hp-finite element methods for singular perturbations:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin ; Heidelberg ; New York ; Barcelona ; Hong Kong ; London
Springer
2002
|
| Schriftenreihe: | Lecture notes in mathematics
1796 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIV, 318 S. graph. Darst. |
| ISBN: | 3540442014 |
Internformat
MARC
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| 100 | 1 | |a Melenk, Jens M. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a hp-finite element methods for singular perturbations |c Jens M. Melenk |
| 264 | 1 | |a Berlin ; Heidelberg ; New York ; Barcelona ; Hong Kong ; London |b Springer |c 2002 | |
| 300 | |a XIV, 318 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Lecture notes in mathematics |v 1796 | |
| 650 | 7 | |a Eindige-elementenmethode |2 gtt | |
| 650 | 4 | |a Perturbations singulières (Mathématiques) | |
| 650 | 7 | |a Singuliere operatoren |2 gtt | |
| 650 | 4 | |a Équations aux dérivées partielles - Solutions numériques | |
| 650 | 4 | |a Differential equations, Partial |x Numerical solutions | |
| 650 | 4 | |a Singular perturbations (Mathematics) | |
| 650 | 0 | 7 | |a Randwertproblem |0 (DE-588)4048395-2 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Finite-Elemente-Methode |0 (DE-588)4017233-8 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Elliptische Differentialgleichung |0 (DE-588)4014485-9 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Singuläre Störung |0 (DE-588)4055100-3 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Partielle Differentialgleichung |0 (DE-588)4044779-0 |2 gnd |9 rswk-swf |
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| 689 | 2 | 1 | |a Randwertproblem |0 (DE-588)4048395-2 |D s |
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Datensatz im Suchindex
| _version_ | 1804129421952548864 |
|---|---|
| adam_text | Contents
Notation XI
1. Introduction 1
1.1 Introduction 1
1.2 Problem class and assumptions 1
1.3 Principal results 4
1.4 Review of existing results 7
1.4.1 Elliptic problems in non smooth domains 7
1.4.2 Regularity in terms of asymptotic expansions 9
1.4.3 hp finite element methods 15
1.4.4 Numerical methods for singular perturbations 16
1.5 Outline of the book 19
Part I. Finite Element Approximation
2. /ip FEM for Reaction Diffusion Problems: Principal Results . 23
2.1 Setting and Introduction 23
2.2 Prelude: the one dimensional case 24
2.2.1 Regularity in one dimension 24
2.2.2 hp FEM in one dimension 25
2.2.3 Numerical examples 30
2.3 Regularity: the two dimensional case 31
2.4 hp FEM approximation 38
2.4.1 /ip meshes and spaces 38
2.4.2 The minimal hp mesh 39
2.4.3 hp FEM 43
2.5 Numerical Examples 45
2.5.1 The classical L shaped domain 46
2.5.2 Robustness with respect to mesh distortion 49
2.5.3 Examples with singular right hand side 50
2.6 /i FEM approximation 54
2.6.1 Approximation on Shishkin meshes in one dimension 55
2.6.2 ft FEM meshes 57
2.6.3 h FEM boundary layer meshes 62
VIII Contents
3. hp Approximation 73
3.1 Motivation and outline 73
3.1.1 General overview of Chapter 3 73
3.1.2 Outline of Chapter 3 75
3.1.3 Robust exponential convergence: key ingredients of proof . 76
3.2 Polynomial approximation results 87
3.2.1 Notation and properties of polynomials 87
3.2.2 Approximation of analytic functions: intervals and squares 90
3.2.3 Approximation of analytic functions on triangles 93
3.2.4 The projector II™ 103
3.2.5 Anisotropic projection operators: 77* °° 105
3.2.6 An optimal error estimate for an H projector 109
3.3 Admissible boundary layer meshes and finite element spaces .... Ill
3.3.1 hp meshes for the approximation of boundary and corner
layers 114
3.3.2 Patchwise structured meshes 116
3.3.3 The p version boundary layer and corner layer patches ... 118
3.3.4 Boundary layer mesh generation via mesh patches 120
3.3.5 Properties of the pull backs to the patches 122
3.4 hp Approximation on minimal meshes 123
3.4.1 Regularity on the reference element 123
3.4.2 Approximation on minimal meshes 127
Part II. Regularity in Countably Normed Spaces
4. The Countably Normed Spaces Blpe 141
4.1 Motivation and outline 141
4.1.1 Motivation 141
4.1.2 Outline of Chapter 4 146
4.2 The spaces H™* and Bl0£ in a Sector 146
4.2.1 Properties of the spaces Hj*{Q) 150
4.2.2 Properties of the countably normed spaces Bl s 154
4.3 Local changes of variables for analytic functions 165
5. Regularity Theory in Countably Normed Spaces 169
5.1 Motivation and outline 169
5.1.1 Motivation 169
5.1.2 Outline of Chapter 5 172
5.2 Analytic regularity results of Babuska and Guo 173
5.3 Analytic regularity: Dirichlet problems 176
5.3.1 Analytic regularity in sectors 177
5.3.2 Regularity in curvilinear polygons 184
5.4 Neumann and transmission problems 188
5.4.1 Neumann and Robin corners 188
Contents IX
5.4.2 Mixed corners 195
5.4.3 Transmission problems 196
5.5 Local regularity 197
5.5.1 Preliminaries: local iJ2 regularity 200
5.5.2 Interior regularity: Proof of Proposition 5.5.1 202
5.5.3 Regularity at the boundary: Proof of Proposition 5.5 2 ... 208
5.5.4 Regularity of transmission problems: Proof of Prop. 5.5.4 . 215
5.5.5 Regularity of Neumann problems: Proof of Prop. 5.5.3 ... 224
Part III. Regularity in Terms of Asymptotic Expansions
6. Exponentially Weighted Coimtably Normed Spaces 227
6.1 Motivation and outline 227
6.1.1 Motivation 227
6.1.2 Outline of Chapter 6 230
6.2 The exponentially weighted spaces H™ e a and Blp e a in sectors .. 231
6.2.1 Properties of the exponentially weighted spaces 231
6.3 Change of variables: from polar to Cartesian coordinates 235
6.4 Analytic regularity in exponentially weighted spaces 238
6.4.1 Transmission problem: problem formulation 238
6.4.2 Transmission problem in exponentially weighted spaces... 243
6.4.3 Analytic regularity in exponentially weighted spaces 246
6.4.4 Analytic regularity for a special transmission problem .... 248
7. Regularity through Asymptotic Expansions 255
7.1 Motivation and outline 255
7.1.1 Motivation 255
7.1.2 Outline of Chapter 7 262
7.2 Regularity of the outer expansion 263
7.3 Regularity of the boundary layer expansion 267
7.3.1 Definition and properties of the boundary layer expansion 267
7.3.2 Proof of Theorem 7.3.3 270
7.4 Regularity through asymptotic expansions 283
7.4.1 Notation and main result 283
7.4.2 Proof of Theorem 7.4.5: smooth and boundary layer parts 288
7.4.3 Proof of Theorem 7.4.5: corner layer and remainder 290
Appendix 297
A.I Some technical lemmata 297
A.I.I Transformations of elliptic equations 297
A.1.2 Leibniz formulas 298
A.1.3 Hardy inequalities 301
A.2 Kondrat ev s theory for a special transmission problem 302
A.2.1 Problem formulation and notation 302
A.2.2 Proof of Proposition A.2.1 303
X Contents
A.3 Stability properties of the Gauss Lobatto interpolant 309
A.4 L°° projectors 310
References 311
Index 317
|
| any_adam_object | 1 |
| author | Melenk, Jens M. |
| author_facet | Melenk, Jens M. |
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| author_sort | Melenk, Jens M. |
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| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.353 |
| dewey-search | 515/.353 |
| dewey-sort | 3515 3353 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| indexdate | 2024-07-09T19:04:53Z |
| institution | BVB |
| isbn | 3540442014 |
| language | English |
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| physical | XIV, 318 S. graph. Darst. |
| publishDate | 2002 |
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| series2 | Lecture notes in mathematics |
| spelling | Melenk, Jens M. Verfasser aut hp-finite element methods for singular perturbations Jens M. Melenk Berlin ; Heidelberg ; New York ; Barcelona ; Hong Kong ; London Springer 2002 XIV, 318 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics 1796 Eindige-elementenmethode gtt Perturbations singulières (Mathématiques) Singuliere operatoren gtt Équations aux dérivées partielles - Solutions numériques Differential equations, Partial Numerical solutions Singular perturbations (Mathematics) Randwertproblem (DE-588)4048395-2 gnd rswk-swf Finite-Elemente-Methode (DE-588)4017233-8 gnd rswk-swf Elliptische Differentialgleichung (DE-588)4014485-9 gnd rswk-swf Singuläre Störung (DE-588)4055100-3 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 s Finite-Elemente-Methode (DE-588)4017233-8 s DE-604 Singuläre Störung (DE-588)4055100-3 s Elliptische Differentialgleichung (DE-588)4014485-9 s Randwertproblem (DE-588)4048395-2 s Lecture notes in mathematics 1796 (DE-604)BV000676446 1796 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009953308&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Melenk, Jens M. hp-finite element methods for singular perturbations Lecture notes in mathematics Eindige-elementenmethode gtt Perturbations singulières (Mathématiques) Singuliere operatoren gtt Équations aux dérivées partielles - Solutions numériques Differential equations, Partial Numerical solutions Singular perturbations (Mathematics) Randwertproblem (DE-588)4048395-2 gnd Finite-Elemente-Methode (DE-588)4017233-8 gnd Elliptische Differentialgleichung (DE-588)4014485-9 gnd Singuläre Störung (DE-588)4055100-3 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| subject_GND | (DE-588)4048395-2 (DE-588)4017233-8 (DE-588)4014485-9 (DE-588)4055100-3 (DE-588)4044779-0 |
| title | hp-finite element methods for singular perturbations |
| title_auth | hp-finite element methods for singular perturbations |
| title_exact_search | hp-finite element methods for singular perturbations |
| title_full | hp-finite element methods for singular perturbations Jens M. Melenk |
| title_fullStr | hp-finite element methods for singular perturbations Jens M. Melenk |
| title_full_unstemmed | hp-finite element methods for singular perturbations Jens M. Melenk |
| title_short | hp-finite element methods for singular perturbations |
| title_sort | hp finite element methods for singular perturbations |
| topic | Eindige-elementenmethode gtt Perturbations singulières (Mathématiques) Singuliere operatoren gtt Équations aux dérivées partielles - Solutions numériques Differential equations, Partial Numerical solutions Singular perturbations (Mathematics) Randwertproblem (DE-588)4048395-2 gnd Finite-Elemente-Methode (DE-588)4017233-8 gnd Elliptische Differentialgleichung (DE-588)4014485-9 gnd Singuläre Störung (DE-588)4055100-3 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| topic_facet | Eindige-elementenmethode Perturbations singulières (Mathématiques) Singuliere operatoren Équations aux dérivées partielles - Solutions numériques Differential equations, Partial Numerical solutions Singular perturbations (Mathematics) Randwertproblem Finite-Elemente-Methode Elliptische Differentialgleichung Singuläre Störung Partielle Differentialgleichung |
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