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Coupling of the Finite Volume Method and the Boundary Element Method [Elektronische Ressource] : Theory, Analysis, and Numerics / vorgelegt von Christoph Erath

161 pages
89069 Ulm | GermanyFakultät für Mathematik und WirtschaftswissenschaftenInstitut für Numerische MathematikDissertationCoupling of the Finite Volume Methodand the Boundary Element MethodTheory, Analysis, and Numericszur Erlangung des Doktorgrades Dr. rer. nat.der Fakultät für Mathematik und Wirtschaftswissenschaftender Universität Ulmvorgelegt vonChristoph Erathaus Schlins, ÖsterreichApril 2010Amtierender 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 2010AbstractWe develop a discretization scheme for the coupling of the finite volume method and theboundary element method in two dimensions, which describes, for example, the transportofaconcentrationinafluid. Thediscretesystemmaintainsnaturallylocalconservation. Ina bounded interior domain we approximate a diffusion convection reaction problem eitherbythefinitevolumeelementmethodorbythecell-centeredfinitevolumemethod,whereasin the corresponding exterior domain the Laplace problem is solved by the boundaryelement method. On the coupling boundary we have appropriate transmission conditions.A weighted upwind scheme guarantees the stability of the method also for convectiondominated problems.
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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 2010Amtierender 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 2010Abstract
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
ofaconcentrationinafluid. Thediscretesystemmaintainsnaturallylocalconservation. In
a bounded interior domain we approximate a diffusion convection reaction problem either
bythefinitevolumeelementmethodorbythecell-centeredfinitevolumemethod,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
thattheydonotdependonthevariationofthemodeldata. Thelocalcontributionsofthe
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-
odemitderRandelementeMethodefürdenzweidimensionalenRaum. EinModellproblem
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
ATypeset: LT X2"EContents
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
iii 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 143Introduction
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-
mericalfluxesandthenaturalformulationofanupwindscheme,whichensuresstabilityfor
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.
Thereexistseveraldifferentfinitevolumeschemes. Inthisthesiswedevelopdiscretization
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.
iiiiv Introduction
ΩC
inΓ
b
Ω
b
outΓ n
in outFigure I. Domains and notation for the model problem with the boundary Γ=Γ [Γ .
Model Problem
2Let Ω R 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( Aru+bu)+cu=f in Ω;
where A is a symmetric diffusion matrix, b is a possibly dominating velocity field, c is a
2reaction function and f is a source term. In the complement Ω =R nΩ, the so calledC
exterior domain, we seek u such thatc
Δu =0 in Ωc C
together with the radiation condition
u (x)=a +b logjxj+o(1) forjxj!1:c 1 1
We can fix either a 2R or b 2R and calculate the other one, that means u behaves1 1 c
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 = @Ω , which is closed and has positive surfaceC
measure. The coupling boundary Γ is divided in an inflow and outflow part, namely
in out

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