Practical Fourier analysis for multigrid methods:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton [u.a.]
Chapman & Hall/CRC
2005
|
| Series: | Numerical insights
4 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | 217 S. graph. Darst. 1 CD-ROM (12 cm) |
| ISBN: | 1584884924 |
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MARC
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Record in the Search Index
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| adam_text | Titel: Practical Fourier analysis for multigrid methods
Autor: Wienands, Roman
Jahr: 2005
Contents
Symbol Description xv
I Practical Application of LFA and xlfa 1
1 INTRODUCTION 3
1.1 SOME NOTATION ....................... 4
1.1.1 Boundary value problems................ 4
1.1.2 Discrete boundary value problems........... 5
1.1.3 Stencil notation...................... 6
1.1.4 Systems of partial differential equations........ 9
1.1.5 Operator versus matrix notation............ 11
1.2 BASIC ITERATIVE SCHEMES................ 12
1.3 A FIRST DISCUSSION OF FOURIER COMPONENTS . . 13
1.3.1 Empirical calculation of convergence factors...... 13
1.3.2 Convergence analysis for the Jacobi method...... 14
1.3.3 Smoothing properties of Jacobi relaxation....... 16
1.4 FROM RESIDUAL CORRECTION TO COARSE-GRID
CORRECTION ......................... 19
1.5 MULTIGRID PRINCIPLE AND COMPONENTS...... 20
1.6 A FIRST LOOK AT THE GRAPHICAL USER INTERFACE 22
2 MAIN FEATURES OF LOCAL FOURIER ANALYSIS FOR
MULTIGRID 29
2.1 THE POWER OF LOCAL FOURIER ANALYSIS...... 29
2.2 BASIC IDEAS.......................... 30
2.2.1 Main goal......................... 30
2.2.2 Necessary simplifications for the discrete problem ... 31
2.2.3 Crucial observation.................... 31
2.2.4 Arising questions..................... 31
2.3 APPLICABILITY OF THE ANALYSIS ........... 32
2.3.1 Type of partial differential equation.......... 33
2.3.2 Type of grid ....................... 33
2.3.3 Type of discretization.................. 34
MULTIGRID AND ITS COMPONENTS IN LFA 35
3.1 MULTIGRID CYCLING .................... 35
3.1.1 Coarse-grid correction operator............. 35
3.1.2 Aliasing of Fourier components............. 36
3.1.3 Correction scheme.................... 37
3.2 FULL MULTIGRID....................... 40
3.3 xlfa FUNCTIONALITY?AN OVERVIEW ......... 42
3.3.1 Menu bar......................... 42
3.3.2 Button bar........................ 43
3.3.3 Parameter display.................... 43
3.3.4 Problem display ..................... 44
3.4 IMPLEMENTED COARSE-GRID CORRECTION
COMPONENTS......................... 44
3.4.1 Discretization and grid structure............ 45
3.4.2 Coarsening strategies................... 46
3.4.3 Coarse-grid operator................... 46
3.4.4 Multigrid cycling..................... 48
3.4.5 Restriction........................ 49
3.4.6 Prolongation....................... 50
3.5 IMPLEMENTED RELAXATIONS .............. 51
3.5.1 Relaxation type and ordering of grid points...... 51
3.5.2 Relaxation methods for systems............. 54
3.5.3 Multistage (MS) relaxations............... 55
USING THE FOURIER ANALYSIS SOFTWARE 57
4.1 CASE STUDIES FOR 2D SCALAR PROBLEMS ...... 59
4.1.1 Anisotropic diffusion equation:
second-order discretization ............... 59
4.1.2 Anisotropic diffusion equation:
fourth-order discretization................ 65
4.1.3 Anisotropic diffusion equation:
Mehrstellen discretization................ 67
4.1.4 Helmholtz equation ................... 69
4.1.5 Biharmonic equation................... 69
4.1.6 Rotated anisotropic diffusion equation......... 70
4.1.7 Convection diffusion equation:
first-order upwind discretization ............ 73
4.1.8 Convection diffusion equation:
higher-order upwind discretization........... 76
4.2 CASE STUDIES FOR 3D SCALAR PROBLEMS ...... 77
4.2.1 Ansiotropic diffusion equation:
second-order discretization ............... 77
4.2.2 Anisotropic diffusion equation:
fourth-order discretization................ 82
4.2.3 Anisotropic diffusion equation:
Mehrstellen discretization................ 82
4.2.4 Helmholtz equation ................... 83
4.2.5 Biharmonic equation................... 83
4.2.6 Convection diffusion equation:
first-order upwind discretization ............ 83
4.3 CASE STUDIES FOR 2D SYSTEMS OF EQUATIONS . . 84
4.3.1 Biharmonic system.................... 84
4.3.2 Stokes equations..................... 86
4.3.3 First-order discretization of the Oseen equations ... 86
4.3.4 Higher-order discretization of the Oseen equations . . 91
4.3.5 Elasticity system..................... 93
4.3.6 A linear shell problem.................. 93
4.4 CREATING NEW APPLICATIONS ............. 94
II The Theory behind LFA 97
5 FOURIER ONE-GRID OR SMOOTHING ANALYSIS 99
5.1 ELEMENTS OF LOCAL FOURIER ANALYSIS ...... 100
5.1.1 Basic definitions..................... 100
5.1.2 Generalization to systems of PDEs........... 102
5.2 HIGH AND LOW FOURIER FREQUENCIES........ 103
5.2.1 Standard and semicoarsening.............. 103
5.2.2 Red-black coarsening and quadrupling......... 104
5.3 SIMPLE RELAXATION METHODS ............. 105
5.3.1 Jacobi relaxation..................... 107
5.3.2 Lexicographic Gauss-Seidel relaxation......... 108
5.3.3 A first definition of the smoothing factor........ 110
5.4 PATTERN RELAXATIONS .................. 113
5.4.1 Red-black Jacobi (RB-JAC) relaxations........ 114
5.4.2 Spaces of 2/i-harmonics................. 115
5.4.3 Auxiliary definitions and relations........... 118
5.4.4 Fourier representation for RB-JAC point relaxation . 120
5.4.5 General definition of the smoothing factor....... 123
5.4.6 Red-black Gauss-Seidel (RB-GS) relaxations ..... 127
5.4.7 Multicolor relaxations.................. 128
5.5 SMOOTHING ANALYSIS FOR SYSTEMS ......... 129
5.5.1 Collective versus decoupled smoothing......... 129
5.5.2 Distributive relaxation.................. 132
5.6 MULTISTAGE (MS) RELAXATIONS ............ 134
5.7 FURTHER RELAXATION METHODS............ 138
5.8 THE MEASURE OF ft-ELLrPTICITY ............ 139
5.8.1 Example 1: anisotropic diffusion equation....... 141
5.8.2 Example 2: convection diffusion equation....... 143
5.8.3 Example 3: Oseen equations .............. 145
6 FOURIER TWO- AND THREE-GRID ANALYSIS 147
6.1 BASIC ASSUMPTIONS .................... 148
6.2 TWO-GRID ANALYSIS FOR 2D SCALAR PROBLEMS . . 149
6.2.1 Spaces of 2/i-harmonics ................. 149
6.2.2 Fourier representation of fine-grid discretization . . . . 151
6.2.3 Fourier representation of restriction .......... 151
6.2.4 Fourier representation of prolongation......... 152
6.2.5 Fourier representation of coarse-grid discretization . . 158
6.2.6 Invariance property of the two-grid operator...... 160
6.2.7 Definition of the two-grid convergence factor ..... 161
6.2.8 Semicoarsening...................... 163
6.3 TWO-GRID ANALYSIS FOR 3D SCALAR PROBLEMS . . 169
6.3.1 Standard coarsening................... 169
6.3.2 Semicoarsening...................... 171
6.4 TWO-GRID ANALYSIS FOR SYSTEMS........... 173
6.5 THREE-GRID ANALYSIS................... 176
6.5.1 Spaces of 4/i-harmonics ................. 177
6.5.2 Invariance property of the three-grid operator..... 179
6.5.3 Definition of three-grid convergence factor....... 180
6.5.4 Generalizations...................... 181
7 FURTHER APPLICATIONS OF LOCAL FOURIER
ANALYSIS 183
7.1 ORDERS OF TRANSFER OPERATORS .......... 184
7.1.1 Polynomial order..................... 184
7.1.2 High- and low-frequency order ............. 185
7.2 SIMPLIFIED FOURIER k-GRID ANALYSIS ........ 187
7.3 CELL-CENTERED MULTIGRID............... 189
7.3.1 Transfer operators.................... 191
7.3.2 Fourier two- and three-grid analysis .......... 192
7.3.3 Orders of transfer operators............... 194
7.3.4 Numerical experiments.................. 195
7.4 FOURIER ANALYSIS FOR MULTIGRID PRECONDITIONED
BYGMRES ........................... 197
7.4.1 Analysis based on the GMRES(m)-polynomial .... 199
7.4.2 Analysis based on the spectrum of the residual
transformation matrix.................. 200
A FOURIER REPRESENTATION OF RELAXATION 203
A.l Two-dimensional case...................... 204
A.2 Three-dimensional case ..................... 204
REFERENCES 207
INDEX 213
|
| any_adam_object | 1 |
| author | Wienands, Roman Joppich, Wolfgang |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| language | English |
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| spelling | Wienands, Roman Verfasser aut Practical Fourier analysis for multigrid methods Roman Wienands ; Wolfgang Joppich Boca Raton [u.a.] Chapman & Hall/CRC 2005 217 S. graph. Darst. 1 CD-ROM (12 cm) txt rdacontent n rdamedia nc rdacarrier Numerical insights 4 Includes bibliographical references and index Fourier, Analyse de Méthodes multigrilles (Analyse numérique) aFourier analysis aMultigrid methods (Numerical analysis) Harmonische Analyse (DE-588)4023453-8 gnd rswk-swf Mehrgitterverfahren (DE-588)4038376-3 gnd rswk-swf Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Harmonische Analyse (DE-588)4023453-8 s Mehrgitterverfahren (DE-588)4038376-3 s Numerisches Verfahren (DE-588)4128130-5 s DE-604 Joppich, Wolfgang Verfasser aut Numerical insights 4 (DE-604)BV012945885 4 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012886588&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Wienands, Roman Joppich, Wolfgang Practical Fourier analysis for multigrid methods Numerical insights Fourier, Analyse de Méthodes multigrilles (Analyse numérique) aFourier analysis aMultigrid methods (Numerical analysis) Harmonische Analyse (DE-588)4023453-8 gnd Mehrgitterverfahren (DE-588)4038376-3 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| subject_GND | (DE-588)4023453-8 (DE-588)4038376-3 (DE-588)4128130-5 |
| title | Practical Fourier analysis for multigrid methods |
| title_auth | Practical Fourier analysis for multigrid methods |
| title_exact_search | Practical Fourier analysis for multigrid methods |
| title_full | Practical Fourier analysis for multigrid methods Roman Wienands ; Wolfgang Joppich |
| title_fullStr | Practical Fourier analysis for multigrid methods Roman Wienands ; Wolfgang Joppich |
| title_full_unstemmed | Practical Fourier analysis for multigrid methods Roman Wienands ; Wolfgang Joppich |
| title_short | Practical Fourier analysis for multigrid methods |
| title_sort | practical fourier analysis for multigrid methods |
| topic | Fourier, Analyse de Méthodes multigrilles (Analyse numérique) aFourier analysis aMultigrid methods (Numerical analysis) Harmonische Analyse (DE-588)4023453-8 gnd Mehrgitterverfahren (DE-588)4038376-3 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| topic_facet | Fourier, Analyse de Méthodes multigrilles (Analyse numérique) aFourier analysis aMultigrid methods (Numerical analysis) Harmonische Analyse Mehrgitterverfahren Numerisches Verfahren |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012886588&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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