Adaptive multilevel methods for mortar edge element methods in IR_1hn3 [Elektronische Ressource] / vorgelegt von Werner Ernst Schabert

Sujets

• Mathematics
• Physik

Informations

• Publié par: universitat_augsburg
• Publié le: 01 janvier 2006
• Nombre de lectures: 38
• Langue: English

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Adaptive Multilevel Methods for
Mortar Edge Element Methods
3in IR
Dissertation
zur Erlangung des akademischen Titels eines
Doktors der Naturwissenschaften
der Mathematisch Naturwissenschaftlichen Fakult¨at
der Universit¨at Augsburg
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vorgelegt von
Dipl. Phys. Werner Ernst Schabert
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E1. Gutachter: Prof. Dr. Ronald H.W. Hoppe
2. Gutachter: Prof. Dr. Kunibert G. Siebert
3. Gutachter: Prof. Dr. Jin-Fa Lee
Tag der mundlic¨ hen Prufung:¨ 9. Februar 2006Abstract
This thesis is concerned with the application of adaptive mortar edge element
methods to the numerical solution of the quasi-stationary limit of Maxwells
equations, also known as the eddy current model, in three space dimensions.
Althougheddycurrentmodelistime-dependent, werestrictouranalysistotime-
independent problems that arise from a time discretization of the partial differ-
ential equations.
For the solution of these equations we consider the mortar approach, which re-
lies on the macro-hybrid variational formulation of the problem with respect to
a geometrically conforming, nonoverlapping decomposition of the computational
domain (cf. [Hop99]). Based on independent, locally quasi-uniform and shape
regular simplicial triangulation of the subdomains we use the lowest order curl-
conforming edge elements of N´ed´elec’s first family for the discretization of the
problem. Due to nonmatching triangulations at the interfaces of adjacent sub-
domains, we have to impose weak continuity constraints on the tangential traces
across the skeleton of the decomposition by means of appropriately chosen La-
grange multipliers.
The mortar edge element discretized problems give rise to indefinite algebraic
saddle point problems. Since the saddle point problem behaves utterly different
on the large kernel of the curl-operator, standard iterative solvers that do not
take care of the the kernel fail in this case. We analyze this problem in great
detailanddevelopamultileveliterativesolverfeaturingahybridsmootherthatis
based on the smoother presented in [Hip98]. The key ingredient of the smoother
is an additional defect correction on the subspace of irrotational vector fields.
However, in order to guarantee convergence of the iterative scheme, we have to
impose compatibility constraints on the triangulations at the interfaces.
To improve the accuracy of the computed solution while keeping the computa-
tional cost as small as possible, we put particular emphasis on mesh adaptivity.
We present an a posteriori error estimator that combines elements of the error
estimatorsgivenin[BHHW00,Woh99c]andreliesonaHelmholtzdecomposition
of the error into an irrotational and weakly solenoidal part. We show that the
error estimator is both efficient and reliable, provided certain assumptions are
fulfilled.
Finally, we demonstrate the convergence properties of the multigrid scheme and
the quality of the error estimator by solving several academic test problems that
cover a wide range of physical applications.Contents
Introduction 11
1 Physical Models of Electromagnetism 15
1.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.1.1 Constitutive Equations . . . . . . . . . . . . . . . . . . . . 17
1.1.2 Interface Conditions . . . . . . . . . . . . . . . . . . . . . 18
1.1.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 19
1.2 Simplified Electromagnetic Models . . . . . . . . . . . . . . . . . 20
1.2.1 Stationary Models: Electrostatics and Magnetostatics . . . 21
1.2.2 The Quasi-Static Model - The Eddy Current Case . . . . . 21
1.2.3 The Time-Harmonic Model . . . . . . . . . . . . . . . . . 22
2 Sobolev and Vector Function Spaces 25
2.1 Standard Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 The Space H(div;Ω) . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3 The Space H(curl;Ω) . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4 Scalar and Vector Potentials . . . . . . . . . . . . . . . . . . . . . 38
2.5 Time-Dependent Function Spaces . . . . . . . . . . . . . . . . . . 39
3 Variational Theory for the Eddy Current Model 41
3.1 Derivation of the Model Problem . . . . . . . . . . . . . . . . . . 41
3.2 Macro-Hybrid Variational Formulation . . . . . . . . . . . . . . . 44
4 The Mortar Edge Element Approximation 49
4.1 Geometrical Setting . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 N´ed´elec’s Lowest Order Curl Conforming Elements . . . . . . . . 51
4.3 The Lagrange Multiplier Space . . . . . . . . . . . . . . . . . . . 53
4.4 Discrete Saddle Point Problem . . . . . . . . . . . . . . . . . . . . 59
5CONTENTS
5 A Priori Error Estimates 67
5.1 Formulation of the Problem Using the Constrained Space . . . . . 67
5.2 Consistency Error . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.3 Approximation Error . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.4 Error of the Lagrange Multiplier . . . . . . . . . . . . . . . . . . . 73
6 Residual-type A Posteriori Error Estimator 75
6.1 Reliability of the Error Estimator . . . . . . . . . . . . . . . . . . 78
6.2 Local Efficiency of the Error Estimator . . . . . . . . . . . . . . . 82
7 Multilevel Based Iterative Solution 91
7.1 Iterative Solver for the Conforming Setting . . . . . . . . . . . . . 92
7.2 Iterative Solver for the Constrained Formulation . . . . . . . . . . 96
7.3 Iterative Solver for the Unco Formulation . . . . . . . . 101
7.3.1 Hybrid Smoother as Preconditioner P . . . . . . . . . . . 103h
7.3.2 Split Preconditioners . . . . . . . . . . . . . . . . . . . . . 104
8 Numerical Results 109
8.1 Multigrid Convergence . . . . . . . . . . . . . . . . . . . . . . . . 109
8.2 Performance of the Error Estimator . . . . . . . . . . . . . . . . . 117
8.3 Optimal Convergence of the Discretization . . . . . . . . . . . . . 128
9 Conclusions 131
Notation 133
6List of Tables
1.1 Electromagnetic field quantities. . . . . . . . . . . . . . . . . . . . 16
8.1 Multigrid convergence rates for Experiment 1. . . . . . . . . . . . 111
8.2id convergence rates for Experiment 2. . . . . . . . . . . . 112
8.3 Multigrid convergence rates for Experiment 3. . . . . . . . . . . . 113
8.4id convergence rates for Experiment 4. . . . . . . . . . . . 113
8.5 Multigrid convergence rates for Experiment 5. . . . . . . . . . . . 114
8.6id convergence rates for Experiments 6 and 7. . . . . . . . 115
8.7 Multigrid convergence rates for Experiments 8 and 9. . . . . . . . 116
8.8 Quality measures for Experiment 10. . . . . . . . . . . . . . . . . 121
8.9 Quality for Experiment 11. . . . . . . . . . . . . . . . . 122
8.10 Quality measures for Experiment 12. . . . . . . . . . . . . . . . . 123
8.11 Quality for Experiment 13. . . . . . . . . . . . . . . . . 124
8.12 Quality measures for Experiment 14. . . . . . . . . . . . . . . . . 125
8.13 Quality for Experiment 15. . . . . . . . . . . . . . . . . 125
8.14 Quality measures for Experiment 16. . . . . . . . . . . . . . . . . 126
78List of Figures
1.1 Situation at the interface between different media. . . . . . . . . . 19
3.1 Geometric decomposition of the domain. . . . . . . . . . . . . . . 45
3.2 Choosing mortar and nonmortar sides at the interfaces. . . . . . . 46
4.1 Triangulations of the subdomains. . . . . . . . . . . . . . . . . . . 50
4.2 Situation at the boundary of the interface δ . . . . . . . . . . . . 56k
ˆ4.3 Affine transformation from the reference triangle K to K.. . . . . 60
7.1 2D computational domain of the mortar example. . . . . . . . . . 98
˜7.2 Scaling of the four smallest eigenvalues of A. . . . . . . . . . . . . 100
8.1 Computational domain for Experiments 1 to 5.. . . . . . . . . . . 110
8.2tional for Experiments 6 and 7. . . . . . . . . . . 116
8.3 Computational domain for Experiments 8 and 9. . . . . . . . . . . 117
8.4tional for Experiment 16. . . . . . . . . . . . . . 127
8.5 True error of the edge element discretization for Experiment 16. . 127
8.6 Error of the edge elementn for Experiment 12. . . . . 128
8.7 Error of the edge element discretization for Experiment 13. . . . . 129
8.8 Error of the edge elementn for Experiments 14 and 15.129
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