Finite Element Methods for Eigenvalue Problems

368 Pages·2017·3.351 MB·English

by  Sunl, Jiguang, Zhou, Aihui

Finite Element
Methods for
Eigenvalue
Problems
MONOGRAPHS AND RESEARCH NOTES IN MATHEMATICS
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MONOGRAPHS AND RESEARCH NOTES IN MATHEMATICS
Finite Element
Methods for
Eigenvalue
Problems
Jiguang Sun
Michigan Technological University
Houghton, USA
Aihui Zhou
Chinese Academy of Sciences
Beijing, China
CRC Press
CRC
Taylor & Francis Group
Boca Raton London New York
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Contents
Preface xi
ListofFigures xv
ListofTables xix
1 FunctionalAnalysis 1
1.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 MetricSpaces,BanachSpacesandHilbertSpaces . . . . . 1
1.1.2 LinearOperators . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.3 SpectralTheoryofLinearOperators . . . . . . . . . . . . . 9
1.2 SobolevSpaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.1 BasicConcepts . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.2 NegativeNorm . . . . . . . . . . . . . . . . . . . . . . . . 16
1.2.3 TraceSpaces . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.3 VariationalFormulation . . . . . . . . . . . . . . . . . . . . . . . 18
1.4 AbstractSpectralApproximationTheories . . . . . . . . . . . . . 20
1.4.1 TheoryofDescloux,Nassif,andRappaz. . . . . . . . . . . 21
1.4.2 TheoryofBabuškaandOsborn . . . . . . . . . . . . . . . 24
1.4.3 VariationallyFormulatedEigenvalueProblems . . . . . . . 30
2 FiniteElements 35
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.1.1 Meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.1.2 LagrangeElements . . . . . . . . . . . . . . . . . . . . . . 37
2.2 QuadratureRules . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.2.1 GaussianQuadratures . . . . . . . . . . . . . . . . . . . . 39
2.2.2 QuadraturesforaTriangle . . . . . . . . . . . . . . . . . . 40
2.2.3 QuadratureRulesforTetrahedra . . . . . . . . . . . . . . . 41
2.3 AbstractConvergenceTheory . . . . . . . . . . . . . . . . . . . . 41
2.3.1 Céa’sLemma . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.3.2 DiscreteMixedProblems. . . . . . . . . . . . . . . . . . . 44
2.3.3 InverseEstimates . . . . . . . . . . . . . . . . . . . . . . . 48
2.4 ApproximationProperties . . . . . . . . . . . . . . . . . . . . . . 50
vii
viii Contents
2.5 Appendix:ImplementationofFiniteElementsin1D . . . . . . . . 53
3 TheLaplaceEigenvalueProblem 59
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2 LagrangeElementsfortheSourceProblem . . . . . . . . . . . . . 61
3.3 ConvergenceAnalysis . . . . . . . . . . . . . . . . . . . . . . . . 65
3.4 NumericalExamples . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.5 Appendix:ImplementationoftheLinearLagrangeElement . . . . 71
3.5.1 TriangularMeshes . . . . . . . . . . . . . . . . . . . . . . 72
3.5.2 MatricesAssembly . . . . . . . . . . . . . . . . . . . . . . 75
3.5.3 BoundaryConditions . . . . . . . . . . . . . . . . . . . . . 79
3.5.4 SampleCodes . . . . . . . . . . . . . . . . . . . . . . . . 79
4 TheBiharmonicEigenvalueProblem 83
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.2 TheArgyrisElement . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.2.1 TheDiscreteProblem . . . . . . . . . . . . . . . . . . . . 89
4.2.2 NumericalExamples . . . . . . . . . . . . . . . . . . . . . 91
4.3 AMixedFiniteElementMethod . . . . . . . . . . . . . . . . . . . 92
4.3.1 AbstractFramework . . . . . . . . . . . . . . . . . . . . . 92
4.3.2 TheCiarlet–RaviartMethod . . . . . . . . . . . . . . . . . 95
4.3.3 NumericalExamples . . . . . . . . . . . . . . . . . . . . . 101
4.4 TheMorleyElement . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.4.1 AbstractTheory . . . . . . . . . . . . . . . . . . . . . . . 102
4.4.2 TheMorleyElement . . . . . . . . . . . . . . . . . . . . . 104
4.4.3 NumericalExamples . . . . . . . . . . . . . . . . . . . . . 108
4.5 ADiscontinuousGalerkinMethod . . . . . . . . . . . . . . . . . . 109
4.5.1 BiharmonicEigenvalueProblems . . . . . . . . . . . . . . 110
4.5.2 C0InteriorPenaltyGalerkinMethod . . . . . . . . . . . . 111
4.5.3 NumericalExamples . . . . . . . . . . . . . . . . . . . . . 115
4.5.4 ComparisonsofDifferentMethods . . . . . . . . . . . . . 122
4.6 C0IPGforaFourthOrderProblem . . . . . . . . . . . . . . . . . 129
4.6.1 TheSourceProblem . . . . . . . . . . . . . . . . . . . . . 131
4.6.2 TheEigenvalueProblem . . . . . . . . . . . . . . . . . . . 134
4.6.3 NumericalExamples . . . . . . . . . . . . . . . . . . . . . 141
4.7 Appendix:MATLABCodefortheMixedMethod . . . . . . . . . 146
5 TheMaxwell’sEigenvalueProblem 149
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.2 TheMaxwell’sEigenvalueProblem . . . . . . . . . . . . . . . . . 152
5.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 152
5.2.2 TheCurl-curlProblem . . . . . . . . . . . . . . . . . . . . 153
Contents ix
5.2.3 Divergence-conformingElements . . . . . . . . . . . . . . 155
5.2.4 Curl-conformingEdgeElements . . . . . . . . . . . . . . . 157
5.2.5 ConvergenceAnalysis . . . . . . . . . . . . . . . . . . . . 161
5.2.6 TheEigenvalueProblem . . . . . . . . . . . . . . . . . . . 163
5.2.7 AnEquivalentEigenvalueProblem . . . . . . . . . . . . . 166
5.2.8 NumericalExamples . . . . . . . . . . . . . . . . . . . . . 167
5.3 TheQuad-curlEigenvalueProblem . . . . . . . . . . . . . . . . . 169
5.3.1 TheQuad-curlProblem. . . . . . . . . . . . . . . . . . . . 170
5.3.2 TheQuad-curlEigenvalueProblem . . . . . . . . . . . . . 179
5.3.3 NumericalExamples . . . . . . . . . . . . . . . . . . . . . 182
6 TheTransmissionEigenvalueProblem 185
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
6.2 ExistenceofTransmissionEigenvalues . . . . . . . . . . . . . . . 188
6.2.1 SphericallyStratifiedMedia . . . . . . . . . . . . . . . . . 188
6.2.2 GeneralMedia . . . . . . . . . . . . . . . . . . . . . . . . 191
6.2.3 Non-existenceofImaginaryTransmissionEigenvalues . . . 192
6.2.4 ComplexTransmissionEigenvalues . . . . . . . . . . . . . 193
6.3 ArgyrisElementforRealTransmissionEigenvalues . . . . . . . . 195
6.3.1 AFourthOrderReformulation . . . . . . . . . . . . . . . . 195
6.3.2 BisectionMethod . . . . . . . . . . . . . . . . . . . . . . . 199
6.3.3 SecantMethod . . . . . . . . . . . . . . . . . . . . . . . . 206
6.3.4 SomeDiscussions . . . . . . . . . . . . . . . . . . . . . . 209
6.4 AMixedMethodUsingTheArgyrisElement . . . . . . . . . . . . 210
6.4.1 TheMixedFormulation . . . . . . . . . . . . . . . . . . . 210
6.4.2 ConvergenceAnalysis . . . . . . . . . . . . . . . . . . . . 212
6.4.3 NumericalExamples . . . . . . . . . . . . . . . . . . . . . 215
6.5 AMixedMethodusingLagrangeElements . . . . . . . . . . . . . 217
6.5.1 AnotherMixedFormulation . . . . . . . . . . . . . . . . . 218
6.5.2 TheDiscreteProblem . . . . . . . . . . . . . . . . . . . . 220
6.5.3 NumericalExamples . . . . . . . . . . . . . . . . . . . . . 222
6.6 TheMaxwell’sTransmissionEigenvalues . . . . . . . . . . . . . . 225
6.6.1 TransmissionEigenvaluesofBalls . . . . . . . . . . . . . . 229
6.6.2 ACurl-conformingEdgeElementMethod. . . . . . . . . . 232
6.6.3 AMixedFiniteElementMethod . . . . . . . . . . . . . . . 235
6.6.4 AnAdaptiveArnoldiMethod . . . . . . . . . . . . . . . . 237
6.6.5 NumericalExamples . . . . . . . . . . . . . . . . . . . . . 239
6.7 Appendix:CodefortheMixedMethod . . . . . . . . . . . . . . . 244
7 TheSchrödingerEigenvalueProblem 247
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
7.2 ApproximationtoGross–PitaevskiiEquation . . . . . . . . . . . . 250
7.2.1 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . 251