Multigrid:
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
San Diego [u.a.]
Elsevier Acad. Press
2007
|
| Ausgabe: | transferred to digital print. |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XV, 631 S. graph. Darst. |
| ISBN: | 012701070X 9780127010700 |
Internformat
MARC
| LEADER | 00000nam a2200000 c 4500 | ||
|---|---|---|---|
| 001 | BV035428956 | ||
| 003 | DE-604 | ||
| 005 | 20210211 | ||
| 007 | t | ||
| 008 | 090408s2007 d||| |||| 00||| eng d | ||
| 020 | |a 012701070X |9 0-12-701070-X | ||
| 020 | |a 9780127010700 |9 978-0-12-701070-0 | ||
| 035 | |a (OCoLC)633667492 | ||
| 035 | |a (DE-599)BVBBV035428956 | ||
| 040 | |a DE-604 |b ger |e rakwb | ||
| 041 | 0 | |a eng | |
| 049 | |a DE-91G |a DE-355 |a DE-20 |a DE-703 |a DE-862 | ||
| 084 | |a SK 920 |0 (DE-625)143272: |2 rvk | ||
| 084 | |a MAT 673f |2 stub | ||
| 100 | 1 | |a Trottenberg, Ulrich |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Multigrid |c U. Trottenberg ; C. W. Oosterlee ; A. Schüller |
| 250 | |a transferred to digital print. | ||
| 264 | 1 | |a San Diego [u.a.] |b Elsevier Acad. Press |c 2007 | |
| 300 | |a XV, 631 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 0 | 7 | |a Mehrgitterverfahren |0 (DE-588)4038376-3 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Mehrgitterverfahren |0 (DE-588)4038376-3 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Oosterlee, Cornelis W. |e Verfasser |4 aut | |
| 700 | 1 | |a Schüller, Anton |e Verfasser |4 aut | |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-0-08-047956-9 |
| 856 | 4 | 2 | |m Digitalisierung UB Regensburg |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017349379&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-017349379 | ||
Datensatz im Suchindex
| DE-BY-FWS_katkey | 739528 |
|---|---|
| _version_ | 1806189821658398720 |
| adam_text | CONTENTS
Preface
xiii
З
З
6
7
9
10
10
12
14
15
15
17
19
20
23
1.6
Intermezzo: Some Basic Facts and Methods
24
1.6.1
Iterative Solvers, Splittings and Preconditioners
24
Basic Multigrid I
28
2.1
Error Smoothing Procedures
28
2.1.1
Jacobi-type Iteration (Relaxation)
29
2.1.2
Smoothing Properties of
ω
-Jacobi
Relaxation
30
2.1.3
Gauss-Seidel-type Iteration (Relaxation)
31
2.1.4
Parallel Properties of Smoothers
33
2.2
Introducing the Two-grid Cycle
34
2.2.1
Iteration by Approximate Solution of the Defect Equation
35
introduction
1.1
Types
ofPDEs
1.2
Grids
and Discretization Approaches
1.2.1
Grids
1.2.2
Discretization Approaches
1.3
Some Notation
1.3.1
Continuous Boundary Value Problems
1.3.2
Discrete Boundary Value Problems
1.3.3
Inner Products and Norms
1.3.4
Stencil Notation
1.4
Poisson s
Equation and Model Problem
1
1.4.1
Matrix Terminology
1.4.2
Poisson
Solvers
1.5
A First Glance at Multigrid
1.5.1
The Two Ingredients of Multigrid
1.5.2
High and Low Frequencies, and Coarse Meshes
1.5.3
From Two Grids to Multigrid
1.5.4
Multigrid Features
1.5.5
Multigrid History
vi MULTIGRID
2.2.2
Coarse
Grid Correction
37
2.2.3
Structure of the Two-grid Operator
39
2.3
Multigrid Components
41
2.3.1
Choices of Coarse Grids
41
2.3.2
Choice of the Coarse Grid Operator
42
2.3.3
Transfer Operators: Restriction
42
2.3.4
Transfer Operators: Interpolation
43
2.4
The Multigrid Cycle
45
2.4.1
Sequences of Grids and Operators
46
2.4.2
Recursive Definition
46
2.4.3
Computational Work
50
2.5
Multigrid Convergence and Efficiency
52
2.5.1
An Efficient 2D Multigrid
Poisson
Solver
52
2.5.2
How to Measure the Multigrid Convergence Factor in
Practice
54
2.5.3
Numerical Efficiency
55
2.6
Full Multigrid
56
2.6.1
Structure of Full Multigrid
57
2.6.2
Computational Work
59
2.6.3
FMG for Poisson s Equation
59
2.7
Further Remarks on Transfer Operators
60
2.8
First Generalizations
62
2.8.1
2D Poisson-like Differential Equations
62
2.8.2
Time-dependent Problems
63
2.8.3
Cartesian Grids in Nonrectangular Domains
66
2.8.4
Multigrid Components for Cell-centered Discretizations
69
2.9
Multigrid in
3D 70
2.9.1
The
3D
Poisson
Problem
70
2.9.2 3D
Multigrid Components
71
2.9.3
Computational Work in
3D 74
Elementary Multigrid Theory
75
3.1
Survey
76
3.2
Why it is Sufficient to Derive Two-grid Convergence Factors
77
3.2.1
ft-Independent Convergence of Multigrid
77
3.2.2
A Theoretical Estimate for Full Multigrid
79
3.3
How to Derive Two-grid Convergence Factors by Rigorous Fourier
Analysis
82
3.3.1
Asymptotic Two-grid Convergence
82
3.3.2
Norms of the Two-grid Operator
83
3.3.3
Results for Multigrid
85
3.3.4
Essential Steps and Details of the Two-grid Analysis
85
3.4
Range of Applicability of the Rigorous Fourier Analysis, Other
Approaches
91
3.4.1
The
3D
Case
91
CONTENTS
vii
3.4.2
Boundary Conditions
93
3.4.3
List of Applications and Limitations
93
3.4.4
Towards Local Fourier Analysis
94
3.4.5
Smoothing and Approximation Property: a Theoretical
Overview
96
4
Local
Fourier
Analysis
98
4.1
Background
99
4.2
Terminology
100
4.3
Smoothing Analysis I
102
4.4
Two-grid Analysis
106
4.5
Smoothing Analysis II
113
4.5.1
Local Fourier Analysis for GS-RB
115
4.6
Some Results, Remarks and Extensions
116
4.6.1
Some Local Fourier Analysis Results for Model Problem
1
117
4.6.2
Additional Remarks
118
4.6.3
Other Coarsening Strategies
121
4.7
ft-Ellipticity
121
4.7.1
The Concept of ft-Ellipticity
123
4.7.2
Smoothing and ft-Ellipticity
126
5
Basic Multigrid
II
130
5.1
Anisotropie
Equations in 2D
131
5.1.1
Failure of Pointwise Relaxation and Standard Coarsening
131
5.1.2
Semicoarsening
133
5.1.3
Line Smoothers
134
5.1.4
Strong Coupling of Unknowns in Two Directions
137
5.1.5
An Example with Varying Coefficients
139
5.2
Anisotropie
Equations in
3D
141
5.2.1
Standard Coarsening for
3D Anisotropie
Problems
143
5.2.2
Point Relaxation for
3D Anisotropie
Problems
145
5.2.3
Further Approaches, Robust Variants
147
5.3
Nonlinear Problems, the Full Approximation Scheme
147
5.3.1
Classical Numerical Methods for Nonlinear PDEs:
an Example
148
5.3.2
Local Linearization
151
5.3.3
Linear Multigrid in Connection with Global Linearization
153
5.3.4
Nonlinear Multigrid: the Full Approximation Scheme
155
5.3.5
Smoothing Analysis: a Simple Example
159
5.3.6
FAS for the Full Potential Equation
160
5.3.7
The (h, ^-Relative Truncation Error and
т
-Extrapolation
163
5.4
Higher Order Discretizations
166
5.4.1
Defect Correction
168
5 4 9
Thf» Mphrctfllfn
Пі
«ra-pti nation for
Poisson
s
Enuation
172
viii MULTIGRID
5.5 Domains
with
Geometric
Singularities
174
5.6
Boundary Conditions and
Singular Systems
177
5.6.1
General Treatment of Boundary Conditions in Multigrid
178
5.6.2
Neumann Boundary Conditions
179
5.6.3
Periodic Boundary Conditions and Global Constraints
183
5.6.4
General Treatment of Singular Systems
185
5.7
Finite Volume Discretization and Curvilinear Grids
187
5.8
General Grid Structures
190
6
Parallel Multigrid in Practice
193
6.1
Parallelism of Multigrid Components
194
6.1.1
Parallel Components for Poisson s Equation
195
6.1.2
Parallel Complexity
196
6.2
Grid Partitioning
197
6.2.1
Parallel Systems, Processes and Basic Rules for
Parallelization
198
6.2.2
Grid Partitioning for Jacobi and Red-Black Relaxation
199
6.2.3
Speed-up and Parallel Efficiency
204
6.2.4
A Simple Communication Model
206
6.2.5
Scalability and the Boundary-volume Effect
207
6.3
Grid Partitioning and Multigrid
208
6.3.1
Two-grid and Basic Multigrid Considerations
208
6.3.2
Multigrid and the Very Coarse Grids
211
6.3.3
Boundary-volume Effect and Scalability in the Multigrid
Context
214
6.3.4
Programming Parallel Systems
215
6.4
Parallel Line Smoothers
216
6.4.1 1
D
Reduction (or Cyclic Reduction) Methods
217
6.4.2
Cyclic Reduction and Grid Partitioning
218
6.4.3
Parallel Plane Relaxation
220
6.5
Modifications of Multigrid and Related Approaches
221
6.5.1
Domain Decomposition Methods: a Brief Survey
221
6.5.2
Multigrid Related Parallel Approaches
225
7
More Advanced Multigrid
227
7.1
The Convection-Diffusion Equation: Discretization I
228
7.1.1
The ID Case
228
7.1.2
Central Differencing
230
7.1.3
First-order Upwind Discretizations and Artificial Viscosity
233
7.2
The Convection-Diffusion Equation: Multigrid I
234
7.2.1
Smoothers for First-order Upwind Discretizations
235
7.2.2
Variable Coefficients
237
7.2.3
The Coarse Grid Correction
239
7.3
The Convection-Diffusion Equation: Discretization II
243
7.3.1
Combining Central and Upwind Differencing
243
7.3.2
Higher Order Upwind Discretizations
244
CONTENTS ¡x
7.4
The Convection-Diffusion Equation: Multigrid II
249
7.4.1
Line Smoothers for Higher Order Upwind Discretizations
249
7.4.2
Multistage Smoothers
253
7.5
ILU
Smoothing Methods
256
7.5.1
Idea of
ILU
Smoothing
257
7.5.2
Stencil Notation
259
7.5.3
ILU
Smoothing for the
Anisotropie
Diffusion Equation
261
7.5.4
A Particularly Robust
ILU
Smoother
262
7.6
Problems with Mixed Derivatives
263
7.6.1
Standard Smoothing and Coarse Grid Correction
264
7.6.2
ILU
Smoothing
267
7.7
Problems with Jumping Coefficients and Galerkin Coarse Grid
Operators
268
7.7.1
Jumping Coefficients
269
7.7.2
Multigrid for Problems with Jumping Coefficients
271
7.7.3
Operator-dependent Interpolation
272
7.7.4
The Galerkin Coarse Grid Operator
273
7.7.5
Further Remarks on Galerkin-based Coarsening
277
7.8
Multigrid as a Preconditioner (Acceleration of Multigrid by Iterant
Recombination)
278
7.8.1
The Recirculating Convection-Diffusion Problem Revisited
278
7.8.2
Multigrid Acceleration by Iterant Recombination
280
7.8.3
Krylov Subspace Iteration and Multigrid Preconditioning
282
7.8.4
Multigrid: Solver versus Preconditioner
287
8
Multigrid for Systems of Equations
289
8.1
Notation and Introductory Remarks
290
8.2
Multigrid Components
293
8.2.1
Restriction
293
8.2.2
Interpolation of Coarse Grid Corrections
294
8.2.3
Orders of Restriction and Interpolation
295
8.2.4
Solution on the Coarsest Grid
295
8.2.5
Smoothers
295
8.2.6
Treatment of Boundary Conditions
296
8.3
LEA for Systems of PDEs
297
8.3.1
Smoothing Analysis
297
8.3.2
Smoothing and A-Ellipticity
300
8.4
The Biharmonic System
301
8.4.1
A Simple Example: GS-LEX Smoothing
302
8.4.2
Treatment of Boundary Conditions
303
8.4.3
Multigrid Convergence
304
8.5
A Linear Shell Problem 307
8.5.1
Decoupled Smoothing
308
8.5.2
Collective versus Decoupled Smoothing
310
8.5.3
Level-dependent Smoothing
311
MULTIGRID
8.6
Introduction to Incompressible
Navier-Stokes
Equations
312
8.6.1
Equations and Boundary Conditions
312
8.6.2
Survey 314
8.6.3
The Checkerboard Instability
315
8.7
Incompressible Navier-Stokes Equations: Staggered Discretizations
316
8.7.1
Transfer Operators
318
8.7.2
Box Smoothing
320
8.7.3
Distributive Smoothing
323
8.8
Incompressible Navier-Stokes Equations: Nonstaggered
Discretizations
326
8.8.1
Artificial Pressure Terms
327
8.8.2
Box Smoothing
328
8.8.3
Alternative Formulations
331
8.8.4
Flux Splitting Concepts
333
8.8.5
Flux Difference Splitting and Multigrid: Examples
338
8.9
Compressible
Euler
Equations
343
8.9.1
Introduction
345
8.9.2
Finite Volume Discretization and Appropriate Smoothers
347
8.9.3
Some Examples
349
8.9.4
Multistage Smoothers in CFD Applications
352
8.9.5
Towards Compressible Navier-Stokes Equations
354
Adaptive Multigrid
356
9.1
A Simple Example and Some Notation
357
9.1.1
A Simple Example
357
9.1.2
Hierarchy of Grids
360
9.2
The Idea of Adaptive Multigrid
361
9.2.1
The Two-grid Case
361
9.2.2
From Two Grids to Multigrid
363
9.2.3
Self-adaptive Full Multigrid
364
9.3
Adaptive Multigrid and the Composite Grid
366
9.3.1
Conservative Discretization at Interfaces of Refinement
Areas
367
9.3.2
Conservative Interpolation
369
9.3.3
The Adaptive Multigrid Cycle
372
9.3.4
Further Approaches
373
9.4
Refinement Criteria and Optimal Grids
373
9.4.1
Refinement Criteria
374
9.4.2
Optimal Grids and Computational Effort
378
9.5
Parallel Adaptive Multigrid Q7Q
9.5.1
Parallelization Aspects
379
9.5.2
Distribution of Locally Refined Grids
381
9.6
Some Practical Results
382
9.6.1
2D
Euler
Flow Around an Airfoil
382
9.6.2
2D and
3D
Incompressible Navier-Stokes Equations
385
CONTENTS xi
10
Some More Multigrid Applications
389
10.1
Multigrid for Poisson-type Equations on the Surface of the Sphere
389
10.1.1
Discretization
390
10.1.2
Specific Multigrid Components on the Surface of a Sphere
391
10.1.3
A Segment Relaxation
393
10.2
Multigrid and Continuation Methods
395
10.2.1
The
Bratu
Problem
396
10.2.2
Continuation Techniques
397
10.2.3
Multigrid for Continuation Methods
398
10.2.4
The Indefinite Helmholtz Equation
400
10.3
Generation of Boundary Fitted Grids
400
10.3.1
Grid Generation Based on
Poisson s
Equation
401
10.3.2
Multigrid Solution of Grid Generation Equations
401
10.3.3
Grid Generation with the Biharmonic Equation
403
10.3.4
Examples
404
10.4
L¡
SS: a
Generic Multigrid Software Package
404
10.4.1
Pre-
and Postprocessing Components
405
10.4.2
The Multigrid Solver
406
10.5
Multigrid in the Aerodynamic Industry
407
10.5.1
Numerical Challenges for the
3D
Compressible
Navier—
Stokes Equations
408
10.5.2
FLOWer
409
10.5.3
Fluid-Structure Coupling
410
10.6
How to Continue with Multigrid
411
Appendixes
A An Introduction to Algebraic Multigrid (by Klaus
Stuben) 413
A.I Introduction
413
A.
1.1
Geometric Multigrid
414
A.
1.2
Algebraic Multigrid
415
A.
1.3
An Example
418
A.
1.4
Overview of the Appendix
420
A.2 Theoretical Basis and Notation
422
A.2.
1
Formal Algebraic Multigrid Components
422
A.2.2 Further Notation
425
A.2.3 Limit Case of Direct Solvers
427
A.2.4 The Variational Principle for Positive Definite Problems
430
A.3 Algebraic Smoothing
432
A.3.
1
Basic Norms and Smooth Eigenvectors
433
A.3.2 Smoothing Property of Relaxation
434
A.3.3 Interpretation of Algebraically Smooth Error
438
A.4 Postsmoothing and Two-level Convergence
444
A.4.1 Convergence Estimate
445
xii
MULTIGRID
A.4.2 Direct Interpolation
447
A.4.3 Indirect Interpolation
459
A.5 Presmoothing and Two-level Convergence
461
A.5.1 Convergence using Mere F-Smoothing
461
A.5.2 Convergence using Full Smoothing
468
A.6 Limits of the Theory
469
A.7 The Algebraic Multigrid Algorithm
472
A.7.1 Coarsening
473
A.7.
2
Interpolation
479
A.7.3 Algebraic Multigrid as Preconditioner
484
A.8 Applications
485
A.8.1 Default Settings and Notation
486
A.8.
2
Poisson-like Problems
487
A.8.3 Computational Fluid Dynamics
494
A.8.4 Problems with Discontinuous Coefficients
503
A.8.5 Further Model Problems
513
A.9 Aggregation-based Algebraic Multigrid
522
A.9.
1
Rescaling of the Galerkin Operator
524
A.9.2 Smoothed Aggregation
526
A.
10
Further Developments and Conclusions
528
В
Subspace Correction Methods and Multigrid Theory {by Peter Oswald)
533
B.I Introduction
533
B.2 Space Splittings
536
B.3 Convergence Theory
551
B.4 Multigrid Applications
556
B.5 A Domain Decomposition Example
561
С
Recent Developments in Multigrid Efficiency in Computational Fluid
Dynamics {by Achi Brandt)
573
C.I Introduction
573
C.2 Table of Difficulties and Possible Solutions
574
References
590
Index
613
|
| any_adam_object | 1 |
| author | Trottenberg, Ulrich Oosterlee, Cornelis W. Schüller, Anton |
| author_facet | Trottenberg, Ulrich Oosterlee, Cornelis W. Schüller, Anton |
| author_role | aut aut aut |
| author_sort | Trottenberg, Ulrich |
| author_variant | u t ut c w o cw cwo a s as |
| building | Verbundindex |
| bvnumber | BV035428956 |
| classification_rvk | SK 920 |
| classification_tum | MAT 673f |
| ctrlnum | (OCoLC)633667492 (DE-599)BVBBV035428956 |
| discipline | Mathematik |
| edition | transferred to digital print. |
| format | Book |
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| id | DE-604.BV035428956 |
| illustrated | Illustrated |
| indexdate | 2024-08-01T14:54:03Z |
| institution | BVB |
| isbn | 012701070X 9780127010700 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-017349379 |
| oclc_num | 633667492 |
| open_access_boolean | |
| owner | DE-91G DE-BY-TUM DE-355 DE-BY-UBR DE-20 DE-703 DE-862 DE-BY-FWS |
| owner_facet | DE-91G DE-BY-TUM DE-355 DE-BY-UBR DE-20 DE-703 DE-862 DE-BY-FWS |
| physical | XV, 631 S. graph. Darst. |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | Elsevier Acad. Press |
| record_format | marc |
| spellingShingle | Trottenberg, Ulrich Oosterlee, Cornelis W. Schüller, Anton Multigrid Mehrgitterverfahren (DE-588)4038376-3 gnd |
| subject_GND | (DE-588)4038376-3 |
| title | Multigrid |
| title_auth | Multigrid |
| title_exact_search | Multigrid |
| title_full | Multigrid U. Trottenberg ; C. W. Oosterlee ; A. Schüller |
| title_fullStr | Multigrid U. Trottenberg ; C. W. Oosterlee ; A. Schüller |
| title_full_unstemmed | Multigrid U. Trottenberg ; C. W. Oosterlee ; A. Schüller |
| title_short | Multigrid |
| title_sort | multigrid |
| topic | Mehrgitterverfahren (DE-588)4038376-3 gnd |
| topic_facet | Mehrgitterverfahren |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017349379&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT trottenbergulrich multigrid AT oosterleecornelisw multigrid AT schulleranton multigrid |