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Digitale Dissertation

Martin Petzoldt :
Regularity and error estimators for elliptic problems with discontinuous coefficients
Regularität und Fehlerschätzer für elliptische Probleme mit unstetigen Diffusionskoeffizienten

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Abstract

We regard linear elliptic equations with a discontinuous diffusion coefficient k in two and three space dimensions. The coefficient k is constant on polygonal (polyhedral) subdomains. These problems are also known as Laplace interface problems. It is known that solutions of these problems have lower regularity due to singularities.

In the second chapter we derive Sobolev H (s)-regularity, where s belongs to (1,2) which hold independently of the shape of the subdomains. We use a known criterion on the structure of coefficients - the quasi-monotonicity condition - to give regularity results in Sobolev spaces H (1+1/4) independent of the jump size of the coefficients. We argue that the quasi-monotonicity is also a necessary condition for higher regularity independent of the jump size of k. Further we give sharp regularity results which depend on the jump size. We show that a checkerboard like distribution of values for the coefficient k leads to the worst possible regularity. For the regularity results in 3D we use the derived 2D results.

In the third chapter of this thesis we discretize the problem with linear finite elements. We propose treatment of the arising singularities by a posteriori mesh refinement on the basis of new a posteriori error estimators. If the quasi-monotonicity condition is fulfilled, we show that the a posteriori error estimators bound the discretization error from above with constants which do not depend on the jump size of the coefficient. For a lower bound of the error the quasi-monotonicity condition is not needed.

In various numerical examples (chapter 4) we confirm the applicability of the derived error estimators to problems with singularities. The examples comprise model problems, problems with real data from groundwater flow and 3D examples. The examples show that mesh refinement lead to error reduction rates in terms of unknowns N to the power of (-1/space dimension), which are expected to be optimal. The ratio of the error estimator and the true error takes on problem independent moderate values.

Table of Contents

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0. Titel page and contents
1. Introduction 3
2. Regularity results for interface problems 7
2.1 Outline 7
2.2 The interface problem for the Laplacian 8
2.3 Notation 9
2.4 Regularity in 2D 11
2.5 The quasi-monotone case 17
2.6 The general case 28
2.7 Regularity in 3D 36
3. Adaptive Finite Element Method 39
3.1 Introduction 39
3.2 Problem setting 40
3.3 Finite Element Method on uniform grids 42
3.4  Finite Element Method on adapted grids 43
3.5 Interpolation operators 44
3.6 Residual based error estimators 53
3.7 Other estimators 59
3.8 Extension to more general problems 61
3.9 Application to transient problems 73
4. Numerical Experiments 77
4.1 Error estimators and adaptive refinement 77
4.2 Implementation issues 78
4.3 Error reduction rates 79
4.4 Robustness 81
4.5 Examples with deteriorating regularity 82
4.6 Examples with real data 88
4.7 Numerical examples in 3D 95
4.8 Example for a parabolic problem 100
4.9 Conclusions for the numerical experiments 101
5. Conclusions, Acknowledgement and Bibliography 103
A Zusammenfassung 111

More Information:

Online available: http://www.diss.fu-berlin.de/2001/111/indexe.html
Language of PhDThesis: english
Keywords: discontinuous coefficients, interface problems, transmission problems, singularities, regularity, a posteriori error estimators
DNB-Sachgruppe: 27 Mathematik
Classification MSC: 35B65 35J25 65N30 65N15
Date of disputation: 30-May-2001
PhDThesis from: Fachbereich Mathematik u. Informatik, Freie Universität Berlin
First Referee: Prof. Dr. Eberhard Bänsch
Second Referee: Prof. Dr. Rüdiger Verfürth
Third Referee: Prof. Dr. Anna-Margarete Sändig
Contact (Author): martin.petzoldt@gmx.de
Contact (Advisor): baensch@wias-berlin.de
Date created:26-Jun-2001
Date available:19-Oct-2001

 

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