Multigrid methods for finite elements:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Dordrecht u.a.
Kluwer
1995
|
| Series: | Mathematics and its applications
318 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Aus dem Russ. übers. |
| Physical Description: | XIV, 331 S. |
| ISBN: | 0792332903 |
Staff View
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Record in the Search Index
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|---|---|
| adam_text | IMAGE 1
MULTIGRID METHODS FOR
FINITE ELEMENTS
BY V. V. SHAIDUROV COMPUTING CENTER, RUSSIAN ACADEMY OF SCIENCES,
SIBERIAN BRANCH, KRASNOYARSK, RUSSIA
KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON
IMAGE 2
CONTENTS
PREFACE IX
INTRODUCTION XI
1 ELLIPTIC BOUNDARY-VALUE PROBLEMS AND BUBNOV-GALERKIN METHOD 1 1.1
STATEMENTS OF DIFFERENTIAL PROBLEMS AND SMOOTHNESS OF SOLUTIONS . . .. 1
1.1.1 MAIN NOTATIONS OF SMOOTHNESS CLASSES 2
1.1.2 SOLVING TWO ABSTRACT PROBLEMS 4
1.1.3 THE DIRICHLET PROBLEM FOR SECOND-ORDER ELLIPTIC EQUATION 7 1.1.4
BOUNDARY-VALUE PROBLEMS OF THIRD KIND 11
1.1.5 THE NEUMANN PROBLEM 14
1.1.6 ELASTICITY PROBLEM 16
1.1.7 THE DIRICHLET PROBLEM FOR BIHARMONIC EQUATION 20
1.1.8 BOUNDARY-VALUE PROBLEMS FOR PLATES 21
1.1.9 THE MIXED METHOD FOR BIHARMONIC EQUATION 23
1.1.10 THE STOKES PROBLEM 26
1.2 THE GENERAL SCHEME OF THE BUBNOV-GALERKIN METHOD 29
1.2.1 SOLVING THE OPERATOR EQUATIONS 29.
1.2.2 THE SPECTRAL PROBLEM 31
1.2.3 THE MIXED METHOD 34
2 GENERAL PROPERTIES OF FINITE ELEMENTS 37
2.1 CLASSIFICATION OF FINITE ELEMENTS 38
2.2 TWO-DIMENSIONAL FINITE ELEMENTS 39
2.2.1 ELEMENTS WITH A TRIANGULAR MESH . I- 39
2.2.2 ELEMENTS WITH A RECTANGULAR MESH 42
2.3 THREE-DIMENSIONAL FINITE ELEMENTS * * * 45
2.3.1 ELEMENTS WITH A TETRAHEDRAL CELL 47
2.3.2 ELEMENTS WITH A MESH SHAPED AS A RECTANGULAR PARALLELEPIPED . . 49
2.4 AFFINE AND ISOPARAMETRIC TRANSFORMATIONS 51
2.4.1 CURVILINEAR ELEMENTS 51
2.4.2 ISOPARAMETRIC TRANSFORMATIONS OF TRIANGLES 52
2.4.3 ISOPARAMETRIC TRANSFORMATION OF A SQUARE 55
2.5 TRIANGULATION OF TWO-DIMENSIONAL DOMAINS 57
V
IMAGE 3
2.5.1 THE BOUNDARY-CORRECTION ALGORITHM 59
2.5.2 THE DEPLETION ALGORITHM 60
2.5.3 THE BREAKING ALGORITHM 61
2.6 TRIANGULATION OF THE THREE-DIMENSIONAL DOMAINS 62
2.6.1 THE BOUNDARY-CORRECTION ALGORITHM 63
2.6.2 THE DEPLETION ALGORITHM 65
2.6.3 THE BREAKING ALGORITHM 68
2.7 OTHER ALGORITHMS AND REFINEMENT OF TRIANGULATION 71
ON THE CONVERGENCE OF APPROXIMATE SOLUTIONS 75
3.1 INTERPOLATION BY FINITE ELEMENTS 76
3.1.1 LOCAL INTERPOLATION 76
3.1.2 GLOBAL INTERPOLATION 78
3.1.3 THE BREAKING ALGORITHM FOR FINITE ELEMENTS AND NESTING OF BASIS
FUNCTIONS SPACE 81
3.2 THE CONVERGENCE OF THE BUBNOV-GALERKIN METHOD 87
3.2.1 THE SOLUTION OF AN OPERATOR EQUATION 87
3.2.2 THE SPECTRAL PROBLEM 90
3.2.3 THE MIXED METHOD 91
3.3 THE USE OF QUADRATURE FORMULAE 92
3.3.1 INFLUENCE OF ISOPARAMETRIC OR AFFINE TRANSFORMATIONS 92
3.3.2 THE ELLIPTICITY AND THE ACCURACY 93
3.3.3 THE TWO-DIMENSIONAL CASE 95
3.3.4 THE THREE-DIMENSIONAL CASE 98
3.4 THE APPROXIMATION OF BOUNDARY CONDITIONS 100
3.4.1 A CONFORMABLE TRIANGULATION AND ISOPARAMETRIC ELEMENTS . . .. 101
3.4.2 AN UNCONFORMABLE TRIANGULATION AND THE PENALTY METHOD . . .. 103
3.4.3 THE MIXED METHOD 105
3.5 THE RICHARDSON EXTRAPOLATION 106
3.6 THE CONDITIONALLY OF LINEAR EQUATIONS SYSTEMS AND DIAGONAL 112
NORMALIZATION
GENERAL DESCRIPTION OF MULTIGRID ALGORITHMS 117
4.1 SOLVING THE PROBLEMS OF PARAMETERS OPTIMIZATION 118
4.2 AN ABSTRACT SETTING THE MULTIGRID ALGORITHMS 125
4.2.1 THE ALGEBRAIC FORMULATION OF THE MULTIGRID ALGORITHMS 126 4.2.2
THE MULTIGRID OPERATOR OF ERROR SUPPRESSION 127
4.3 THE CONVERGENCE IN THE SYMMETRIC VARIANT 129
4.3.1 THE REGULAR CASE 129
4.3.2 THE VERIFICATION OF THE CONVERGENCE CRITERION 132
4.3.3 THE NON-REGULAR CASE 135
4.3.4 THE MORE REGULAR CASE 139
4.4 A CONVERGENCE FOR SYMMETRIC SIGN-INDEFINITE OPERATORS 141
4.4.1 THE CONVERGENCE THEOREM 141
4.4.2 CONVERGENCE IN AN ORDINARY SITUATION 144
4.5 GENERAL CASE. CONVERGENCE IN INITIAL NORM 146
4.5.1 CONVERGENCE CRITERIA 146
VI
IMAGE 4
4.5.2 A CASE OF ORDINARY ACCURACY 148
4.6 THE FORMULATION OF THE CONVERGENCE CONDITIONS FOR THE LAGRANGE
FINITE ELEMENTS .* 151
4.6.1 A SYMMETRIC POSITIVE-DEFINITE BILINEAR FORM 151
4.6.2 SYMMETRIC SIGN-INDEFINITE BILINEAR FORM 160
4.6.3 AN ASYMMETRIC SIGN-INDEFINITE BILINEAR FORM 161
4.7 SIMULTANEOUS APPLICATION OF THE MULTIGRID ALGORITHMS AND THE
RICHARDSON EXTRAPOLATION 162
4.7.1 CONVERGENCE IN ENERGY NORM 164
4.7.2 CONVERGENCE IN L2- N RM 167
4.8 SOLVING THE SINGULAR PROBLEMS 168
4.9 SOLVING THE SPECTRAL PROBLEMS 179
4.10 THE ALGORITHM IN THE SYMMETRIZED CASE 192
4.10.1 CONVERGENCE IN Z/2-NORM 192
4.10.2 CONVERGENCE IN ENERGY NORM 198
4.11 THE ALGORITHM FOR THE MIXED METHOD 200
4.11.1 CONVERGENCE IN L2-NORM 200
4.11.2 CONVERGENCE IN ENERGY NORM 203
REALIZATION OF THE ALGORITHMS FOR SECOND-ORDER EQUATIONS 207
5.1 THE TWO-DIMENSIONAL DIRICHLET PROBLEM 208
5.1.1 PROPERTIES OF A DIFFERENTIAL AND A DIFFERENCE PROBLEMS 208
5.1.2 A CALCULATION OF THE NUMBER OF ARITHMETIC OPERATIONS 210
5.1.3 A NUMERICAL EXPERIMENT 214
5.1.4 SOME STATEMENTS FOR THE OTHER SITUATIONS 215
5.1.5 GRAPHIC REPRESENTATION OF ALGORITHMS 216
5.2 A MODIFIED ALGORITHM FOR DOMAINS WITH A CURVILINEAR BOUNDARY 217 5.3
THE PROBLEM WITH A POINTWISE SINGULARITY 224
5.4 THE THREE-DIMENSIONAL DIRICHLET PROBLEM 230
5.5 THE SPECTRAL PROBLEM 233
5.6 THE BOUNDARY VALUE PROBLEMS OF SECOND AND THIRD KIND 243
5.6.1 THE THIRD BOUNDARY VALUE PROBLEM ON A SUBORDINATE TRIANGULATION
244 5.6.2 THE NEUMANN PROBLEM 246
5.6.3 THE THIRD BOUNDARY VALUE PROBLEM ON REGULAR GRIDS 247
SOLVING NONLINEAR PROBLEMS AND SYSTEMS OF EQUATIONS 251
6.1 NONLINEAR PROBLEMS WITH AN OPERATOR OF MONOTONE TYPE 251
6.2 SOLUTION OF QUASI-LINEAR EQUATIONS OF ORDER 2 255
6.2.1 THE EQUATION WITH A WEAK NONLINEARITY 255
6.2.2 THE NEWTON METHOD AND THE MULTIGRID ALGORITHM 259
6.2.3 THE MAGNETOSTATIC PROBLEM 264
6.2.4 THE SIMPLEST TWO-STAGE ITERATIVE PROCESS 267
6.2.5 OTHER LINEARIZATIONS 270
6.3 SOLVING THE ELASTICITY PROBLEM 272
6.3.1 THE ELASTICITY PLANE PROBLEM 272
6.3.2 PROJEOTIVE-DIFFERENCE PROBLEM 273
VII
IMAGE 5
6.3.3 THE MULTIGRID ITERATIVE ALGORITHM 274
6.3.4 THE NUMERICAL EXPERIMENT 274
6.3.5 POSSIBLE GENERALIZATIONS 275
6.4 THE BIHARMONIC EQUATION 276
6.4.1 THE MIXED FORMULATION 276
6.4.2 THE MULTIGRID ALGORITHM 280
6.4.3 ANOTHER APPROACH 286
6.5 THE STATIONARY STOKES PROBLEM 288
6.5.1 THE FORMULATION OF THE DIFFERENTIAL PROBLEM 288
6.5.2 THE DISCRETE PROBLEM 291
6.5.3 THE MULTIGRID ALGORITHM 295
6.5.4 OTHER COMBINATIONS OF FINITE ELEMENTS 300
6.6 THE STATIONAIY NAVIER-STOKES PROBLEM 301
6.6.1 THE FORMULATION OF A DIFFERENTIAL PROBLEM 302
6.6.2 THE DISCRETE PROBLEM 303
6.6.3 THE NEWTON METHOD AND THE MULTIGRID ALGORITHM 303
6.6.4 THE MAIN RESULT 307
BIBLIOGRAPHY 313
SUBJECT INDEX 327
VIII
|
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| spelling | Šaidurov, Vladimir V. Verfasser aut Mnogosetocnye metody konecnych elementov Multigrid methods for finite elements by V. V. Shaidurov Dordrecht u.a. Kluwer 1995 XIV, 331 S. txt rdacontent n rdamedia nc rdacarrier Mathematics and its applications 318 Aus dem Russ. übers. Mehrgitterverfahren (DE-588)4038376-3 gnd rswk-swf Finite-Elemente-Methode (DE-588)4017233-8 gnd rswk-swf Mehrgitterverfahren (DE-588)4038376-3 s Finite-Elemente-Methode (DE-588)4017233-8 s DE-604 Mathematics and its applications 318 (DE-604)BV008163334 318 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006776722&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Šaidurov, Vladimir V. Multigrid methods for finite elements Mathematics and its applications Mehrgitterverfahren (DE-588)4038376-3 gnd Finite-Elemente-Methode (DE-588)4017233-8 gnd |
| subject_GND | (DE-588)4038376-3 (DE-588)4017233-8 |
| title | Multigrid methods for finite elements |
| title_alt | Mnogosetocnye metody konecnych elementov |
| title_auth | Multigrid methods for finite elements |
| title_exact_search | Multigrid methods for finite elements |
| title_full | Multigrid methods for finite elements by V. V. Shaidurov |
| title_fullStr | Multigrid methods for finite elements by V. V. Shaidurov |
| title_full_unstemmed | Multigrid methods for finite elements by V. V. Shaidurov |
| title_short | Multigrid methods for finite elements |
| title_sort | multigrid methods for finite elements |
| topic | Mehrgitterverfahren (DE-588)4038376-3 gnd Finite-Elemente-Methode (DE-588)4017233-8 gnd |
| topic_facet | Mehrgitterverfahren Finite-Elemente-Methode |
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