Multi-grid methods and applications: with 48 tab.
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2003
|
| Edition: | 2. print. |
| Series: | Springer series in computational mathematics
4 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 377 S. graph. Darst. |
| ISBN: | 3540127615 0387127615 |
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MARC
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| 245 | 1 | 0 | |a Multi-grid methods and applications |b with 48 tab. |c Wolfgang Hackbusch |
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| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2003 | |
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| 490 | 1 | |a Springer series in computational mathematics |v 4 | |
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Record in the Search Index
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| adam_text | Contents
1. Preliminaries 1
1.1 Introduction 1
1.2 Notation 2
1.3 Some Elements from Linear Algebra 5
1.3.1 Analysis of Iterative Processes 5
1.3.2 Norms 6
1.3.3 Symmetric Matrices 8
1.4 Some Elements from Functional Analysis 9
1.4.1 Continuous Functions and Holder Spaces 9
1.4.2 Sobolev Spaces 10
1.4.3 Dual Spaces 10
1.4.4 Hilbert Scales 11
1.4.5 Sesqui-Linear Forms 14
1.5 Exercises 15
2. Introductory Model Problem 17
2.1 One-Dimensional Model Problem 17
2.2 Smoothing Effect of Classical Iterations 18
2.3 Two-Grid Method 21
2.4 Convergence of the Two-Grid Iteration 25
2.5 Multi-Grid Method 30
2.6 Comments 34
2.6.1 Variants of the Two- and Multi-Grid Iteration 34
2.6.2 Contraction Numbers with Respect to Other Norms 35
2.6.3 Measuring of the Smoothing Effect 36
2.6.4 The Multi-Grid Iteration as a New Type of Iteration 37
2.6.5 Historical Comments 38
2.7 Exercises 39
3. General Two-Grid Method 41
3.1 The Boundary Value Problem and Its Discretisation 41
3.1.1 The Boundary Value Problem 41
VIII Contents
3.1.2 Finite Difference Discretisation 43
3.1.3 Conforming Finite Element Discretisation 46
3.2 Two-Grid Algorithm 48
3.3 Smoothing Iterations 49
3.5.1 GauB-Seidel Iteration 49
3.3.2 Modifications of Jacobi s Iteration 52
3.3.3 Blockwise Iterations 54
3.3.4 ADI as a Smoothing Procedure 55
3.3.5 Semi-Iterative Methods 56
3.3.6 Conjugate Gradient Methods 57
3.3.7 Other Smoothing Iterations 58
3.4 Prolongation 58
3.4.1 Fine and Coarse Grid 58
3.4.2 Piecewise Linear Interpolation as Prolongation 59
3.4.3 Interpolation of Higher Order 61
3.4.4 Modifications 63
3.5 Restriction . . . T 64
3.6 Canonical Prolongations and Restrictions for Finite Element
Equations 66
3.7 Coarse-Grid Matrix 68
3.8 Comments 69
3.8.1 Separated Discrete Boundary Conditions 69
3.8.2 Construction of a Finite Element Hierarchy 71
3.8.3 A Further Variant of the Two-Grid Iteration 74
3.8.4 The Case of Q^t £ G, 76
3.8.5 Comment on Weighted Restrictions 77
3.9 Exercises 78
4. General Multi-Grid Iteration 80
4.1 Multi-Grid Algorithm 80
4.2 Convergence of the Multi-Grid Iteration 85
4.3 Computational Work 86
4.4 Examples of Multi-Grid Iterations and Numerical Results ... 89
4.4.1 A Multi-Grid Program for Solving Poisson s Equation .... 89
4.4.2 Poisson s Equation in a General Domain 94
4.4.3 Other Boundary Conditions 95
4.4.4 Further Poisson Solvers 96
5. Nested Iteration Technique 98
5.1 Algorithm 98
5.2 Analysis of the Nested Iteration 100
5.3 Computational Work and Efficiency 103
5.4 Nested Iteration with Extrapolation 104
5.5 Numerical Example 107
Contents IX
5.6 Comments 109
5.7 Exercises 110
6. Convergence of the Two-Grid Iteration 112
6.1 Sufficient Conditions for Convergence 112
6.1.1 The Two-Grid Iteration Matrix 112
6.1.2 A Factorisation of M, 113
6.1.3 Smoothing and Approximation Properties 113
6.1.4 A General Convergence Theorem 114
6.1.5 The Case of v2 * 0 115
6.2 The Smoothing Property 116
6.2.1 Preparatory Lemmata 117
6.2.2 Criteria 120
6.2.3 The Smoothing Property of the Jacobi Iterations 122
6.2.3.1 The Simple Damped Jacobi Iteration 122
6.2.3.2 The Damped Jacobi Iteration (3.3.6) 123
6.2.3.3 Jacobi s Iteration with Squared Matrix 123
6.2.3.4 The Case of Non-Hermitian L, 124
6.2.4 The Smoothing Property of the GauB-Seidel Iteration . . . .125
6.2.4.1 The 2-Cyclic GauB-Seidel Iteration 125
6.2.4.2 The Lexicographical GauB-Seidel Iteration 128
6.2.4.3 The Symmetric GauB-Seidel Iteration 129
6.2.4.4 Estimates with Respect to Other Norms 130
6.2.4.5 The Case of Non-Hermitian L, 131
6.2.5 The Smoothing Property of Semi-Iterative Methods 131
6.2.5.1 Richardson s Iteration 131
6.2.5.2 The Accelerated Symmetric GauB-Seidel Iteration 133
6.2.5.3 The Alternating Direction Method 134
6.3 The Approximation Property 134
6.3.1 The Approximation Property of Finite Element Equations . .135
6.3.1.1 The Definition of the Problem 135
6.3.1.2 The Standard Case 137
6.3.1.3 The Less Regular Case 140
6.3.1.4 The Very Regular Case 143
6.3.1.5 The Smoothing Property Revisited 144
6.3.2 The Approximation Property of Finite Difference
Schemes 145
6.3.2.1 The Discrete Regularity 145
6.3.2.2 Criteria 147
6.4 The Symmetric Case 150
6.4.1 Quantitative Analysis 150
6.4.2 Estimates Without Regularity Assumptions 154
6.5 Comments 155
6.6 Exercises 157
X Contents
7. Convergence of the Multi-Grid Iteration 160
7.1 The General Convergence Theorem 160
7.2 The Symmetric Case 164
7.3 Comments 167
7.4 Exercises 167
8. Fourier Analysis 169
8.1 Model Examples 169
8.1.1 Poisson s Equation in a Square 169
8.1.2 Periodic Boundary Conditions 173
8.1.3 The Case of Q = Rd 175
8.2 Local Mode Analysis 176
8.2.1 Neglect of the Boundary Conditions 177
8.2.2 Treatment of Variable Coefficients . 177
8.2.3 Idealisation of the Coarse-Grid Correction and Smoothing
Rates 178
9. Nonlinear Multi-Grid Methods 181
9.1 The Nonlinear Problem 181
9.2 Newton-Multi-Grid Iteration 182
9.3 The Nonlinear Multi-Grid Iteration 183
9.3.1 Nonlinear Smoothing Iterations 184
9.3.2 The Nonlinear Two-Grid Iteration 185
9.3.3 The Nonlinear Multi-Grid Algorithm 187
9.3.4 The Nonlinear Nested Iteration 188
9.4 Numerical Example: Natural Convection in an
Enclosed Cavity 189
9.5 Convergence Analysis 192
9.5.1 Convergence of the Nonlinear Two-Grid Iteration 192
9.5.2 Convergence of the Nonlinear Multi-Grid Iteration 196
9.5.3 Analysis of the Nonlinear Nested Iteration 198
9.6 Exercises 199
10. Singular Perturbation Problems 201
10.1 Anisotropic Elliptic Equations 201
10.1.1 A Model Problem 201
10.1.2 Smoothing with Respect to Alternating Directions 204
10.1.3 The Incomplete LU Decomposition as a Smoother 205
10.2 Indefinite Problems 209
10.2.1 Strongly Positive Definite Problems 209
10.2.2 Weakly Indefinite Problems 210
10.3 Interface Problems 212
Contents XI
10.3.1 A One-Dimension Model Example 212
10.3.2 Two-Dimensional Interface Problems 215
10.4 Convection Diffusion Equation 217
10.4.1 The Continuous Problem 217
10.4.2 Stable Discretisations 218
10.4.3 Multi-Grid Algorithms 220
10.4.3.1 The Case of the Stable Discretisation (5b) 220
10.4.3.2 Wesseling s Multi-Grid Algorithm 222
10.4.3.3 The Case of Scheme (5 a) with Artificial Viscosity 223
10.5 Comments 226
10.6 Exercises 229
11. Elliptic Systems 231
11.1 Examples 231
11.1.1 Systems of Coupled Elliptic Equations 231
11.1.2 Examples: Systems of Cauchy-Riemann, Stokes, and
Navier-Stokes 233
11.1.3 Ellipticity of a General System 234
11.1.4 Variational Formulation 235
11.1.5 Discretisation 236
11.2 A Multi-Grid Approach to a General System 237
11.2.1 Appropriate Norms and Scales 237
11.2.1.1 The Continuous Case 237
11.2.1.2 Discrete Spaces 238
11.2.1.3 Approximation Property 239
11.2.2 A Smoothing Iteration for a General System 242
11.2.3 Nonlinear Systems 245
11.3 The Distributive Relaxation Smoother 245
11.4 An Elliptic Problem Augmented by an Algebraic Equation . .248
11.4.1 The Linear Case 248
11.4.2 The Nonlinear Case 250
12. Eigenvalue Problems and Singular Equations 252
12.1 Discussion of the Problems 252
12.1.1 Eigenvalue Problem 252
12.1.2 Singular Equations 253
12.2 Reformulation of the Problems 255
12.2.1 Reformulation of the Eigenvalue Problem as a Nonlinear
Equation 255
12.2.2 Reformulation of the Singular Equation 256
12.3 Direct Multi-Grid Approach to Eigenvalue Problems 257
12.3.1 Derivation of a Two-Grid Iteration 257
12.3.2 Modifications and Generalisations 260
XII Contents
12.3.3 Multi-Grid Iteration for the Eigenvalue Problem and the
Nested Iteration 261
12.3.4 Ritz Projection 264
12.3.5 Direct Multi-Grid Solution of the Singular Equation 265
12.3.6 Simultaneous Solution of n Singular Equations 266
12.4 The Generalised Eigenvalue Problem 267
12.5 The Nonlinear Eigenvalue Problem 269
13. Continuation Techniques 270
13.1 Continuous Continuation Problem 270
13.2 Modified Nested Iteration 271
13.2.1 Treatment of Turning Points 271
13.2.2 Modified Nested Iteration 272
13.2.3 Modifications 274
13.2.4 Frozen Truncation Error Technique 275
13.3 Parabolic Initial-Boundary Value Problems 276
14. Extrapolation and Defect Correction Techniques 277
14.1 Extrapolation 277
14.1.1 Richardson s Extrapolation 277
14.1.2 Truncation Error Extrapolation 278
14.1.3 r-Extrapolation 278
14.2 Defect Correction Techniques 282
14.2.1 The Iterative Defect Correction 282
14.2.2 The Defect Correction as a Finite Process 283
14.3 Combinations of the Multi-Grid Method and the Defect
Correction Principle 284
14.3.1 The Multi-Grid Algorithm as a Secondary Iteration 284
14.3.2 The Defect Correction with Additional Smoothing 285
14.3.3 The Defect Correction Inside the Multi-Grid Process 287
14.3.3.1 Algorithms 287
14.3.3.2 Convergence Analysis of the Multi-Grid Iteration with
Defect Correction 288
14.3.3.3 Error Estimates of the Limit m, 289
14.3.3.4 Extensions 291
14.3.4 An Application to the Convection Diffusion Equation . . . .292
15. Local Techniques 293
15.1 The Local Defect Correction 293
15.2 A Multi-Grid Iteration with a Local Grid Refinement . . . .297
15.3 Domain Decomposition Methods 299
15.3.1 Domain Decomposition 299
Contents XIII
15.3.2 The Schwarz Iteration 300
15.3.3 Multi-Grid Methods with Domain Decompositions 302
16. The Multi-Grid Method of the Second Kind 305
16.1 Fredholm s Integral Equation of the Second Kind 305
16.1.1 Historical Comments 305
16.1.2 Fredholm s Integral Equation of the Second Kind 305
16.1.3 Discretisation 306
16.2 Multi-Grid Iteration 309
16.2.1 First Variant 309
16.2.1.1 Algorithm 309
16.2.1.2 Computational Work 310
16.2.1.3 Convergence Analysis 311
16.2.2 Second Variant 313
16.2.2.1 Algorithm 313
16.2.2.2 Computational Work 314
16.2.2.3 Convergence Analysis 314
16.2.3 Third Variant 315
16.2.3.1 Algorithm 315
16.2.3.2 Computational Work 315
16.2.3.3 Convergence Analysis 315
16.3 Numerical Results 316
16.3.1 Comparison of the Three Variants 316
16.3.2 The Integral Equation Method for Elliptic Problems 319
16.4 Nested Iteration 321
16.4.1 Algorithm 321
16.4.2 Accuracy 322
16.4.3 Nested Iteration with Nystrom s Interpolation 322
16.5 Comments and Modifications 324
16.5.1 More than One Picard Iteration .• 324
16.5.2 Other Smoothing Iterations 324
16.5.3 Application of the Integral Equation Method to the
Calculation of Circulatory Flow around an Aerofoil 325
16.5.4 Construction of the Coarse-Grid Discretisation 327
16.5.5 Further Modifications 327
16.6 The Nonlinear Equation of the Second Kind 328
16.7 Nonlinear Multi-Grid Method of the Second Kind 330
16.7.1 Two-Grid Iteration 330
16.7.2 Multi-Grid Iteration 331
16.7.3 Nested Iteration 333
16.7.4 Convergence 333
16.7.5 Computational Work 334
16.8 Application to Nonlinear Elliptic Equations 335
16.8.1 Reformulation of the Continuous Boundary Value Problem . .335
XIV Contents
16.8.2 Discretisation 336
16.8.3 Computational Work 337
16.8.4 Numerical Example 338
16.9 Multi-Grid Methods with Iterative Computation of Jf;(u() . . .338
16.9.1 Notation 338
16.9.2 Algorithm 339
16.9.3 Convergence Analysis 341
16.10 Applications to Elliptic Boundary Value and Eigenvalue
Problems 345
16.10.1 Elliptic Boundary Value Problems 345
16.10.2 Eigenvalue Problems 347
16.10.3 First Example: Steklov Eigenvalue Problem 348
16.10.4 Second Example: Nonlinear Eigenvalue Problem 349
16.10.5 Elliptic Boundary Value Problems (Revisited) 349
16.11 Application to Parabolic Problems 351
16.11.1 Time-Periodic Parabolic Problems 351
16.11.2 Optimal Control Problems for Parabolic Equations 352
Bibliography 354
Subject Index 375
|
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| ctrlnum | (OCoLC)248851208 (DE-599)BVBBV017129704 |
| discipline | Mathematik |
| edition | 2. print. |
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| id | DE-604.BV017129704 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T19:14:03Z |
| institution | BVB |
| isbn | 3540127615 0387127615 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-010327448 |
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| physical | XIV, 377 S. graph. Darst. |
| publishDate | 2003 |
| publishDateSearch | 2003 |
| publishDateSort | 2003 |
| publisher | Springer |
| record_format | marc |
| series | Springer series in computational mathematics |
| series2 | Springer series in computational mathematics |
| spelling | Hackbusch, Wolfgang 1948- Verfasser (DE-588)115588582 aut Multi-grid methods and applications with 48 tab. Wolfgang Hackbusch 2. print. Berlin [u.a.] Springer 2003 XIV, 377 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Springer series in computational mathematics 4 Mehrgitterverfahren Mehrgitterverfahren (DE-588)4038376-3 gnd rswk-swf Mehrschrittverfahren (DE-588)4198527-8 gnd rswk-swf Mehrschrittverfahren (DE-588)4198527-8 s DE-604 Mehrgitterverfahren (DE-588)4038376-3 s 1\p DE-604 Springer series in computational mathematics 4 (DE-604)BV000012004 4 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010327448&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Hackbusch, Wolfgang 1948- Multi-grid methods and applications with 48 tab. Springer series in computational mathematics Mehrgitterverfahren Mehrgitterverfahren (DE-588)4038376-3 gnd Mehrschrittverfahren (DE-588)4198527-8 gnd |
| subject_GND | (DE-588)4038376-3 (DE-588)4198527-8 |
| title | Multi-grid methods and applications with 48 tab. |
| title_auth | Multi-grid methods and applications with 48 tab. |
| title_exact_search | Multi-grid methods and applications with 48 tab. |
| title_full | Multi-grid methods and applications with 48 tab. Wolfgang Hackbusch |
| title_fullStr | Multi-grid methods and applications with 48 tab. Wolfgang Hackbusch |
| title_full_unstemmed | Multi-grid methods and applications with 48 tab. Wolfgang Hackbusch |
| title_short | Multi-grid methods and applications |
| title_sort | multi grid methods and applications with 48 tab |
| title_sub | with 48 tab. |
| topic | Mehrgitterverfahren Mehrgitterverfahren (DE-588)4038376-3 gnd Mehrschrittverfahren (DE-588)4198527-8 gnd |
| topic_facet | Mehrgitterverfahren Mehrschrittverfahren |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010327448&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000012004 |
| work_keys_str_mv | AT hackbuschwolfgang multigridmethodsandapplicationswith48tab |