. But first, we define for each element T ∈ T the refinement

indicator

η2T := µ2

T ‖RIm‖2L2(T ) +

1

2

∑

E∈EI∩ET

α−1/2µE‖JIm‖2L2(E) +

∑

E∈EΓ∩ET

α−1/2µE‖JIm‖2L2(E)

+∑

E∈EΓ∩ET

hE‖∂uh,Γ/∂s− ∂/∂s (u0 − Vφh + (1/2 + K)ξh) ‖2L2(E) (3.63)

+∑

E∈EΓ∩ET

hE‖Wξh + (1/2 + K∗)φh‖2L2(E)

with RIm and JIm from (3.55) and (3.56), respectively. The following theorem states an

upper bound for the Morley error u− Imuh in the energy norm and the errors ξ − ξh and

φ− φh from the exterior problem.

Theorem 3.2.15 (Reliability). Let u ∈ H1(Ω), ξ ∈ H1/2∗ (Γ), φ ∈ H−1/2(Γ) be the

solution of our model problem in Definition 2.0.2 and uh ∈ S1(T ), ξh ∈ S1∗ (EΓ), φh ∈

P0(EΓ) the discrete solution of the discrete problem in Definition 2.4.4, and Imuh the

Morley interpolant defined in Subsection 3.2.1. There is a constant Crel, which depends

only on the shape of the elements in T but neither on the size, the number of elements

nor on the model data such that

|||u− Imuh|||Ω + ‖ξ − ξh‖H1/2(Γ) + ‖φ− φh‖H−1/2(Γ) ≤ Crel

(∑

T ∈Tη2

T

)1/2

.

Proof. In this proof we write for the Morley error e := u− Imuh ∈ H1(Ω), for the trace

error δ := ξ − ξh ∈ H1/2∗ (EΓ) and for the conormal error ǫ := φ − φh ∈ H−1/2(EΓ). By

definition of the energy norm and the coercivity property (2.4) we start with

|||e|||2Ω ≤ (α∇e,∇e)Ω + (ce, e)Ω

= (α∇u,∇e)Ω + (cu, e)Ω + (−α∇Imuh,∇e)Ω − (c Imuh, e)Ω

= (f − c Imuh, e)Ω + 〈t0, e〉Γ + 〈φ, e〉Γ +∑

T ∈T

∫

T(−α∇Imuh) · ∇e dx

= (f − c Imuh, e)Ω + 〈t0, e〉Γ + 〈φ, e〉Γ +∑

T ∈T

∫

∂T(−α∇Imuh) · n e ds

−∑

T ∈T

∫

Tdiv(−α∇Imuh)e dx,

where we have used the weak form (2.8a) and integration by parts. Furthermore, we

obtain

|||e|||2Ω ≤∑

T ∈T

∫

TRIme dx+ 〈t0, e〉Γ + 〈φ, e〉Γ +

∑

E∈EI

∫

E[[−α∇Imuh]] · n e ds

+∑

E∈EΓ

∫

E(−α∇Imuh) · n e ds

=∑

T ∈T

∫

TRIm(e− eT ) dx+

∑

E∈E

∫

EJIm(e− eE) ds+ 〈φ− φh, e〉Γ ,


where we have applied the orthogonalities (3.58) for the integral mean eT =

|T |−1∫

T e dx, (3.59) and (3.60) for eE = h−1E

∫E e ds, respectively. If we use the Cauchy-

Schwarz inequality and the estimates (3.61) and (3.62) we get

|||e|||2Ω .

(∑

T ∈Tµ2

T ‖RIm‖2L2(T )

)1/2

+

(∑

E∈Eα−1/2µE‖JIm‖2

L2(E)

)1/2 |||e|||Ω

+ 〈φ− φh, e〉Γ .

(3.64)

It remains to estimate the last term. We observe that

〈φ− φh, u− Imuh〉Γ = 〈φ− φh, u− uh,Γ〉Γ + 〈φ− φh, uh,Γ − Imuh〉Γ .

The error on the boundary can be written by u − uh,Γ = −p0 − Vǫ + (1/2 + K)δ. Thus,

we estimate

〈φ− φh, u− uh,Γ〉Γ = 〈ǫ,−p0〉Γ + 〈ǫ,−Vǫ〉Γ + 〈ǫ, (1/2 + K)δ〉Γ

≤ ‖ǫ‖H−1/2(Γ)‖p0‖H1/2(Γ) − 〈ǫ,Vǫ〉Γ + 〈p1, δ〉Γ − 〈Wδ, δ〉Γ

≤ ‖ǫ‖H−1/2(Γ)‖p0‖H1/2(Γ) + ‖δ‖H1/2(Γ)‖p1‖H−1/2(Γ)

− 〈ǫ,Vǫ〉Γ − 〈Wδ, δ〉Γ

and Corollary 3.2.9 shows

〈φ− φh, uh,Γ − Imuh〉Γ = 0.

Then the same arguments as in the proof for Theorem 3.1.19 leads to

|||e|||2Ω .

(∑

T ∈Tµ2

T ‖RIm‖2L2(T )

)1/2

+

(∑


L2(E)

)1/2 |||e|||Ω

+ ‖ǫ‖H−1/2(Γ)

∑

E∈EΓ

hE‖∂p0/∂s‖2L2(E)

1/2

+ ‖δ‖H1/2(Γ)

∑

E∈EΓ

hE‖p1‖2L2(E)

1/2

− 〈ǫ,Vǫ〉Γ − 〈Wδ, δ〉Γ .

Since V and W are positive definite on H−1/2(Γ) and H1/2∗ (Γ), respectively, we can put

the last two terms to the left-hand side and write

|||e|||2Ω + ‖δ‖2H1/2(Γ) + ‖ǫ‖2

H−1/2(Γ)

.

(∑

T ∈Tµ2

T ‖RIm‖2L2(T )

)1/2

+

(∑


L2(E)

)1/2 |||e|||Ω

+

∑

E∈EΓ

hE‖∂p0/∂s‖2L2(E)

1/2

‖ǫ‖H−1/2(Γ) +

∑

E∈EΓ

hE‖p1‖2L2(E)

1/2

‖δ‖H1/2(Γ).


If we use again the Cauchy-Schwarz inequality we arrive at

|||e|||2Ω + ‖δ‖2H1/2(Γ) + ‖ǫ‖2

H−1/2(Γ)

.∑

T ∈Tµ2

T ‖RIm‖2L2(T ) +

∑


L2(E)

+∑

E∈EΓ

hE‖∂/∂s (uh,Γ − u0 + Vφh + (1/2 + K)ξh) ‖2L2(E)

+∑

E∈EΓ

hE‖Wξh + (1/2 + K∗)φh‖2L2(E),


Remark 3.2.16. If we define new conforming finite elements of Morley-type, where Corol-

lary 3.2.9 remains not valid, we only have to ensure uh,Γ(ai) = Imuh(ai) for ai ∈ NΓ. This

property is enough to prove reliability. The only difference to the above proof is that the

term 〈φ− φh, uh,Γ − Imuh〉Γ does not vanish. Because of uh,Γ(ai) = Imuh(ai) for ai ∈ NΓ

we can apply (3.18) of Lemma 3.1.16. See [15, Remark 5] that there must be at least one

root on the edge. We estimate

〈φ− φh, uh,Γ − Imuh〉Γ ≤ ‖φ− φh‖H−1/2(Γ)‖uh,Γ − Imuh‖H(1/2)(Γ)

. ‖ǫ‖H−1/2(Γ)

∑

E∈EΓ

hE‖∂/∂s(uh,Γ − Imuh)‖2L2(E)

1/2

and get an additional quantity.

3.2.3 Local Efficiency of the Error Estimator

As in Subsection 3.1.4 we assume Assumption 3.1.24 on the data and a little bit more

regularity for the solution and the jump terms of the weak form in Definition 2.2.2, namely

u ∈ H1(Ω) with γ0u ∈ H1(Γ),

ξ ∈ H1∗ (Γ), φ ∈ L2(Γ),

u0 ∈ H1(Γ), t0 ∈ L2(Γ).

As usual, we describe by uh ∈ P0(T ), uh,Γ ∈ S1(EΓ), ξ ∈ S1∗ (EΓ) and φh ∈ P0(EΓ) the

computed discrete solutions of Definition 2.4.4 with b = 0 and Imuh is the appropriate

Morley interpolant defined in Subsection 3.2.1.

Theorem 3.2.17. The error indicator ηT is a local estimator. Theorem 3.1.38 holds

analogously for the error u− Imuh, ξ− ξh, φ−φh and the refinement indicator ηT defined

in (3.63).

Proof. Note that the Claims 3.1.33–3.1.37 hold because the inverse inequalities in

Lemma 3.1.26–3.1.29 are proven for piecewise polynomials and thus appropriate for the


Morley interpolant Im. We only take a closer look to the tangential jump on the boundary.

We estimate

h1/2E ‖∂uh,Γ/∂s− ∂/∂s (u0 − Vφh + (1/2 + K)ξh) ‖L2(E)

≤ h1/2E ‖∂Imuh/∂s− ∂/∂s (u0 − Vφh + (1/2 + K)ξh) ‖L2(E)

+ h1/2E ‖∂/∂s(uh,Γ − Imuh)‖L2(E).

The first term on the right-hand side can be estimate with Claim 3.1.37. Thus, we only

have to estimate ‖∂/∂s(uh,Γ − Imuh)‖L2(E). This term vanishes according to Corol-

lary 3.2.9. That means, all quantities of (3.63) can be estimated in the same way as

for Theorem 3.1.38.

Chapter 4

Numerical Experiments

In this chapter we study the accuracy of the derived discrete systems in Chapter 2 and

the corresponding a posteriori error estimates in Chapter 3 for the model problem in

Definition 2.0.2. We have a special focus on adaptive mesh-refining, which is steered by the

local refinement indicators ηT of the appropriate estimators in (3.20), (3.23) and (3.63),

respectively. All computations are done in MatlabR© (R2009b) and the programming

language C used by the MEX-interface of MatlabR©.

The first section gives a brief collection of the numerical implementation for the discrete

systems, especially the boundary integral operators, the quadrature rules, the refinement

indicators and the used adaptive algorithm for the mesh-refinement. The second section

provides three numerical examples for the coupling of the finite volume element method

and the boundary element method, whereas in the third section we discuss three examples

for the coupling with the cell-centered finite volume method. As usual let u ∈ H1(Ω),

ξ ∈ H1/2∗ (Γ), φ ∈ H−1/2(Γ) be the solution of the weak form in Definition 2.2.2 and

uh ∈ S1(T ), ξh ∈ S1∗ (EΓ), φh ∈ P0(EΓ) the solution either of the discrete problem with the

finite volume element method in Definition 2.3.2 or with the finite volume element method

with upwinding in Definition 2.3.7. For the discrete problem with the cell-centered finite

volume method in Definition 2.4.4 we use the same notation. Note that there holds

uh ∈ P0(T ) and uh,Γ ∈ S1(EΓ). The notation is clear from the context.

4.1 Implementation Aspects

In this section we give a summary on the implemented methods used for the numerical ex-

amples in Section 4.2 and Section 4.3. For the numerical approximation over an triangle we

use a Gauss quadrature of degree five. On the reference triangle T =(0, 0), (1, 0), (0, 1)

we have the following evaluation points xi = (xi,1, xi,2) ∈ R × R and weights wi ∈ R:

x1,1 = 1/3, x1,2 = 1/3, w1 = 270/2400,

x2,1 = (6 +√

15)/21, x2,2 = (6 +√

15)/21, w2 = (155 + 2√

15)/2400, ⊲

95

96 Chapter 4. Numerical Experiments

x3,1 = (9 − 2√

15)/21, x3,2 = (6 +√

15)/21, w3 = (155 + 2√

15)/2400,

x4,1 = (6 +√

15)/21, x4,2 = (9 − 2√

15)/21, w4 = (155 + 2√

15)/2400,

x5,1 = (6 −√

15)/21, x5,2 = (6 −√

15)/21, w5 = (155 − 2√

15)/2400,

x6,1 = (9 + 2√

15)/21, x6,2 = (6 −√

15)/21, w6 = (155 − 2√

15)/2400,

x7,1 = (6 −√

15)/21, x7,2 = (9 + 2√

15)/21, w7 = (155 − 2√

15)/2400.

We stress that no evaluation point lies on the edge or in a corner of T . Thus, with an

appropriate triangulation of Ω, this quadrature is useful to calculate the integral over the

elements T , even if the function is not smooth, i.e. the singularity should be on an edge

or in the corner of T . Numerical approximation over an edge is done by a six point Gauss

(Legendre) quadrature. On the reference interval (−1, 1) we have the following evaluation

points xi ∈ (−1, 1) and weights wi ∈ R:

x1 = −0.932469514203152, w1 = 0.171324492379170,

x2 = −0.661209386466265, w2 = 0.360761573048139,

x3 = −0.238619186083197, w3 = 0.467913934572691,

x4 = 0.238619186083197, w4 = 0.467913934572691,

x5 = 0.661209386466265, w5 = 0.360761573048139,

x6 = 0.932469514203152, w6 = 0.171324492379171,

where again no evaluation point lies at an endpoint of (−1, 1).

Remark 4.1.1. We use the above quadratures for every integral over a triangle or line,

respectively. Note that in some cases, e.g. if a function is constant over the triangle or

line, we do not need these quadratures.

4.1.1 The Discrete Systems

The discrete systems in Definition 2.3.2, Definition 2.3.7 and Definition 2.4.4 can be divided

in assembling of block matrices for the finite volume methods, the boundary integral

operators and matrices, which consider the interface of both methods. We remind that

for our closed coupling boundary Γ there holds #EΓ = #NΓ. But we will differ between

the notation, to indicate if a matrix comes from an edge or node based calculation.

The Finite Volume Element Method. The discretization is based on the dual mesh

T ∗. We do not have to construct T ∗, and we can use an element based implementation

over the primal mesh T , see Figure 4.1. Note that the discrete unknowns are located

at the nodes N of the primal mesh and we use the property of flux conservation for

the implementation. We consider the implementation for T -piecewise constant α. The

quadrature over a triangle described above is used to integrate the right-hand side f

and the reaction term c on the triangle Z ∈ ZVi , which gives us a part of the integrals

of f and c over Vi for the ith line in the system. We calculate the integral mean of

4.1. Implementation Aspects 97

T

Z Vi

xT

xEi

xEj

xEk

ai

aj ak

Figure 4.1. The discretization by the finite volume element method follows directly from the

primal mesh T . We naturally get the triangle Z ∈ ZV .

b · n on the lines xEixT , xEjxT and xEkxT with the six point Gauss quadrature rule. If

#N is the number of unknowns, we get from the discrete system (2.17a) or the discrete

system with upwinding (2.27a) the sparse matrix AF ∈ R#N ×#N and the right-hand

side Ff + Ft0 ∈ R#N ×1. Additionally, we get the matrix C ∈ R

#N ×#EΓ , because of the

coupling with φh.

Remark 4.1.2. We stress that for α ∈ R or if α is T -piecewise constant, it is well-

known that the finite volume element matrix and the Galerkin matrix of the diffusion part

coincide, see Lemma 2.3.9 and [7, 44, 49, 52].

Galerkin Entries of the Boundary Integral Operators. Let us denote by χi ∈P0(EΓ) the characteristic function of Ei ∈ EΓ, then the set χ1, . . . , χ#EΓ

is a basis of

P0(EΓ). For each node ai ∈ NΓ let ηi ∈ S1(EΓ) be the hat function associated with the

node ai, i.e. ηi(aj) = δij . Then, the set η1, . . . , η#NΓ is a basis of S1(EΓ). In this work

we defined the discrete problems in Definition 2.3.2, 2.3.7 and 2.4.4. In all these problems

we have to build the matrices of

V ∈ R#EΓ×#EΓ , Vij = 〈Vχi, χj〉Γ , K ∈ R

#EΓ×#NΓ , Kij = 〈Kηi, χj〉Γ ,

K∗ ∈ R#NΓ×#EΓ , K∗

ij = 〈K∗χi, ηj〉Γ , W ∈ R#NΓ×#NΓ , Wij = 〈Wηi, ηj〉Γ ,

which follows easily from the corresponding ansatz spaces. With the identity 〈Wψ, θ〉Γ =

〈V(∂ψ/∂s), ∂θ/∂s〉Γ for ψ, θ ∈ H1/2(Γ), see also Theorem 1.2.9, it is clear that the entries

of W can be calculated by V. Since K∗ is the adjoint of K, we follow K∗ = K′, i.e. K∗

is the transpose of the matrix K. The calculation of the Galerkin matrix V follows the

result of [21, 56] by use of analytic anti-derivatives. There, an analytical expression for the

entries of the Galerkin matrix is provided. The same technique is available for the matrix

K. We stress that V and K are dense matrices and thus the assembly is of quadratic

complexity.

Remark 4.1.3. These implementations are stable enough for our purpose, at least for the

examples provided in this thesis. In [40, 41] we used a different approach. For example, the


entries of the Galerkin matrix V are essentially of the type Iij :=∫

Ei

∫Ej

log |x−y| dsx dsy,

for two edges Ei, Ej ∈ EΓ. However, we found that it is an issue of stability to use

numerical quadrature for certain far field entries: To be more precise, let xEi , xEj ∈ R2 be

the midpoints of the edges Ei and Ej and hEi , hEj > 0 the corresponding edges lengths.

Provided |xEi − xEj | > 16 minhEi , hEj , we computed Iij by a 16 × 16 point tensorial

Gauss quadrature. Otherwise, we used the analytic formulae of [21, 56], which appear to

become numerically unstable for hEi ≪ hEj due to cancellation effects.

Remark 4.1.4. Recently a package called Hilbert [5] is available, which is free for

academic use. It is a MatlabR© library, where certain entries of the Galerkin matrices are

computed by use of numerical quadrature as well, see the documentation in [5]. We only

remark that this package works also with the MEX-interface of MatlabR©.

Discretization of the Coupling with the Finite Volume Element Method. The

discretization of the boundary element equations (2.17b), (2.17c), (2.27b) and (2.27c)

follows from the above matrices, i.e. V, K, K∗ and W. Note that 〈uh, ψh〉Γ leads to

the transpose matrix of C from the finite volume element part 〈φh, v∗〉Γ, see (2.17a)

and (2.27a), respectively. The right-hand side gives Fu0 ∈ R#NΓ×1. To fix the constant

in ξh ∈ S1(EΓ) we add in (2.17c) and (2.27c), respectively, 〈λ, θh〉Γ with λ ∈ R and add a

fourth line with 〈ξh, µ〉Γ = 0 for all µ ∈ R. Thus, the integral mean of ξh is chosen to be

zero on Γ. We refer to Remark 2.2.1 for more details. This gives us an additional matrix

D ∈ R#NΓ×1 for 〈λ, θh〉Γ, and 〈ξh, µ〉Γ = 0 leads to the transpose matrix of D. If we define

the unknowns ua1 , . . . , ua#N for uh, φ1, . . . , φ#EΓ for φh and ξ1, . . . , ξ#NΓ

for ξh we

get the following system of linear equations, if we assume that a∞ = 0 in the radiation

condition (2.1c):

#N #EΓ #NΓ 1

#N

AF −C 0 0

−C′ −V K 0

0 K′ W D

0 0 D′ 0

·

ua1

...

...

ua#N

φ1...

φ#EΓ

ξ1...

ξ#NΓ

λ

=

Ff

+

Ft0

−Fu0

0

0

#E Γ

#N

Γ1

Note that AF and C are sparse. If we want to fix b∞ in the radiation condition (2.1c),

Lemma 1.2.14 proves that the integral mean of φ over Γ has to be 2πb∞. This leads to an

additional line 〈φh, µb∞〉Γ = 2πb∞ in the discrete model problem in Definition 2.3.2 and


Definition 2.3.7 with µb∞ ∈ R and an additional term 〈λb∞ , ψh〉Γ with λb∞ ∈ R in (2.17b)

and (2.27b), respectively.

The Cell-Centered Finite Volume Method. We consider only the extended discrete

problem in Definition 2.4.4. We use again a T -element based discretization for the cell-

centered method, which mainly results in the terms FDT,E(uh), FC

T,E(uh) and FRT (uh) of

Subsection 2.4.2. Here the unknown uh is piecewise constant, uh|T := uT for T ∈ T . Note

that we calculate the diffusion flux FDT,E(uh) and convection flux FC

T,E(uh) on the edges only

once because of the flux conservation property. For the diffusion flux we use the diamond-

path technique (2.54), where we need the unknowns uT1 , . . . , uT#NΓ and values on the

nodes a ∈ N . We express the unknowns of the interior nodes by values of the cell-centered

finite volume solution uh via (2.46). For the nodes on the boundary Γ we add additional

unknowns ua1 , . . . , ua#NΓ, which gives us the extended solution uh,Γ ∈ S1(EΓ). Thus, the

term∑

E∈ET \Γ FDT,E(uh) +

∑E∈ET \Γin FC

T,E(uh) + FRT (uh) results in two different matrixes

AF,I ∈ R#T ×#T and AF,B ∈ R

#T ×#NΓ if we rearrange it according to the unknowns.

To complete the discretization of (2.45a) the coupling term gives C ∈ R#T ×#NΓ and

the right-hand side Ff + Ft0 ∈ R#T ×1, where f is now integrated over T ∈ T . Note

that (2.45b) follows from (2.46). Thus, ua gives a matrix CF ∈ R#NΓ×#T , ςa,h leads

to PC ∈ R#NΓ×#EΓ and we have an additional contribution Cin ∈ R

#NΓ×#T in case of

b · n 6= 0 on Γin. Furthermore, ςa,t0 leads to the right-hand side FC,t0 ∈ R#NΓ×#NΓ .

Discretization of the Coupling with the Cell-Centered Finite Volume Method.

The coupling matrices of (2.45c) and (2.45d) are the same as above, namely V, K, K∗

and W. We stress that 〈uh,Γ, ψh〉Γ of (2.45c) gives a matrix AC ∈ R#EΓ×#NΓ . Here, it is

not the transpose of C. All things considered we get the following discrete system, if we

assume that a∞ = 0 in the radiation condition (2.1c):

#T #NΓ #EΓ #NΓ 1

#T

AF,I AF,B −C 0 0

CF Cin − I PC 0 0

0 −AC −V K 0

0 0 K′ W D

0 0 0 D′ 0

·

uT1

...

...

uT#T

ua1...ua#NΓ

φ1...

φ#EΓ

ξ1...

ξ#NΓ

λ

=

Ff

+

Ft0

−FC,t0

−Fu0

0

0

#N

Γ#

E Γ#

NΓ

1


4.1.2 Implementation of the Error Estimators

The calculation of the refinement indicators ηT in (3.20) and ηT,up in (3.23) for the coupling

with the finite volume element method is done by use of the introduced quadrature rules.

Note that uh ∈ S1(T ) and thus ∇uh is piecewise constant. For T -piecewise constant α

the quantity of the interior jump J can be calculated exact. We also need values for V,

K and K∗ for the six point Gaussian quadrature rule on an edge E ∈ EΓ. The analytical

calculation uses again the results of [21, 56], see the discussion for generating the Galerkin

entries for the boundary integral operators in Subsection 4.1.1. For the hypersingular

operator W we use the relation Wvh = − ∂∂sV ∂

∂svh, vh ∈ S1(EΓ). This holds because of

Theorem 1.2.9, at least in a distributional sense. Note that for a closed Γ we have

∫

Γvh(y)

∂

∂swh(y) dsy = −

∫

Γ

∂

∂svh(y)wh(y) dsy

for wh ∈ S1(EΓ). Thus, we can use the implementation of K∗, if we replace the normal

vector nx by the tangential vector tx, i.e. tx is chosen orthogonal to nx in mathematical

positive sense. We approximate the arc-length derivative ∂/∂s on an edge E ∈ EΓ of the

jump u0 and the operators V and K by a central difference quotient, i.e. for x ∈ E and

v ∈ C(E) we use ∂v/∂s =(v(x2) − v(x1)

)/|x2 −x1| with a distance |x2 −x1| = hE/20 and

x = (x2 + x1)/2, where the points x1 and x2 on E are ordered in a mathematical positive

sense with respect to the coupling boundary Γ. We stress that the points of the central

difference quotient from the used six point Gaussian quadrature, are in the interior of the

edge E.

The refinement indicator ηT of (3.63) for the coupling with the cell-centered finite volume

method needs the Morley interpolant Imuh introduced in Subsection 3.2.1. As mentioned

there, we need for every problem a different finite element (T,PT ,ΣT ), i.e. a different ap-

proximation space PT and different functionals ΣT . We only have an a posteriori estimator

for diffusion and diffusion convection problems, see Remark 3.2.1. We found that it is an

issue of stability, especially for adaptive mesh-refinements, to use barycentric coordinates

as basis functions for PT . Since the discrete solutions are known, it is easy to calculate

the appropriate functionals ΣT , which are given in (3.48)–(3.54).

The residual-based error estimator η is then given in a general form by the ℓ2-sum

η =

(∑

T ∈Tη2

T

)1/2

(4.1)

and additional for the finite volume element method with upwinding we have

η =

(∑

T ∈T

(η2

T + η2T,up

))1/2

. (4.2)


4.1.3 Implementation of the Energy Norm

In numerical examples, where we know the analytical solution u, we can calculate the error

in the energy norm |||u−uh|||Ω and |||u−Imuh|||Ω, respectively, according to (2.3), where we

use the Gaussian quadrature on each triangle. The error norms of the trace ‖ξ−ξh‖H1/2(Γ)

and of the conormal ‖φ− φh‖H−1/2(Γ) are replaced by their equivalent energy norms, i.e.

‖ξ − ξh‖2H1/2(Γ) ∼ |||ξ − ξh|||2W := 〈W(ξ − ξh), ξ − ξh〉Γ ,

‖φ− φh‖2H−1/2(Γ) ∼ |||φ− φh|||2V := 〈V(φ− φh), φ− φh〉Γ .

We follow again the ideas of [21] to calculate |||φ−φh|||V , which leads to an approximation

of a double integral. That means for v ∈ C(Γ) we have

|||v|||V = − 1

2π

∫

Γv(x)

∫

Γv(y) log |x− y| dsy dsx

= − 1

2π

∑

Ei∈EΓ

(∫

Ei

v(x)∑

Ej∈EΓ

∫

Ej

v(y) log |x− y| dsy

)dsx.

For the outer integral we use a 32 point Gaussian quadrature [72] on each Ei ∈ EΓ, whereas

for the interior integral we differ two cases. The point x is one of the 32 Gaussian points of

the edge Ei ∈ EΓ. If x is not on the edge Ej ∈ EΓ we use again the 32 Gaussian quadrature.

Otherwise, we divide and transform the integral such that the singular point x lies at the

end of the unit interval. Let us denote by a1 and a2 the start and endpoint of the edge

Ej ∈ EΓ. Note that there holds x 6= a1 and x 6= a2. Then the transformation leads to

∫ a2

a1

v(y) log |x− y| dsy =|a2 − x|

2log |a2 − x|

∫ 1

−1v((a2 − x)(t+ 1)/2 + x

)dt

+ |a2 − x|∫ 1

0v((a2 − x)t+ x

)log |t| dt

+|a1 − x|

2log |a1 − x|

∫ 1

−1v((a1 − x)(t+ 1)/2 + x

)dt

+ |a1 − x|∫ 1

0v((a1 − x)t+ x

)log |t| dt.

We approximate the integrals over (−1, 1) again by the 32 point Gaussian quadrature. For

the integrals over (0, 1), where we have a log term, we use a 8 point Gaussian quadrature

rule with logarithmic weights [72]. That means

−∫ 1

0v(x) log |x| dx =

8∑

i=1

wiv(xi)


with xi ∈ (0, 1) and wi ∈ R:

x1 = 1.332024456670017 · 10−2, w1 = 1.644166085401345 · 10−1,

x2 = 7.975043120810019 · 10−2, w2 = 2.375256130444862 · 10−1,

x3 = 1.978710336952959 · 10−1, w3 = 2.268419844405181 · 10−1,

x4 = 3.541540000330712 · 10−1, w4 = 1.757540769143812 · 10−1,

x5 = 5.294585807343734 · 10−1, w5 = 1.129240278141036 · 10−1,

x6 = 7.018145339840138 · 10−1, w6 = 5.787220912785310 · 10−2,

x7 = 8.493793226047583 · 10−1, w7 = 2.097907312279870 · 10−2,

x8 = 9.533264507337711 · 10−1, w8 = 3.686406995724206 · 10−3.

The energy norm |||ξ − ξh|||W is calculated by the relation between the single layer and

hypersingular integral operator given in Theorem 1.2.9.

Finally, we define for the coupling of the finite volume element method and the boundary

element method the total energy norm by

Eh :=(|||u− uh|||2Ω + |||φ− φh|||2V + |||ξ − ξh|||2W

)1/2

or in the case of the coupling with the cell-centered finite volume method and the post

processed Morley interpolant Imuh

Em :=(|||u− Imuh|||2Ω + |||φ− φh|||2V + |||ξ − ξh|||2W

)1/2.

4.1.4 Adaptive Algorithm and Mesh-Refinement

Let us denote the error estimator by η, see Subsection 4.1.2 and (4.1) or (4.2), which was

written in the general form

η :=

(∑

T ∈T

(η2

T (+η2T,up)

))1/2

.

Throughout, we run the following standard algorithm, where we use θ = 1 for uniform

and θ = 1/2 for adaptive mesh-refinement, respectively.

Algorithm 4.1.5. Given an initial mesh T (0), k = 0, and 0 ≤ θ ≤ 1, do the following:

1. Compute the discrete solution uh ∈ S1(T (k)) or uh ∈ P0(T (k)), ξh ∈ S1∗ (EΓ) and

φh ∈ P0(EΓ) for the current mesh T (k) = T1, . . . , T#T . For the cell-centered coupling

compute also the appropriate Morley interpolant Imuh defined in Subsection 3.2.1.

2. Compute the refinement indicators ηT (ηT,up)) for all elements Tj ∈ T (k).

4.2. Examples for the Coupling with the Finite Volume Element Method 103

3. Construct a minimal subset M(k) of T (k) such that

θ∑

T ∈T (k)

(η2

T (+η2T,up)

) ≤∑

T ∈M(k)

(η2

T (+η2T,up)

), (4.3)

and mark all elements in M(k) for refinement.

4. Refine at least all marked elements T ∈ M(k) and generate a new mesh T (k+1).

5. Update k 7→ k + 1 and go to 1.

Remark 4.1.6. The marking criterion (4.3) was introduced in [36] to prove convergence of

an adaptive algorithm for some piecewise linear conforming finite element method for the

Laplace problem. Despite convergence, even the question of optimal convergence rates of

the adaptive finite element method based on residual error estimators is well-understood,

see the work [26] for a precise statement of optimality and the history of mathematical

arguments.

In all experiments, the initial mesh T (0) is a uniform and regular triangulation, where

all of the elements are triangles. We use a red-green-blue strategy to obtain T (k+1) from

T (k), i.e. marked elements are uniformly refined and the obtained mesh is regularized by

a green-blue closure [73]. Therefore, in all our examples the shape regularity constant is

bounded and the ratio between two neighbor boundary edges in our examples is bounded

by four, we refer to Remark 1.3.2, Remark 1.3.3, and Remark 3.1.18 for more details.

4.2 Examples for the Coupling with the Finite Volume El-

ement Method

In this section we describe three numerical examples for the coupling of the finite volume

element method and the boundary element method, which prove the validity of the discrete

system and the a posteriori estimator. In example one and two we know the explicit

solution u and this allows us to compare the error estimator with the exakt energy norm

error. In example three we simulate the stationary concentration of a chemical dissolved

and distributed in different fluids. We write x = (x1, x2) ∈ R × R for a point in the plane

and define N := #T for the number of elements on the triangulation T .

4.2.1 Diffusion Reaction Problem with a Generic Singularity

We consider the model problem of Definition 2.0.2 on the square domain

Ω = (−1/4, 1/4) × (−1/4, 1/4) (4.4)

with the coupling boundary Γ = ∂Ω as shown in Figure 4.2(a). By (r, ϕ), r ∈ R+0 ,

ϕ ∈ [0, 2π[, we denote the polar coordinates with (x1, x2) = r(cosϕ, sinϕ) ∈ R × R. Let


x1

x2

α = α2 from

0.01, . . . , 108

α = α1 = 1

−0.25 0 0.25−0.25

0

0.25

(a) Distribution of α. (b) Initial mesh T (0).

Figure 4.2. Domain Ω = (−1/4, 1/4) × (−1/4, 1/4) for the example in Subsection 4.2.1. The

distribution of α in (a). Note that α1 = 1 is fixed, whereas α2 can vary. The initial mesh T (0)

consists of 16 triangles in (b).

Ω be decomposed into two parts, namely Ω2 :=(x1, x2) ∈ Ω

∣∣ 0 ≤ ϕ(x1, x2) ≤ π/2

and

Ω1 = Ω\Ω2. The diffusion coefficient is piecewise constant and is given by

α : R × R → R : (x1, x2) 7→

1 for (x1, x2) ∈ Ω1,

α2 for (x1, x2) ∈ Ω2.

See Figure 4.2(a). Additionally, we choose the reaction term c = 100 and the convection

vector b = (0, 0)T . For the interior domain Ω we prescribe the solution by

u(x1, x2) :=

rζ1ζ2 cos(ζ1(π − (ϕ− π/2)

))for (x1, x2) ∈ Ω1,

rζ1 cos(ζ1(ϕ− π/2)

)for (x1, x2) ∈ Ω2

(4.5)

with

ζ1 =4

πarctan

(√3 + α2

1 + 3α2

)and ζ2 = −α2

sin(ζ1

π4

)

sin(ζ1

3π4

) .

Therefore, the right-hand side is f = c u(x1, x2). This solution u(x1, x2) in Ω is motivated

by [63, 64]. Note that for α2 ≤ 1 there holds u ∈ H2(Ω) and for α2 > 1 we have

u ∈ H2−ε(Ω), where ε > 0 depends on ζ1. According to the above formula we see that

ζ1 is monotonically decreasing with α2 to the value 2/3. Therefore, the solution belongs

at least to u ∈ H1+2/3(Ω). We refer to [63] for more details, there one can find a general

description even if Ω2 is not the whole upper right corner. For the exterior domain ΩC we

choose

uc(x1, x2) = log√x2

1 + x22. (4.6)


α2 = 0.01:

Eh (uni.)

η (uni.)

Eh (ada.)

η (ada.)

α2 = 1000: Eh (uni.)

η (uni.)

Eh (ada.)

η (ada.)

11/2

13/4

1 1/3

number of elements

erro

ran

des

tim

ator

101 102 103 104 105 106

10−4

10−3

10−2

10−1

100

101

102

103

104

Figure 4.3. Energy error Eh =(

|||u − uh|||2Ω + |||φ − φh|||2V + |||ξ − ξh|||2W)1/2

, as well as the

corresponding error estimator η in the example in Subsection 4.2.1 for uniform (uni.) and

adaptive (ada.) mesh-refinement for α2 = 0.01 and α2 = 1000.

Note that this solution fulfills the model problem equation (2.1b) and the radiation con-

dition (2.1c) with a∞ = 0 and b∞ = 1. The jumps u0 and t0 are computed from the given

exact solutions. The initial mesh T (0) consists of 16 triangles, see Figure 4.2(b). Figure 4.3

shows curves of the total energy error Eh as well as the curves of the error estimator η

with respect to uniform and adaptive mesh-refinement over the number of elements. In

the lower half we see the curves for α2 = 0.01 and in the upper half for α2 = 1000. For

both mesh-refining strategies, the error estimator η is observed to be reliable and efficient.

We plot the experimental results over the number of elements, where both axes are scaled

logarithmically. Therefore, a straight line g with slope −p corresponds to a dependence

g = O(N−p), where N = #T denotes the number of elements. Note that for uniform

mesh-refinement, the order O(N−p) with respect to N corresponds to O(h2p) with respect

to the maximal mesh size h := maxT ∈T

hT . In the case α2 = 0.01 we observe experimentally

some super convergence almost of order O(N−3/4). This holds for both, uniform and

adaptive mesh-refinement, where the absolute values of the adaptive error Eh are slightly

better. Note that in this case both solutions, the interior (4.5) and exterior (4.6), are

smooth and thus Theorem 2.3.10 guarantees a convergence order of at least O(N−1/2). In

the upper half of Figure 4.3 we see the results for α2 = 1000. This chosen α2 leads to a

singularity in the origin and ζ1 = 0.6674. Thus, u ∈ H1+0.6674−ε(Ω) with ε > 0, see [63],

whereas the exterior solution uc is still smooth. For uniform mesh-refinement the energy


(a) #T (4) = 590. (b) #T (6) = 3120.

(c) #T (8) = 15357.

x1x2

−0.25 −0.125 0 0.125 0.25−250

−200

−150

−100

−50

0

−0.25−0.125

00.1250.25

−250

−200

−150

−100

−50

0

(d) Interior solution for T (6).

Figure 4.4. Adaptively generated meshes T (k) for k = 4, 6, 8 in (a)–(c) in the example in

Subsection 4.2.1 for α2 = 1000. In figure (d) we see the solution in the interior domain Ω for

T (6) for α2 = 1000.

error Eh decreases like O(N−1/3) as it can be expected from the coupling of finite element

method and boundary element method. The adaptive algorithm leads to an improved

order of convergence O(N−1/2). For both mesh-refining strategies, the error estimator η

is observed to be reliable and efficient and robust against the model parameters α and c.

A sequence of adaptively generated meshes for α2 = 1000 is provided in Figure 4.4(a)–(c).

Figure 4.4(d) plots the solution in the interior domain for the adaptively refined mesh

T (6). In Figure 4.5 we zoom in the adaptively generated mesh T (9) with a factor 3 near

the origin. Figure 4.6 shows a comparison of interior energy norm |||u − uh|||Ω and the

energy norms of the Cauchy data |||φ−φh|||V and |||ξ− ξh|||W for adaptive mesh-refinement.

In the case α2 = 1000 we can see the optimal convergence rate for the interior energy norm

O(N−1/2), whereas the energy norms for the exterior problem have a rate of approximately


Zoom: 3

ce

Figure 4.5. Adaptively generated mesh T (9) with 33533 elements for the example in Subsec-

tion 4.2.1 for α2 = 1000 with a zoom area (factor 3).

O(N−1). Nevertheless the overall convergence rate is dominated by the interior energy

norm. Therefore, a separate handling of the coupling boundary elements and the finite

volume elements could be done, since we do not need the number of unknowns for the

exterior solution as it is forced by the refinement of the interior domain. And because

the contribution of the exterior problem to the linear equation system is not a sparse

matrix, this would lead to an effort in solving the system. A related work in the context

of the coupling of finite elements method and boundary element method is [59], where

they get an a posteriori error estimator derived by using hierarchical basis techniques.

To demonstrate that our error estimator works, we plot the energy norms for α2 = 1 in

Figure 4.6. Note that the interior solution is linear and therefore, a finite volume method

with Dirichlet boundary conditions would give us the exact solution. For the coupling

problem we therefore expect a higher convergence rate for the interior energy norm than

for the exterior energy norms. This is, in fact, observed. Moreover, in Figure 4.7(a) we see

that the refinement is dominated at the coupling boundary in this case. In Figure 4.7(b)

we plot the interior and exterior solution in the square (−1/2, 1/2) × (−1/2, 1/2), where

we recognize the jumps u0 and t0 at the coupling boundary. We plot the interior solution

for T (8) with 2576 elements and α2 = 1. For the exterior solution we evaluate the exterior

representation formula (1.12) on a uniform grid with 1536 triangles on each node with the

Cauchy data from T (8). For points on the boundary Γ coming from the exterior domain,


α2 = 1000:

|||u− uh|||Ω|||φ− φh|||V|||ξ − ξh|||W

α2 = 1:

|||u− uh|||Ω|||φ− φh|||V|||ξ − ξh|||W

11

11/2

number of elements

erro

ran

des

tim

ator

101 102 103 104 105 106

10−6

10−5

10−4

10−3

10−2

10−1

100

101

102

Figure 4.6. Comparison of the energy errors |||u−uh|||Ω, |||φ−φh|||V and |||ξ−ξh|||W for adaptive

mesh-refinement α2 = 1000 (top) and α2 = 1 (bottom) in the example in Subsection 4.2.1.

(a) #T (8) = 2576.

x1

x2

−0.5−0.25

00.25

0.5

−1.25

−1

−0.75

−0.5

−0.25

0

0.25

−0.5

−0.25

0

0.25

0.5−1.5

−1

−0.5

0

0.5

(b) Interior and exterior solution.

Figure 4.7. Adaptively generated mesh T (8) in the example in Subsection 4.2.1 for α2 = 1

in (a) and the solution in the interior and exterior domain in (b).

we use the exterior trace of (1.12). Note that this trace reads

ξh(x) = −(Vφh)(x) +((K +

ϕ

2π

)ξh

)(x) + a∞

for a point evaluation x ∈ Γ. We refer to Lemma 1.2.4 for the definition of the angle

ϕ. According to Remark 2.2.1 the left-hand side gives us the approximative value of

ξ. The last Figure 4.8 for this example shows the robustness of our estimator against

α and c. The efficiency index η/Eh, which measures how many times we have over-


0.01

0.1

0.5

1

3

10

100

1000

106

108

number of elements

effici

ency

index

101 102 103 104 105 1061.5

2

2.5

3

3.5

4

4.5

5

Figure 4.8. Efficiency index η/Eh for adaptively generated meshes for different α2 =

0.01, 0.1, 0.5, 1, 3, 10, 100, 1000, 106, 108 in the example in Subsection 4.2.1.

estimated the actual error, tends to 4 for adaptively generated meshes for different

α2 = 0.01, 0.1, 0.5, 1, 3, 10, 100, 1000, 106, 108. Note that c = 100 and thus we have

also covered the case of reaction dominated problems.


x1

x2

0 0.25 0.50

0.25

0.5

Figure 4.9. Domain Ω = (0, 1/2) × (0, 1/2) for the example in Subsection 4.2.2. The initial

mesh T (0) consists of 16 triangles.

4.2.2 Diffusion Convection Problem

We consider the model problem of Definition 2.0.2 on the square domain

Ω = (0, 1/2) × (0, 1/2) (4.7)

with the coupling boundary Γ = ∂Ω. We choose a fixed diffusion coefficient of α ∈0.05, 0.1, 0.5, 1, 10, 100, 1000, the convection field b = (100x1, 0)T and the reaction co-

efficient c = 0. Note that for this problem the coupling condition (2.1e) does not occur,

i.e. we have no inflow boundary Γin. For all calculations we use the upwind discrete

coupling of Definition 2.3.7 with the weighting function Φ defined in (2.24). We prescribe

an analytical solution

u(x1, x2) = 0.5

(1 − tanh

(0.25 − x1

0.02

))

for the interior domain Ω and

uc(x1, x2) = log√

(x1 − 0.25)2 + (x2 − 0.25)2

similar as in the example in Subsection 4.2.1 for the exterior domain ΩC with a∞ = 0

and b∞ = 1. We calculate the right-hand side f and the jumps u0 and t0 appropriate.

Figure 4.9 shows the initial mesh of 16 triangles. In Figure 4.10 we plot the convergence

rate for uniform and adaptive mesh-refinement for α = 0.05 (lower part) and α = 1000

(upper part) with respect to the number of elements. Since the interior and exterior

solution are smooth, we observe the expected convergence rate of O(N−1/2) in both cases,

see Theorem 2.3.15. In the case α = 0.05, which is convection dominated, we also see that

for adaptive mesh-refinement the elements in the refined shock start to leave the convection

dominated regime at around 300 elements, where the estimator starts to decrease, whereas


α = 0.05:

Eh (uni.)

η (uni.)

Eh (ada.)

η (ada.)

α = 1000: Eh (uni.)

η (uni.)

Eh (ada.)

η (ada.)

11/2

11/2

number of elements

erro

ran

des

tim

ator

101 102 103 104 105 106

10−3

10−2

10−1

100

101

102

103

Figure 4.10. Energy error Eh =(

|||u − uh|||2Ω + |||φ − φh|||2V + |||ξ − ξh|||2W)1/2

, as well as the

corresponding error estimator η in the example in Subsection 4.2.2 for uniform (uni.) and

adaptive (ada.) mesh-refinement for α = 0.05 and α2 = 1000.

(a) #T (6) = 3788.

x1x2 0

0.1250.25

0.3750.5

0

0.2

0.4

0.6

0.8

1

00.125

0.250.375

0.50

0.2

0.4

0.6

0.8

1

(b) Interior solution for T (6).

Figure 4.11. Adaptively generated mesh T (6) in the example in Subsection 4.2.2 for α = 0.05

in (a) and the associated solution in the interior domain Ω in (b).

for uniform refinement we need more than 1000 elements. We stress that in both cases the

adaptive generated solutions lead to a better energy norm Eh, since the adaptive algorithm

leads to a expected refinement at the shock x1 = 1/4 of the solution u. In Figure 4.11 we

show an adaptively generated mesh T (6) in (a) and its associated solution in (b). Finally,

Figure 4.12 plots the efficiency index η/Eh, which tends to 6 and thus our estimator is

robust with respect to the ratio of α and ‖b‖L∞(Ω).


0.05

0.1

0.5

1

10

100

1000

number of elements

effici

ency

index

101 102 103 104 105 1062

4

6

8

10

12

14

16

Figure 4.12. Efficiency index η/Eh for adaptively generated meshes for different α =

0.05, 0.1, 0.5, 1, 10, 100, 1000 in the example in Subsection 4.2.2.


x1

x2

α = 10−4

α = 5·10−4 α = 10−3

−0.25 0 0.25−0.25

0

0.25

(a) Distribution of α and initial mesh T (0).

x1

x2

f = 5

−0.2 −0.1 0 0.25

−0.2

−0.05

0.25

(b) Volume force f .

Figure 4.13. Domain Ω = (−1/4, 1/4)2\([0, 1/4] × [−1/4, 0]

)in the example in Subsec-

tion 4.2.3. The distribution of α and the initial mesh T (0) consists of 12 triangles in (a). In (b)

the volume force f has the value 5 in the gray rectangle, otherwise it is 0.

4.2.3 Convection Dominated Problem

We consider the model problem of Definition 2.0.2 on the classical L-shaped domain

Ω = (−1/4, 1/4)2\([0, 1/4] × [−1/4, 0])

(4.8)

with the coupling boundary Γ = ∂Ω as shown in Figure 4.13(a). The diffusion coefficient

in Ω is piecewise constant and is given by

α : R × R → R : (x1, x2) 7→

10−4 for x2 ≤ 0,

10−3 for x1 > 0,

5 · 10−4 else,

see also Figure 4.13(a). Additionally, we choose b = (15, 10)T and c = 10−2. We have a

volume force f in the lower square, i.e.

f =

5 for − 0.2 ≤ x1 ≤ −0.1, −0.2 ≤ x2 ≤ −0.05,

0 else,

see also Figure 4.13(b). We prescribe the jumps u0 = 0 and t0 = 0 and and the radiation

condition b∞ = 0. We use the full upwind scheme for the approximation of the convection

term. This model can describe the stationary concentration of a chemical dissolved and

distributed in different fluids, where we have a convection dominated problem in the

interior Ω and a diffusion distribution in the exterior domain ΩC . The solution of such a

problem may have local phenomena such as injection wells and will lead to step layers on

the boundary (0, 0) to (0,−0.25), due to the convection in this direction and the different


x1

x2−0.250 0.25

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

−0.250

0.25−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

(a) Solution with oscillations.

x1

x2−0.250 0.25

0.005

0.01

0.015

0.02

0.025

0.03

−0.250

0.25−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

(b) Solution with full upwinding.

Figure 4.14. Strong oscillations of the interior solution by approximation without upwinding

in (a) compared with the full upwind scheme solution in (b) in the example in Subsection 4.2.3

for uniform mesh-refinement with 3072 elements.

x1

x2

−0.5

−0.25

0

0.25

0.5

0.005

0.01

0.015

0.02

0.025

0.03

−0.5

−0.25

0

0.25

0.50

0.01

0.02

0.03

0.035

Figure 4.15. Interior and exterior solution for the adaptively generated mesh T (7) with 4201

elements in the example in Subsection 4.2.3.

diffusion coefficient of the interior and exterior problem. The initial mesh T (0) consists

of 12 triangles, see Figure 4.13(a). Note that this problem is convection dominated and

we only could get a uniform or adaptive mesh solution without oscillation by use of the

full upwind scheme, i.e. the weighting function (2.23), instead of partly upwinding by

the weighting function (2.24). The reason might be the steep layer on the boundary from

(0, 0) to (0,−0.25), Figure 4.14(a) shows the interior approximation without an upwind

scheme for the convection part on a uniform generated mesh with 3072 elements, which

leads to strong oscillations, whereas in Figure 4.14(b) we plot the solution with the full


η (uni.)

η (ada.)1

1/2

1

1/4

number of elements

esti

mat

or

101 102 103 104 105 106

10−1

100

101

(a) Error estimator.

x1

x2

−0.3 −0.15 0 0.15 0.3

−0.3

−0.15

0

0.15

0.3

(b) Contour lines.

Figure 4.16. Error estimator η in the example in Subsection 4.2.3 for uniform (uni.) and

adaptive (ada.) mesh-refinement in (a). We generate the contour lines in (b) from the solution

of the adaptively generated mesh T (13) with 286625 elements.

upwind scheme. Figure 4.15 shows the interior and exterior solution for T (7) with 4201

elements, where we plot the exterior solution with the representation formula (1.12) on a

uniformly refined mesh similar as in the example in Subsection 4.2.1. We can see that the

solution is continuous but has sharp layers on the boundary due to the convection term and

the jump of the diffusion coefficient. In Figure 4.16(a) we see the predicted convergence

rate for uniform and adaptive mesh-refinement. For uniform mesh-refinement the energy

error Eh decreases like O(N−1/4). The adaptive algorithm leads to an improved order of

convergence O(N−1/2). We stress that we do not know anything on the regularity of u and

uc. Figure 4.16(b) plots the contour lines of the solution from the last adaptive generated

mesh T (13). We see a significant transport from the square f = 5 in the direction of the

convection vector b. Since u0 = 0 the contour lines are continuous on the boundary, which

can be seen on (0, 0) to (0.25, 0). On (0, 0) to (0,−0.25) and (0.25, 0.25) to (0.25, 0) the

resolution is not high enough. We remark that the exterior problem has only a diffusion

term, which can be seen on the circular contour lines. Figure 4.17 presents a sequence of

adaptively generated meshes. As expected, the refinement occurs on the boundary (0, 0)

to (0,−0.25) and (0.25, 0.25) to (0.25, 0), in the direction of the vector b and mainly near

the volume source f = 5. Finally, in Figure 4.18 we provide two zoom areas with factor 2

of the adaptively generated mesh T (9).


(a) #T (4) = 426. (b) #T (6) = 2115.

(c) #T (7) = 4201. (d) #T (8) = 8529.

Figure 4.17. Adaptively generated meshes T (k) for k = 4, 6, 7, 8 in the example in Subsec-

tion 4.2.3.


Zoom:2

ce

Zoom:2

ce

Figure 4.18. Adaptively generated mesh T (9) with 17077 elements in the example in Subsec-

tion 4.2.3 with two zoom areas (factor 2).


Em (uni.)

η (uni.)

Em (ada.)

η (ada.)

11/3

11/2

number of elements

erro

ran

des

tim

ator

101 102 103 104 105 10610−4

10−3

10−2

10−1

100

101

Figure 4.19. Energy error Em =(

|||u− Imuh|||2Ω + |||φ− φh|||2V + |||ξ− ξh|||2W)1/2

, as well as the

corresponding error estimator η in the example in Subsection 4.3.1 for uniform and adaptive

mesh-refinement.

4.3 Examples for the Coupling with the Cell-Centered Fi-

nite Volume Method

We provide three numerical examples for the coupling of the cell-centered finite volume

method and the boundary element method. We write again x = (x1, x2) ∈ R × R for a

point in the plane and the notation N := #T and O(N−p) is the same as in Section 4.2.

4.3.1 L-Shaped Problem

We consider the model problem of Definition 2.0.2 on the classical L-shaped domain defined

in (4.8) with the coupling boundary Γ = ∂Ω as shown in Figure 4.13(a). Here, we set

α = 1, b = (0, 0)T and c = 0. The given exact solution is the harmonic function u(x1, x2) =

Im((x1 + ix2)2/3

)and reads in polar coordinates (r, ϕ) with r ∈ R

+0 and ϕ ∈ [0, 2π[

u(x1, x2) = r2/3 sin(2ϕ/3) with (x1, x2) = r(cosϕ, sinϕ).

Thus, the right-hand side f = 0. Note that u has a generic singularity at the reentrant

corner (0, 0), which leads to u ∈ H1+2/3−ε(Ω) for all ε > 0. Therefore, a conforming finite

element method with polynomial ansatz space leads to convergence of order O(h2/3) for

the finite element error in the H1-norm, where h denotes the uniform mesh-size. This

4.3. Examples for the Coupling with the Cell-Centered Finite Volume Method 119

(a) #T (16) = 7346.

x1

x2−0.25

00.25

0

0.125

0.25

0.375

0.5

−0.25

0

0.25

0

0.25

0.5

(b) Solution for T (14).

Figure 4.20. Adaptively generated mesh T (16) in (a) and interior solution for T (14) with 2536

elements in (b) in the example in Subsection 4.3.1.

corresponds to order O(N−1/3) with respect to the number of elements. For the exterior

problem (2.1b) we use

uc(x1, x2) = log√

(x1 + 0.125)2 + (x2 − 0.125)2

similar as in the example in Subsection 4.2.1 with the radiation condition (2.1c) a∞ = 0 and

b∞ = 1. The jumps u0 and t0 are calculated appropriately. This model problem describes

the classical Laplace problem in the interior and exterior domain and we want to verify our

discretization ansatz in Definition 2.4.4 and the Morley interpolant of Definition 3.2.2 for

pure diffusion problems. We find this problem in several works [25, 18] to mention only a

few but not all and therefore, one can see this as a benchmark problem to test new discrete

systems. The initial mesh is shown in Figure 4.13(a) of Example 4.2.3. Figure 4.19 plots

the experimental results for the energy error Em and the corresponding error estimator η

over the number of elements N and confirms the reliability and efficiency of the estimator.

For uniform mesh-refinement, the energy error Em converges with a suboptimal order of

O(N−1/3). The adaptive algorithm leads to an improved order of convergence O(N−1/2).

Figure 4.20(a) shows the adaptively refined mesh after 16 refinements, where we observed a

finer grid near the singularity at the origin. In Figure 4.20(b) we plot the piecewise constant

interior solution for the adaptively generated mesh T (14) and Figure 4.21 provides the

interior and exterior solution on the square (−1/2, 1/2) × (−1/2, 1/2), which is generated

by the solution of the adaptive mesh T (12) similar as in the example in Subsection 4.2.1.


x1

x2

−0.5

−0.25

0

0.25

0.5

−2

−1.5

−1

−0.5

0

0.5

−0.5

−0.25

0

0.25

0.5

−2

−1

00.5

Figure 4.21. Interior and exterior solution for T (12) with 914 elements in the example in

Subsection 4.3.1.


c = 10 :

Em (uni.)

η (uni.)

Em (ada.)

η (ada.)

c = 105 : Em (uni.)

η (uni.)

Em (ada.)

η (ada.)

11/2

11/2

number of elements

erro

ran

des

tim

ator

101 102 103 104 105 10610−3

10−2

10−1

100

101

102

103

104

Figure 4.22. Energy error Em =(

|||u − Imuh|||2Ω + |||φ − φh|||2V + |||ξ − ξh|||2W)1/2

, as well as

the corresponding error estimator η in the example in Subsection 4.3.2 for uniform (uni.) and

adaptive (ada.) mesh-refinement for c = 10 and c = 105.

4.3.2 Diffusion Reaction Problem

We consider the model problem of Definition 2.0.2 on a square domain (0, 0) × (1/2, 1/2)

with the coupling boundary Γ = ∂Ω. Figure 4.9 shows the initial mesh. Here, we set the

diffusion coefficient α = 1, the convection vector b = (0, 0)T and the reaction coefficient

c =1, 5, 10, 50, 102, 103, 104, 105

. The given exact solution in the interior domain Ω is

u(x1, x2) = e−x1√

c.

For the exterior solution uc we choose the same as in the example in Subsection 4.2.2 and

calculate the missing data f , u0 and t0 appropriately. Note that the interior solution u

exhibits an exponential boundary layer for c → ∞ on the x2-axis. Figure 4.22 shows curves

of the total energy error Em as well as the curves of the error estimator η with respect

to uniform and adaptive mesh-refinement over the number of elements. In the lower half

we see the curves for c = 10 and in the upper half for c = 105. For both mesh-refining

strategies, the error estimator η is observed to be reliable and efficient. In both cases the

expected convergence order of O(N−1/2) is reached for N big enough, since the solution

is smooth. More precisely, for c = 10, where the interior solution is almost linear, the

uniform and adaptive mesh-refinement strategy leads to the same absolute values in the

energy norm, whereas for c = 105 the adaptive strategy is significantly better with respect

to the energy norm. The error estimator η for c = 105 marks the point, where the system


Zoom: 6

ce

Figure 4.23. Adaptively generated mesh T (12) with 15583 elements in the example in Sub-

section 4.3.2 (c = 105) with a zoom area (factor 6).

leaves the singular perturbed regime, which is approximately by 4000 elements in the case

of adaptive mesh-refinement and more than 105 elements for uniform mesh-refinement.

Figure 4.23 shows the adaptively generated mesh T (12) for c = 105 with a zoom area of

factor 6. The mesh-refinement occurs more or less only near the x2-axis and the interior

solution in Figure 4.24 shows the boundary layer on the x2-axis and the jump relation of

the interior solution u and the exterior solution uc. In Figure 4.25 we plot the efficiency

index η/Eh, which converges to 24 and shows the robustness of the error estimator with

respect to the diffusion coefficient α and the reaction term c.


x1

x2−0.25

00.25

0.50.75

−1.25

−0.75

−0.25

0.25

0.75

−0.25

0

0.25

0.5

0.75

−1

−0.5

0

0.5

1

Figure 4.24. Solution for c = 105 and the adaptively refined mesh T (12) with 15583 elements

in the example in Subsection 4.3.2.

1

5

10

50

100

103

104

105

number of elements

effici

ency

index

101 102 103 104 105 10616

18

20

22

24

26

28

Figure 4.25. Efficiency index η/Em for adaptively generated meshes for different c =

1, 5, 10, 50, 100, 103, 104, 105 in the example in Subsection 4.3.2.


x1

x2

−0.3 −0.15 0 0.15 0.3

−0.3

−0.15

0

0.15

0.3

Figure 4.26. We generate the contour lines for the example in Subsection 4.2.3 from the

solution of the uniformly refined mesh T (7) with 196608 elements.

4.3.3 Problem with Convection

We consider the model problem of Definition 2.0.2 on the classical L-shaped domain of

the example in Subsection 4.2.3 with the coupling boundary Γ = ∂Ω. We choose a

constant diffusion coefficient α = 0.1. The other model data are identical to the problem

in Subsection 4.2.3. Namely, b = (15, 10)T and c = 10−2. The volume force f is in the

lower square, i.e.

f =

5 for − 0.2 ≤ x1 ≤ −0.1, −0.2 ≤ x2 ≤ −0.05,

0 else,

see also Figure 4.13(b). We prescribe the jumps u0 = 0 and t0 = 0 and the radiation

condition b∞ = 0. We use the weighted upwind scheme for the approximation of the

convection term with the weighting function (2.24), see equation (2.55) in Subsection 2.4.2.

In Figure 4.26 we plot the contour lines from the solution of the uniformly refined mesh

T (7) with 196608 elements. We see a significant transport from the square f = 5 in the

direction of the convection vector b. Since u0 = 0 the contour lines are continuous on the

boundary. We remark that the exterior problem has only a diffusion term, which can be

seen on the circular contour lines. Figure 4.27 shows the solution, which confirms that

the coupling method with the cell-centered finite volume method works also for problems

with convection. The piecewise constant interior solution is plotted on a uniformly refined

mesh with 768 elements. Similar to the example in Subsection 4.2.3 we see a step layer on

the boundary (0, 0) to (0,−0.25), which occurs because of the convection in this direction

and the different diffusion coefficient of the interior and exterior problem.


x1

x2

−0.5

−0.25

0

0.25

0.5

0.005

0.01

0.015

0.02

0.025

0.03

−0.5

−0.25

0

0.25

0.50

0.01

0.02

0.03

0.035

Figure 4.27. Interior and exterior solution for the uniformly generated mesh T (4) with 768

elements in the example in Subsection 4.3.3.


Conclusion

This thesis represents a research in construction, theoretical analysis, practical implemen-

tation and testing for the coupling of the finite volume method and the boundary element

method. A diffusion convection reaction problem was approximated either by the finite

volume element method or the cell-centered finite volume method in a bounded interior

domain, whereas the Laplace problem was solved by the boundary element method in the

corresponding exterior domain. Our approach is very attractive in fluid dynamics with a

dominated convection term in an interior domain, where one can use an upwind scheme,

and for a diffusion process in a possibly unbounded exterior domain. For an overview of

the main results and used techniques of this thesis we refer to the Introduction, see also

Figure IV in the Introduction. The thesis may be seen as the starting point of gener-

alizations in various directions for this kind of numerical coupling, which features local

conservation and stability with respect to the convection term.

We provided a complete a priori and a posteriori analysis for the coupling of the finite

volume element method and the boundary element method defined in Definition 2.3.2.

and with upwinding in Definition 2.3.7. It will be a future work to extend our analy-

sis to nonlinear diffusion coefficients and to consider other exterior problems, which can

be solved by the boundary element method. But the theoretical proof for robust a pri-

ori estimates with respect to the model data of this coupling method is still open. The

a posteriori analysis focused on the piecewise constant diffusion coefficients and that the

estimator was robust with respect to the model data. The analysis for a diffusion matrix

A of Assumption 2.0.1 can be done by an obvious modification. An open question is still

the (optimal) convergence of the adaptive scheme. We point out that this question is even

open for the pure adaptive finite volume element method.

The situation for the coupling of the cell-centered finite volume method and the boundary

element method is more complicated. We provided two discrete systems, namely in Def-

inition 2.4.1 and an extended version in Definition 2.4.4. For both discrete cell-centered

coupling systems there is neither an existence proof nor an a priori result available. The

reason might be the fact that the analysis of the cell-centered finite volume method is

based on the consistency of the numerical fluxes, whereas the boundary element method

uses the properties of a Galerkin scheme. The second approach was more convenient for

using local refined meshes and constructing the Morley interpolant for the a posteriori

127

128 Conclusion

analysis. Since we think in contrast to [61] that the Morley interpolant is not convenient

for convection problems using the upwind approximation, see Remark 3.2.1, one can adapt

the non-conforming interpolant of [76] for this coupling method.

All numerical examples in this thesis have academic character. Thus, it is an interest-

ing task to test both coupling methods on real problems and to implement the system

in three dimensions. Since the boundary element convergence rate, i.e. the rate of the

Cauchy data, is usually higher than the convergence rate of the finite volume solution,

another task is also the separate handling of the coupling elements and the finite volume

elements for the mesh-refinement algorithm. A related work in the context of the coupling

of finite elements method and boundary element method is [59], where they get an a pos-

teriori error estimator derived by using hierarchical basis techniques.

To conclude we emphasize that this thesis provides the first mathematical analysis and jus-

tification on the coupling of the finite volume method and the boundary element method.

The numerical experiments show that our coupling methods and the a posteriori error

estimators work in practice and our approach is also appropriate for convection and/or

reaction dominated problems.

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Index of Notation

Common Notation

N, N0 natural numbers, N ∪ 0R, R+ real numbers, positive real numbers

Rd, Rd×m Euclidean d-space, space of d×m-matrices with real entries

curl v curl operator, curl v = (∂v/∂x2,−∂v/∂x1)T in R2

∆v Laplace operator, ∆v = ∂2v/∂x21 + ∂2v/∂x2

2 in R

∇v gradient as column vector, ∇v = (∂v/∂x1, ∂v/∂x2)T in R2

div v divergence, div v = ∂v1/∂x1 + ∂v2/∂x2 in R

either d-dimensional Lebesgue measure, the Euclidean norm in Rd,

| · | absolute value or the cardinality of a set, if it is not clear in the

context, it is explained explicitly

∂/∂s arc length derivative

x · y Euclidean scalar product of x,y ∈ Rd

int(K), ∂K, K interior, boundary and closure of a domain K ⊂ R2

conv(X) convex hull of the elements of the set X

span(X) collection of all linear combinations of the elements of the set X

L(X;Y

)space of linear bounded operators from X to Y

Pm space of (algebraic) polynomials of total degree ≤ m

I identity matrix

o(1) Landau symbol, f(x) ∈ o(1) for |x| → ∞, then lim|x|→∞

|f(x)| = 0

O(g) Landau symbol, g is asymptotically an upper bound

. v . w is equivalent to v ≤ Cw with the constant C > 0

v ∼ w is equivalent to C1w ≤ v ≤ C2w with the constants∼

C1, C2 > 0

Notation for the Model Problem

Ω, ΩC interior and exterior domain in R2

Γ coupling boundary, Γ = ∂Ω ∩ ∂ΩC

Γin, Γout inflow and outflow coupling boundary

Continued on the next page

135

136 Index of Notation

Notation for the Model Problem

u, uc weak solution in Ω, ΩC

f right-hand side of the interior problem

A, b, c diffusion matrix, convection vector, reaction coefficient

u0, t0 jumps on Γ

a∞, b∞ real numbers for the radiation condition

ξ, φ Cauchy data, 24

Notation for Analytical Basics

Ck(Ω) space of k-times continuously differentiable functions on Ω, 1

C(Ω) C0(Ω)

Ck(Ω) Ck(Ω) :=v|Ω

∣∣ v ∈ Ck(R2), k ∈ N0 ∪ ∞, 1

C∞(Ω) space of infinitely differentiable functions, 1

Ck,λ(Ω) space of all Hölder continuous functions on Ω, 4

Ckc (Ω) space of all Ck(Ω) functions with compact support in Ω, 2

C∞c (Ω) space of all C∞(Ω) functions with compact support in Ω, 2

Lp(Ω) Lebesgue space of pth power integrable functions on Ω, 2

Lpℓoc(Ω) Lebesgue space of pth power locally integrable functions on Ω, 2

L∞(Ω) Lebesgue space of essentially bounded functions on Ω, 2

Lp(Ω)2 space of Lp functions on Ω with values in R2, 2

Hm(Ω) Sobolev space on Ω for m ≥ 0, 3

H−m(Ω) dual space of Hm(Ω), m ≥ 0, 4

Hmℓoc(Ω) Sobolev space of all local Hm(Ω) functions, 3

Hm(Γ) Sobolev space on the boundary Γ, 5

Hm∗ (Γ) Hm

∗ (Γ) :=ψ ∈ Hm(Γ)

∣∣ 〈ψ, 1〉L2(Γ) = 0

, 5

W 1,p(Ω) Sobolev space of Lp(Ω) functions, whose gradient are in Lp(Ω), 4

W 1,p(Ω)2 space of all W 1,p(Ω) functions on Ω with values in R2

‖v‖W 1,p(Ω) norm of W 1,p(Ω), 4

(·, ·)Ω , (·, ·)L2(Ω) scalar product in L2(Ω), 2

〈·, ·〉Ω , 〈·, ·〉L2(Ω) extended scalar product in L2(Ω), 4

(·, ·)Hm(Ω) scalar product in Hm(Ω), 3

‖ · ‖Lp(Ω), ‖ · ‖Lp(Γ) Lp(Ω) and Lp(Γ)-norm, 2

‖·‖Hm(Ω), ‖·‖Hm(Γ) Hm(Ω) and Hm(Γ)-norm, 3, 5

| · |s,Ω Sobolev Slobodeckij seminorm, 3

γ0, γint0 , γext

0 trace operators, skip index, if the side is clear, 5

γ1, γint1 , γext

1 conormal derivative, skip index if the side is clear, 6


Notation for Boundary Integral Equations

N , V, K Newton (1.1), single layer (1.2) and double layer potential (1.3), 7

single layer, double layer, adjoint double layer, hypersingularV, K, K∗, W

integral operator, 8

S int, Sext, S Pioncaré Steklov operator, 13

Notation for the Triangulation

T , N , E triangulation of the domain Ω, set of all nodes, edges of T , 14

T , E element (triangle), edge of T , 14

∂T boundary of T

a, x node of T , a ∈ N , point in Ω

N number of elements, N := #TNT , ET set of all nodes, elements of T ∈ T , 14

NΓ, NI set of all coupling, interior nodes, 14

NM set of all midpoints of an edge E , 14

Ni index set, set of all neighbors of a node ai, 14

EΓ, E inΓ , Eout

Γ set of all edges on Γ, Γin, Γout, 14

EV all parts of edges in V ∈ T ∗, 16

τij , τTij interface between two control volumes of T ∗ (on T ), 16

DT set of all τTij in T , 16

ZV set of triangles in V ∈ T ∗, 16

ωa, (ωa) patch (elementwise) for the node a ∈ N , 18

ωE , (ωE) patch (elementwise) of the edge E ∈ E , 18

ωT , (ωT ) patch (elementwise) of the element T ∈ T , 18

ωE , ω

T patch ωE and ωT with the nodes, 18

hT , hV Euclidean diameter of T , V , 14, 16

h h := maxT ∈T hT , 14

hE length of the edge E, 14

hT , hE global mesh size function, 14

normal vector (depending on x) points outward with respect to then (nx)

corresponding domain, 17

ni normal vector points outward with respect to the element Vi ∈ T ∗

t tangential vector of E ∈ E orthogonal to nE (math. positive), 17

Notation for the Discrete Analysis

Pp(T ), Pp(EΓ) space of T - and EΓ-piecewise polynomials with degree p ∈ N0, 19

P0(T ∗) space of T ∗-piecewise constant functions, 19

Continued on the next page


Notation for the Discrete Analysis

S1(T ) space of T -piecewise affine and global continuous function, 18

Hm(EΓ) ‘broken Sobolev spaces’, piecewise Hm(E), 19

vK integral mean of v ∈ L2(K) over K ⊂ R2, 21

ηi standard nodal linear basis function, 19

χi characteristic function of the volume Vi ∈ T ∗, 19

I∗h T ∗-piecewise interpolation operator on constant functions, 19

[[·]]E jump over E, 21

bT , bE , bE,κ classical and squeezed bubble functions on T and E, 22, 73

Fext extension operator, 22

Notation for the Coupling Problem

CA,1, CA,2 positive constant with CA,1|v|2 ≤ vT A(x)v ≤ CA,2|v|2, 23

Cbc,1 coercivity constant with (div b)/2 + c ≥ Cbc,1 ≥ 0, 23

A(·, ·),CA,2,CA,2′ bilinear form (interior problem) and continuity constants, 25

||| · |||Ω natural energy (semi-) norm, 25

B(·, ·),Ccont,Ccoer problem bilinear form, continuity and coercivity constant, 27

AV (·, ·), AupV (·, ·) finite volume element bilinear form (upwind) 30, (34)

F (·) right-hand side to AV (·, ·), AupV (·, ·), 30

uh finite volume element or cell-centered finite volume solution, 30, 44

ξh, φh discrete Cauchy data, 30, 45

uTh,ij discrete weighted upwind function, 33

uT uT = uh|T for the cell-centered finite volume method, 44

ua, uh,Γ,

ςa = ςa,h + ςa,t0

approximation on Γ for the cell-centered finite volume solution, 45

ΥT (a) interpolation weight of T with respect to the node a, 47

FDT,E(uh) discrete diffusion over E of T , 44

FCT,E(uh) discrete convection flux over E of T , 44

FRT (uh) approximated reaction term of T , 44

e, δ, ǫ interior error, (exterior) trace error, (exterior) conormal error, 35

p0, p1 orthogonalities of the error, 35

constant diffusion coefficient for the cell-centered finiteα

volume method, 44

βTij , βT,E integral mean of b · n over τT

ij and E, 33, 51

Pe|T Péclet number, 31

Φ(·) weighting function, 33


Notation for the A Posteriori Error Estimate (see also Figure A)

piecewise constant diffusion coefficient for the finite volumeα

element method, 55

constant diffusion coefficient for the cell-centered finiteα

volume method

Ih interpolation operator Ih : H1(Ω) → S1(T ), 58

eh, e∗h discrete errors eh = Ihe, e

∗h = I∗

heh, 59

R, J residual and edge-residual jump, 59

ηT refinement indicator, 66

ηT,up measures the upwind error, 68

Imuh Morley interpolant, 84

RIm , JIm residual and edge-residual jump for the Morley interpolant, 88

Notation for the Numerical Experiments

η error estimator, η :=(∑

T ∈T η2T (+η2

T,up))1/2

, 100

||| · |||V , ||| · |||W equivalent energy norm to ‖ · ‖H−1/2 , ‖ · ‖H1/2 , 101

total energy norm for the coupling with the finite volume elementEh method, 102

total energy norm for the coupling with the cell-centered finiteEm volume method, 102

x = (x1, x2) point in the plane R × R


αT := α|T

βT := minx∈T

12 div b(x) + c(x)

αE := maxαT1, αT2

αE := αT

βE := minβT1, βT2

βE := βT

µE := minβ

−1/2E , hEα

−1/2E

µE := hEα−1/2E if βE = 0

µT := minβ

−1/2T , hTα

−1/2T

µT := hTα−1/2T if βT = 0

κE := min

1,α

1/2

E

β1/2

EhE

κE := 1 if βE = 0

E ∈ EI , E ∈ ET1∩ ET2

E ∈ EΓ, E ∈ ET

E ∈ EI , E ∈ ET1∩ ET2

E ∈ EΓ, E ∈ ET

Figure A. Overview of the quantities to get a robust a posteriori error estimator with respect

to the model data α, b and c. For the coupling with the finite volume element method α is

T -piecewise constant, whereas for the coupling with the cell-centered finite volume method we

have α ∈ R+.

List of Figures

Introduction

I Model domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

II Motivation problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

III Different meshes for finite volume methods. . . . . . . . . . . . . . . . . . . vi

IV Road map and main results of this thesis. . . . . . . . . . . . . . . . . . . . xii

Chapter 1

1.1 Notation for the trace of K and conormal derivative of V for pointvalues. . . 9

1.2 Construction of the dual mesh T ∗. . . . . . . . . . . . . . . . . . . . . . . . 16

1.3 Some important sets and lines on the dual mesh T ∗. . . . . . . . . . . . . . 17

1.4 The five patches introduced in Subsection 1.3.3. . . . . . . . . . . . . . . . . 18

1.5 The operator I∗h. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.6 Bubble functions on a triangle T and an edge patch ωE . . . . . . . . . . . . 22

Chapter 2

2.1 Notation for upwind scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.2 Illustration of equation (2.36) for the proof to Lemma 2.3.9. . . . . . . . . . 36

2.3 Notation to the proof of Lemma 2.3.14. . . . . . . . . . . . . . . . . . . . . 42

2.4 The different cases for calculating ua with a ∈ NN ∪ NΓ. . . . . . . . . . . . 47

2.5 Illustration of the admissible condition. . . . . . . . . . . . . . . . . . . . . 49

2.6 Diamond-path. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Chapter 3

3.1 The main steps to the proof of Theorem 3.1.19. . . . . . . . . . . . . . . . . 54

3.2 The partition of Ω into subdomains Ωℓ. . . . . . . . . . . . . . . . . . . . . 55

3.3 Quasi-monotone and not quasi-monotone. . . . . . . . . . . . . . . . . . . . 56

3.4 Main steps to the proof of Theorem 3.1.38. . . . . . . . . . . . . . . . . . . 72

3.5 Squeezed bubble function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.6 Main steps to the proof of Theorem 3.2.15. . . . . . . . . . . . . . . . . . . 84

3.7 Notation for the construction of the Morley interpolant. . . . . . . . . . . . 87

Chapter 4

4.1 The discretization by the finite volume element method. . . . . . . . . . . . 97

141

142 List of Figures

4.2 Initial mesh and distribution of α in the example in Subsection 4.2.1. . . . . 104

4.3 Comparison of energy error and estimator for uniform and adaptive mesh

refinement in the example in Subsection 4.2.1. . . . . . . . . . . . . . . . . . 105

4.4 Adaptively generated meshes and the interior solution in the example in

Subsection 4.2.1 for α2 = 1000. . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.5 Zoom area of an adaptively generated mesh for the example in Subsec-

tion 4.2.1 for α2 = 1000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.6 Comparison of the energy errors in the example in Subsection 4.2.1. . . . . 108

4.7 Adaptively generated mesh and solution in the example in Subsection 4.2.1

for α2 = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

4.8 Efficiency index in the example in Subsection 4.2.1. . . . . . . . . . . . . . . 109

4.9 The initial mesh in the example in Subsection 4.2.2. . . . . . . . . . . . . . 110



4.11 Adaptively generated mesh in the example in Subsection 4.2.2 for α = 0.05. 111


4.13 Initial mesh and distribution of the data in the example in Subsection 4.2.3. 113

4.14 Comparison of the solution without and with upwind in the example in

Subsection 4.2.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

4.15 Solution for the example in Subsection 4.2.3. . . . . . . . . . . . . . . . . . 114

4.16 Error estimator and contour lines in the example in Subsection 4.2.3. . . . . 115

4.17 Adaptively generated meshes for the example in Subsection 4.2.3. . . . . . . 116

4.18 Zoom area of an ad adaptively generated mesh for the example in Subsec-

tion 4.2.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117



4.20 Adaptively generated mesh and interior solution in the example in Subsec-

tion 4.3.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119




4.23 Adaptively generated in the example in Subsection 4.3.2. . . . . . . . . . . . 122



4.26 Contour lines for the example in Subsection 4.3.3. . . . . . . . . . . . . . . 124


Index of Notation

A Quantities for the a posteriori error estimator. . . . . . . . . . . . . . . . . . 140

Curriculum Vitæ

Personal Data

Name Christoph Erath

Date of Birth 2. October 1979

Place of Birth Feldkirch, Austria

Nationality Austrian

Marital Status unmarried

E-Mail [email protected]

Current Position

Since Oct 2005 University Assistant with teaching practice, Institute of

Numerical Mathematics, University of Ulm, Germany

Education and Studies

Since Oct 2005 Doctoral Studies, Institute of Numerical Mathematics,

University of Ulm, Germany

Oct 1999 – Oct 2005 Studies of Mathematics in Computer Science, Vienna

University of Technology, Austria, Final exam with

distinction

Jul 2003 – Jun 2004 Studies abroad, Norwegian University of Science and

Technology (NTNU), Trondheim, Norway

Sep 1994 – Jun 1999 Technical College for Electronics/Telecommunication

engineering in Rankweil, Austria, school leaving exam

with distinction

Sep 1986 – Jul 1994 Primary and Secondary School

143

144 Curriculum Vitæ

Grants and Awards

Mar 2007 – Feb 2010 Scholarship for postgraduate students from the federal

state Baden-Württemberg, Germany

2006 Award 2006 from the Austrian Mathematical Society

for the Diploma Thesis, Austria

2000 – 2003 Scholarship awarded by Hilti AG’s “Fund for the

promotion of young academics in the fields of science

and economics”, Liechtenstein

1999 Award by Hilti AG, Thüringen, Austria

Other Scientific Activities

Since 2008 Referee for the journals “Numerische Mathematik” and

“Applied Numerical Mathematics”

Ulm, 28. April 2010

Publications

• C. Erath, S. A. Funken, P. Goldenits, and D. Praetorius. Simple error estimators

for the Galerkin BEM for some hypersingular integral equation in 2D. Submitted to

SIAM J. Numer. Anal., 2009.

Available online at http://numerik.uni-ulm.de/preprints/2009/hypsing2d.pdf.

• C. Erath, S. Ferraz-Leite, S. A. Funken, and D. Praetorius. Energy norm based a

posteriori error estimation for boundary element methods in two dimensions. Appl.

Numer. Math., 59(11):2713–2734, 2009.

• C. Erath and D. Praetorius. A posteriori error estimate and adaptive mesh refine-

ment for the cell-centered finite volume method for elliptic boundary value problems.

SIAM J. Numer. Anal., 47(1):109–135, 2008.

• C. Erath, S. A. Funken, and D. Praetorius. Adaptive cell-centered finite volume

method, Finite Volumes for Complex Applications V, 359-366, Wiley, 2008

• C. Erath. Adaptive Finite Volumen Methode. Diploma thesis in German, Vienna

University of Technology, Vienna, Austria, 2005.

Available online at http://www.ub.tuwien.ac.at/dipl/2005/AC04915569.pdf.

Erklärung

Ich, Christoph Erath, versichere hiermit, dass ich die Arbeit selbständig angefertigt habe

und keine anderen als die angegebenen Quellen und Hilfsmittel benutzt sowie die wörtlich

oder inhaltlich übernommenen Stellen als solche kenntlich gemacht habe. Ich erkläre

außerdem, dass diese Dissertation bisher weder im In- noch im Ausland in dieser oder

ähnlicher Form in einem anderen Promotionsverfahren vorgelegt wurde.

Ulm, den 28. April 2010

Documents

Coupling of the Finite Volume Method and the Boundary