Numerical solution of nonlinear elliptic problems via preconditioning operators: theory and applications
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York
Nova Science Publ.
2002
|
| Schriftenreihe: | Advances in computation
11 |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVIII, 402 S. graph. Darst. |
| ISBN: | 1590333764 |
Internformat
MARC
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| 100 | 1 | |a Faragó, István |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Numerical solution of nonlinear elliptic problems via preconditioning operators |b theory and applications |c I. Faragó and J. Karátson |
| 264 | 1 | |a New York |b Nova Science Publ. |c 2002 | |
| 300 | |a XVIII, 402 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Advances in computation |v 11 | |
| 700 | 1 | |a Karátson, János |e Verfasser |4 aut | |
| 830 | 0 | |a Advances in computation |v 11 |w (DE-604)BV026418425 |9 11 | |
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Datensatz im Suchindex
| _version_ | 1804144936883322880 |
|---|---|
| adam_text | IMAGE 1
ADVANCES IN COMPUTATION: THEORY AND PRACTICE
VOLUME 11
NUMERICAL SOLUTION OF
NONLINEAR ELLIPTIC PROBLEMS VIA PRECONDITIONING OPERATORS:
THEORY AND APPLICATIONS
I. FARAG6 AND J. KARATSON
NOVA SCIENCE PUBLISHERS, INC. NEW YORK
IMAGE 2
CONTENTS
PREFACE IX
INTRODUCTION XI
I MOTIVATION 1
1 NONLINEAR ELLIPTIC EQUATIONS IN MODEL PROBLEMS 3
1.1 ELASTO-PLASTIC TORSION OF RODS 4
1.2 ELECTROMAGNETIC FIELD THEORY (NONLINEAR MAXWELL EQUATIONS) 7
1.3 NONLINEAR ELASTICITY 9
1.4 ELASTO-PLASTIC BENDING OF CLAMPED PLATES 10
1.5 SEMILINEAR EQUATIONS 11
1.6 SOME OTHER EXAMPLES 13
1.6.1 FLOW MODELS 13
1.6.2 NON-POTENTIAL PROBLEMS 14
1.7 WEAK FORMULATIONS 15
2 LINEAR ALGEBRAIC SYSTEMS 21
2.1 CONDITIONING OF SYSTEMS OF LINEAR ALGEBRAIC EQUATIONS 21
2.1.1 WELL-POSEDNESS 21
2.1.2 THE CONDITION NUMBER AND ITS PROPERTIES 22
2.1.3 COMPUTER IMPLEMENTATION 24
2.1.4 AN EXAMPLE OF IMPROVED CONDITIONING 27
2.1.5 SOME ITERATIVE METHODS AND THEIR CONVERGENCE 29
2.2 PROBLEMS WITH LARGE CONDITION NUMBERS 31
2.2.1 DISCRETIZED ELLIPTIC PROBLEMS 31
2.2.2 DIFFICULTIES ARISING FROM LARGE CONDITION NUMBERS 32 2.3
PRECONDITIONING OF LINEAR ALGEBRAIC SYSTEMS 33
2.3.1 PRECONDITIONING AS THE MAIN TOOL OF IMPROVING THE CONDITION NUMBER
33
IMAGE 3
VI I. FARAGD AND J. KARDTSON
2.3.2 SPECTRAL EQUIVALENCE AND PRECONDITIONING 36
2.3.3 SOME IMPORTANT PRECONDITIONING TECHNIQUES 37
3 LINEAR ELLIPTIC PROBLEMS 41
3.1 SOME PROPERTIES OF LINEAR OPERATORS IN HILBERT SPACE 42
3.1.1 ENERGY SPACES 42
3.1.2 SPECTRAL EQUIVALENCE AND CONTRACTIVITY 43
3.2 WELL-POSEDNESS OF LINEAR ELLIPTIC PROBLEMS 46
3.2.1 WEAK SOLUTIONS 46
3.2.2 REGULARITY 48
3.3 STANDARD SOLUTION METHODS FOR SOME LINEAR ELLIPTIC PROBLEMS 49 3.3.1
FINITE ELEMENT DISCRETIZATION 49
3.3.2 EFFICIENT SOLUTION ALGORITHMS FOR GENERAL LINEAR ELLIPTIC PROBLEMS
52 3.3.3 FAST SOLVERS FOR PROBLEMS IN SPECIAL FORM 53
3.4 ITERATION AND PRECONDITIONING IN SOBOLEV SPACE 55
3.4.1 THE CONDITION NUMBER OF LINEAR OPERATORS 55
3.4.2 PRECONDITIONED GRADIENT TYPE METHODS AND MESH INDEPENDENCE 56
3.4.3 SOME REMARKS ON TWO-SIDED PRECONDITIONING 61
4 NONLINEAR ALGEBRAIC SYSTEMS AND PRECONDITIONING 63
4.1 THE CONDITION NUMBER OF NONLINEAR ALGEBRAIC SYSTEMS 65
4.2 SOME ITERATIVE METHODS FOR NONLINEAR ALGEBRAIC SYSTEMS 66
4.2.1 GRADIENT TYPE METHODS 66
4.2.2 NEWTON S METHOD 67
4.3 PRECONDITIONING FOR NONLINEAR ALGEBRAIC SYSTEMS 71
4.3.1 PRECONDITIONED SIMPLE ITERATIONS FOR NONLINEAR ALGEBRAIC SYS- TEMS
71
4.3.2 VARIABLE PRECONDITIONING AND NEWTON S METHOD 72
4.3.3 THE PROBLEM OF PRECONDITIONING FOR DISCRETIZED NONLINEAR EL-
LIPTIC EQUATIONS 73
II THEORETICAL BACKGROUND 75
5 NONLINEAR EQUATIONS IN HILBERT SPACE 77
5.1 POTENTIALS AND MONOTONE OPERATORS 78
5.2 ITERATIVE METHODS FOR SMOOTH MAPPINGS 85
5.2.1 SIMPLE ITERATIONS (GRADIENT METHOD) 85
5.2.2 NEWTON-LIKE METHODS 92
5.3 PRECONDITIONING BY LINEAR OPERATORS IN HILBERT SPACE 101
5.3.1 DEFINITION AND PROPERTIES OF THE CONDITION NUMBER 102 5.3.2 FIXED
PRECONDITIONING OPERATORS 105
5.3.3 VARIABLE PRECONDITIONING OPERATORS I LL
IMAGE 4
CONTENTS VII
5.4 PRECONDITIONING AND GRADIENTS 121
6 SOLVABILITY OF NONLINEAR ELLIPTIC PROBLEMS 125
6.1 SOME PROPERTIES OF THE GENERALIZED DIFFERENTIAL OPERATORS 125 6.2
WEAK SOLUTIONS 139
6.2.1 GENERAL WELL-POSEDNESS RESULTS 140
6.2.2 VARIOUS SOLVABILITY THEOREMS 143
6.2.3 NON-POTENTIAL PROBLEMS 150
6.3 QUALITATIVE PROPERTIES 152
6.3.1 REGULARITY OF THE SOLUTION 152
6.3.2 POSITIVITY OF THE SOLUTION 153
6.4 EXAMPLES 154
III ITERATIVE SOLUTION OF NONLINEAR ELLIPTIC BOUNDARY VALUE PROB- LEMS
161
7 ITERATIVE METHODS IN SOBOLEV SPACE 163
7.1 SIMPLE ITERATIONS 166
7.1.1 SECOND ORDER DIRICHLET PROBLEMS 166
7.1.2 MIXED AND HIGHER ORDER PROBLEMS 173
7.2 NEWTON-LIKE METHODS 183
7.2.1 THE GENERAL DAMPED NEWTON ALGORITHM 185
7.2.2 SECOND ORDER PROBLEMS: INNER-OUTER ITERATIONS 190
7.2.3 SECOND ORDER PROBLEMS: VARIABLE PRECONDITIONING 198 7.2.4 OTHER
PROBLEMS 199
7.3 PRECONDITIONING AND SOBOLEV GRADIENTS 202
7.4 SOME MORE ITERATIVE METHODS IN SOBOLEV SPACE 210
7.4.1 THE NONLINEAR CONJUGATE GRADIENT METHOD 210
7.4.2 FROZEN COEFFICIENT ITERATIONS 213
7.4.3 DOUBLE SOBOLEV GRADIENTS _ 215
7.4.4 SYMMETRIC PART PRECONDITIONING 220
7.4.5 SOME REMARKS ON MULTISTEP ITERATIONS 224
8 PRECONDITIONING STRATEGIES FOR DISCRETIZED NONLINEAR ELLIPTIC PROB-
LEMS BASED ON PRECONDITIONING OPERATORS 227
8.1 SOME GENERAL PROPERTIES OF THE SOBOLEV SPACE PRECONDITIONERS . . ..
232 8.1.1 STIFFNESS MATRICES: SOLVABILITY, UPDATING AND STRUCTURE
CHARAC- TERISTICS 232
8.1.2 MESH INDEPENDENT CONDITIONING PROPERTIES 234
8.2 VARIOUS PRECONDITIONING STRATEGIES BASED ON THE SOBOLEV SPACE
BACKGROUND 240
8.2.1 DISCRETE LAPLACIAN PRECONDITIONER 242
IMAGE 5
VIII I. FARAGD AND J. KARDTSON
8.2.2 CONSTANT COEFFICIENT PRECONDITIONERS 246
8.2.3 SEPARABLE PRECONDITIONERS 249
8.2.4 LINEAR PRINCIPAL PART PRECONDITIONER 252
8.2.5 FROZEN COEFFICIENT PRECONDITIONER 254
8.2.6 INITIAL SHAPE PRECONDITIONERS 256
8.2.7 PRECONDITIONERS USING DOMAIN DECOMPOSITION 260
8.2.8 DIAGONAL COEFFICIENT PRECONDITIONERS 267
8.2.9 DOUBLE SOBOLEV GRADIENT PRECONDITIONER 269
8.2.10 SYMMETRIC PART PRECONDITIONERS 273
8.2.11 INCORPORATING BOUNDARY CONDITIONS 275
8.2.12 NON-INJECTIVE PROBLEMS 279
8.2.13 DISCRETE BIHARMONIC PRECONDITIONER 280
8.2.14 DOUBLE DIAGONAL COEFFICIENT PRECONDITIONERS 282
8.2.15 DECOUPLED LAPLACIAN PRECONDITIONERS FOR SYSTEMS 285
9 ALGORITHMIC REALIZATION OF ITERATIVE METHODS BASED ON PRECONDI-
TIONING OPERATORS 289
9.1 GENERAL ALGORITHMS 292
9.1.1 SIMPLE ITERATIONS 294
9.1.2 NEWTON-LIKE ITERATIONS 297
9.1.3 SOME CONVERGENCE RESULTS 305
9.2 FINITE ELEMENT REALIZATION 308
9.2.1 THE GRADIENT-FINITE ELEMENT METHOD 309
9.2.2 THE NEWTON-FINITE ELEMENT METHOD . 311
9.2.3 CONVERGENCE ESTIMATES AND MESH INDEPENDENCE 312 9.2.4 MULTILEVEL
TYPE ITERATIONS 316
9.3 SOME REMARKS ON THE USE OF LAPLACIAN PRECONDITIONERS 320 9.4 ON
TIME-DEPENDENT PROBLEMS 327
10 SOME NUMERICAL ALGORITHMS FOR NONLINEAR ELLIPTIC PROBLEMS IN PHYSICS
331
10.1 ELASTO-PLASTIC TORSION OF RODS 332
10.2 ELECTROMAGNETIC FIELD EQUATION 340
10.3 ELASTO-PLASTIC BENDING OF CLAMPED PLATES 351
10.4 NONLINEAR ELASTICITY 355
10.5 RADIATIVE COOLING 357
10.6 ELECTROSTATIC POTENTIAL IN A BALL 363
11 APPENDIX 375
BIBLIOGRAPHY 381
INDEX 399
|
| any_adam_object | 1 |
| author | Faragó, István Karátson, János |
| author_facet | Faragó, István Karátson, János |
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| dewey-tens | 510 - Mathematics |
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| illustrated | Illustrated |
| indexdate | 2024-07-09T23:11:29Z |
| institution | BVB |
| isbn | 1590333764 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-021988768 |
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| physical | XVIII, 402 S. graph. Darst. |
| publishDate | 2002 |
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| series | Advances in computation |
| series2 | Advances in computation |
| spelling | Faragó, István Verfasser aut Numerical solution of nonlinear elliptic problems via preconditioning operators theory and applications I. Faragó and J. Karátson New York Nova Science Publ. 2002 XVIII, 402 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Advances in computation 11 Karátson, János Verfasser aut Advances in computation 11 (DE-604)BV026418425 11 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=021988768&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Faragó, István Karátson, János Numerical solution of nonlinear elliptic problems via preconditioning operators theory and applications Advances in computation |
| title | Numerical solution of nonlinear elliptic problems via preconditioning operators theory and applications |
| title_auth | Numerical solution of nonlinear elliptic problems via preconditioning operators theory and applications |
| title_exact_search | Numerical solution of nonlinear elliptic problems via preconditioning operators theory and applications |
| title_full | Numerical solution of nonlinear elliptic problems via preconditioning operators theory and applications I. Faragó and J. Karátson |
| title_fullStr | Numerical solution of nonlinear elliptic problems via preconditioning operators theory and applications I. Faragó and J. Karátson |
| title_full_unstemmed | Numerical solution of nonlinear elliptic problems via preconditioning operators theory and applications I. Faragó and J. Karátson |
| title_short | Numerical solution of nonlinear elliptic problems via preconditioning operators |
| title_sort | numerical solution of nonlinear elliptic problems via preconditioning operators theory and applications |
| title_sub | theory and applications |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=021988768&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV026418425 |
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