Mathematical aspects of discontinuous Galerkin methods:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2012
|
| Series: | Mathématiques et Applications
69 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | XVII, 384 S. Ill., graph. Darst. |
| ISBN: | 9783642229794 3642229794 |
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MARC
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| 245 | 1 | 0 | |a Mathematical aspects of discontinuous Galerkin methods |c Daniele Antonio Di Pietro ; Alexandre Ern |
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IMAGE 1
CONTENTS
BASIC CONCEPTS 11.1 WELL-POSEDNESS FOR LINEAR MODEL PROBLEMS 21.1.1 THE
BANACH-NECAS-BABUSKA THEOREM 21.1.2 THE LAX-MILGRAM LEMMA 31.1.3
LEBESGUE AND SOBOLEV SPACES 41.2 THE DISCRETE SETTING 71.2.1 THE DOMAIN
Q 71.2.2 MESHES 81.2.3 MESH FACES, AVERAGES, AND JUMPS 91.2.4 BROKEN
POLYNOMIAL SPACES 121.2.5 BROKEN SOBOLEV SPACES 131.2.6 THE FUNCTION
SPACE IF(DIV; FI) AND ITS BROKEN VERSION 161.3 ABSTRACT NONCONFORMING
ERROR ANALYSIS 181.3.1 THE DISCRETE PROBLEM 181.3.2 DISCRETE STABILITY
191.3.3 CONSISTENCY 211.3.4 BOUNDEDNESS 211.3.5 ERROR ESTIMATE 221.4
ADMISSIBLE MESH SEQUENCES 221.4.1 SHAPE AND CONTACT REGULARITY 231.4.2
GEOMETRIC PROPERTIES 241.4.3 INVERSE AND TRACE INEQUALITIES 261.4.4
POLYNOMIAL APPROXIMATION 301.4.5 THE ONE-DIMENSIONAL CASE 34 SCALAR
FIRST-ORDER PDES 35 STEADY ADVECTION-REACTION 372.1 THE CONTINUOUS
SETTING 382.1.1 ASSUMPTIONS ON THE DATA 382.1.2 THE GRAPH SPACE 392.1.3
TRACES IN THE GRAPH SPACE 40XIII BIBLIOGRAFISCHE INFORMATIONEN
HTTP://D-NB.INFO/1013326997 DIGITALISIERT DURCH
IMAGE 2
XIV
CONTENTS
2.1 A WEAK FORMULATION AND WELL-POSEDNESS 42
2.1.5 PROOF OF MAIN RESULTS 43
2.1.6 NONHOMOGENEOUS BOUNDARY CONDITION 46
2.2 CENTERED FLUXES 47
2.2.1 HEURISTIC DERIVATION 50
2.2.2 ERROR ESTIMATES 53
2.2.3 NUMERICAL FLUXES 55
2.3 UPWINDING 56
2.3.1 TIGHTENED STABILITY USING PENALTIES 57
2.3.2 ERROR ESTIMATES BASED ON COERCIVITY 58
2.3.3 ERROR ESTIMATES BASED ON INF-SUP STABILITY 61
2.3.4 NUMERICAL FLUXES 65
UNSTEADY FIRST-ORDER PDES 67
3.1 UNSTEADY ADVECTION-REACTION 68
3.1.1 THE CONTINUOUS SETTING 69
3.1.2 SPACE SEMIDISCRETIZATION 72
3.1.3 TIME DISCRETIZATION 74
3.1.4 MAIN CONVERGENCE RESULTS 78
3.1.5 ANALYSIS OF FORWARD EULER AND FINITE VOLUME SCHEMES 83
3.1.6 ANALYSIS OF EXPLICIT RK2 SCHEMES 89
3.2 NONLINEAR CONSERVATION LAWS 98
3.2.1 THE CONTINUOUS SETTING 99
3.2.2 NUMERICAL FLUXES FOR SPACE SEMIDISCRETIZATION 100 3.2.3 TIME
DISCRETIZATION 106
3.2.4 LIMITERS 108
II SCALAR SECOND-ORDER PDES 117
4 PDES WITH DIFFUSION 119
4.1 PURE DIFFUSION: THE CONTINUOUS SETTING 120
4.1.1 WEAK FORMULATION AND WELL-POSEDNESS 120
4.1.2 POTENTIAL AND DIFFUSIVE FLUX 121
4.2 SYMMETRIC INTERIOR PENALTY 122
4.2.1 HEURISTIC DERIVATION 122
4.2.2 OTHER BOUNDARY CONDITIONS 126
4.2.3 BASIC ENERGY-ERROR ESTIMATE 128
4.2.4 L 2 -NORM ERROR ESTIMATE 133
4.2.5 ANALYSIS FOR LOW-REGULARITY SOLUTIONS 134
4.3 LIFTINGS AND DISCRETE GRADIENTS 137
4.3.1 LIFTINGS: DEFINITION AND STABILITY 138
4.3.2 DISCRETE GRADIENTS: DEFINITION AND STABILITY 140
4.3.3 REFORMULATION OF THE SIP BILINEAR FORM 140
4.3.4 NUMERICAL FLUXES 142
IMAGE 3
CONTENTS XV
4.4 MIXED DG METHODS 143
4.4.1 THE SIP METHOD AS A MIXED DG METHOD 143
4.4.2 NUMERICAL FLUXES 145
4.4.3 HYBRID MIXED DG METHODS 148
4.5 HETEROGENEOUS DIFFUSION 150
4.5.1 THE CONTINUOUS SETTING 151
4.5.2 DISCRETIZATION 153
4.5.3 ERROR ESTIMATES FOR SMOOTH SOLUTIONS 156
4.5.4 ERROR ESTIMATES FOR LOW-REGULARITY SOLUTIONS 160 4.5.5 NUMERICAL
FLUXES 162
4.5.6 ANISOTROPY 163
4.6 DIFFUSION-ADVECTION-REACTION 163
4.6.1 THE CONTINUOUS SETTING 163
4.6.2 DISCRETIZATION 165
4.6.3 ERROR ESTIMATES 166
4.6.4 LOCALLY VANISHING DIFFUSION 171
4.7 AN UNSTEADY EXAMPLE: THE HEAT EQUATION 176
4.7.1 THE CONTINUOUS SETTING 176
4.7.2 DISCRETIZATION 177
4.7.3 ERROR ESTIMATES 179
4.7.4 BDF2 TIME DISCRETIZATION 182
4.7.5 IMPROVED C(L 2 (FI))-ERROR ESTIMATE 184
ADDITIONAL TOPICS ON PURE DIFFUSION 187
5.1 DISCRETE FUNCTIONAL ANALYSIS 188
5.1.1 THE BV SPACE AND THE ||-|| D G,P-NORMS 188
5.1.2 DISCRETE SOBOLEV EMBEDDINGS 190
5.1.3 DISCRETE COMPACTNESS 193
5.2 CONVERGENCE TO MINIMAL REGULARITY SOLUTIONS 195
5.2.1 CONSISTENCY REVISITED 195
5.2.2 CONVERGENCE 197
5.3 VARIATIONS ON SYMMETRY AND PENALTY 198
5.3.1 VARIATIONS ON SYMMETRY 199
5.3.2 VARIATIONS ON PENALTY 203
5.3.3 SYNOPSIS 206
5.4 CELL-CENTERED GALERKIN 207
5.5 LOCAL POSTPROCESSING 211
5.5.1 LOCAL RESIDUALS 211
5.5.2 POTENTIAL RECONSTRUCTION 214
5.5.3 DIFFUSIVE FLUX RECONSTRUCTION BY PRESCRIPTION 217 5.5.4 DIFFUSIVE
FLUX RECONSTRUCTION BY SOLVING LOCAL PROBLEMS 222
5.6 A POSTERIORI ERROR ESTIMATES 227
5.6.1 OVERVIEW 228
5.6.2 ENERGY-NORM ERROR UPPER BOUNDS 229
5.6.3 ERROR LOWER BOUNDS 236
IMAGE 4
XVI CONTENTS
III SYSTEMS 239
6 INCOMPRESSIBLE FLOWS 241
6.1 STEADY STOKES FLOWS 243
6.1.1 THE CONTINUOUS SETTING 243
6.1.2 EQUAL-ORDER DISCONTINUOUS VELOCITIES AND PRESSURES . . . 248 6.1.3
CONVERGENCE TO SMOOTH SOLUTIONS 255
6.1.4 CONVERGENCE TO MINIMAL REGULARITY SOLUTIONS 260 6.1.5 FORMULATIONS
WITHOUT PRESSURE STABILIZATION 266 6.2 STEADY NAVIER-STOKES FLOWS 267
6.2.1 THE CONTINUOUS SETTING 267
6.2.2 THE DISCRETE SETTING 271
6.2.3 CONVERGENCE ANALYSIS 276
6.2.4 A CONSERVATIVE FORMULATION 280
6.3 THE UNSTEADY CASE 283
6.3.1 THE CONTINUOUS SETTING 284
6.3.2 THE PROJECTION METHOD 284
7 FRIEDRICHS' SYSTEMS 293
7.1 BASIC INGREDIENTS AND EXAMPLES 294
7.1.1 BASIC INGREDIENTS 294
7.1.2 EXAMPLES 296
7.2 THE CONTINUOUS SETTING 301
7.2.1 FRIEDRICHS' OPERATORS 302
7.2.2 THE CONE FORMALISM 304
7.2.3 THE BOUNDARY OPERATOR M 306
7.2.4 THE WELL-POSEDNESS RESULT 308
7.2.5 EXAMPLES 309
7.3 DISCRETIZATION 312
7.3.1 ASSUMPTIONS ON THE DATA AND THE EXACT SOLUTION 312 7.3.2 DESIGN OF
THE DISCRETE BILINEAR FORM 314
7.3.3 CONVERGENCE ANALYSIS 318
7.3.4 NUMERICAL FLUXES 320
7.3.5 EXAMPLES 320
7.4 TWO-FIELD SYSTEMS 322
7.4.1 THE CONTINUOUS SETTING 323
7.4.2 DISCRETIZATION 324
7.4.3 PARTIAL COERCIVITY 329
7.5 THE UNSTEADY CASE 331
7.5.1 EXPLICIT RUNGE-KUTTA SCHEMES 331
7.5.2 EXPLICIT LEAP-FROG SCHEMES FOR TWO-FIELD SYSTEMS . . . 334
APPENDIX A: IMPLEMENTATION 343
A.I MATRIX ASSEMBLY 343
A.1.1 BASIC NOTATION 343
A.1.2 MASS MATRIX 344
A.1.3 STIFFNESS MATRIX 344
IMAGE 5
CONTENTS XVII
A.2 CHOICE OF BASIS FUNCTIONS 347
A.2.1 REQUIREMENTS 347
A.2.2 JACOBI AND LEGENDRE POLYNOMIALS 349
A.2.3 NODAL BASIS FUNCTIONS 350
A.2.4 MODAL BASIS FUNCTIONS 352
BIBLIOGRAPHY 355
AUTHOR INDEX 375
INDEX 381 |
| any_adam_object | 1 |
| author | Di Pietro, Daniele Antonio Ern, Alexandre 1967- |
| author_GND | (DE-588)1020155698 (DE-588)114186839 |
| author_facet | Di Pietro, Daniele Antonio Ern, Alexandre 1967- |
| author_role | aut aut |
| author_sort | Di Pietro, Daniele Antonio |
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| dewey-full | 518.63 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 518 - Numerical analysis |
| dewey-raw | 518.63 |
| dewey-search | 518.63 |
| dewey-sort | 3518.63 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| spelling | Di Pietro, Daniele Antonio Verfasser (DE-588)1020155698 aut Mathematical aspects of discontinuous Galerkin methods Daniele Antonio Di Pietro ; Alexandre Ern Berlin [u.a.] Springer 2012 XVII, 384 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Mathématiques et Applications 69 Diskontinuierliche Galerkin-Methode (DE-588)4588309-9 gnd rswk-swf Diskontinuierliche Galerkin-Methode (DE-588)4588309-9 s DE-604 Ern, Alexandre 1967- Verfasser (DE-588)114186839 aut Mathématiques et Applications 69 (DE-604)BV024973330 69 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3850528&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024592092&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Di Pietro, Daniele Antonio Ern, Alexandre 1967- Mathematical aspects of discontinuous Galerkin methods Mathématiques et Applications Diskontinuierliche Galerkin-Methode (DE-588)4588309-9 gnd |
| subject_GND | (DE-588)4588309-9 |
| title | Mathematical aspects of discontinuous Galerkin methods |
| title_auth | Mathematical aspects of discontinuous Galerkin methods |
| title_exact_search | Mathematical aspects of discontinuous Galerkin methods |
| title_full | Mathematical aspects of discontinuous Galerkin methods Daniele Antonio Di Pietro ; Alexandre Ern |
| title_fullStr | Mathematical aspects of discontinuous Galerkin methods Daniele Antonio Di Pietro ; Alexandre Ern |
| title_full_unstemmed | Mathematical aspects of discontinuous Galerkin methods Daniele Antonio Di Pietro ; Alexandre Ern |
| title_short | Mathematical aspects of discontinuous Galerkin methods |
| title_sort | mathematical aspects of discontinuous galerkin methods |
| topic | Diskontinuierliche Galerkin-Methode (DE-588)4588309-9 gnd |
| topic_facet | Diskontinuierliche Galerkin-Methode |
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