Practical Fourier Analysis for Multigrid Methods (Google eBook)

Front Cover
CRC Press, Oct 28, 2004 - Mathematics - 240 pages
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Before applying multigrid methods to a project, mathematicians, scientists, and engineers need to answer questions related to the quality of convergence, whether a development will pay out, whether multigrid will work for a particular application, and what the numerical properties are. Practical Fourier Analysis for Multigrid Methods uses a detailed and systematic description of local Fourier k-grid (k=1,2,3) analysis for general systems of partial differential equations to provide a framework that answers these questions.

This volume contains software that confirms written statements about convergence and efficiency of algorithms and is easily adapted to new applications. Providing theoretical background and the linkage between theory and practice, the text and software quickly combine learning by reading and learning by doing. The book enables understanding of basic principles of multigrid and local Fourier analysis, and also describes the theory important to those who need to delve deeper into the details of the subject.

The first chapter delivers an explanation of concepts, including Fourier components and multigrid principles. Chapter 2 highlights the basic elements of local Fourier analysis and the limits to this approach. Chapter 3 examines multigrid methods and components, supported by a user-friendly GUI. Chapter 4 provides case studies for two- and three-dimensional problems. Chapters 5 and 6 detail the mathematics embedded within the software system. Chapter 7 presents recent developments and further applications of local Fourier analysis for multigrid methods.
  

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Contents

INTRODUCTION
3
11 SOME NOTATION
4
112 Discrete boundary value problems
5
113 Stencil notation
6
114 Systems of partial differential equations
9
115 Operator versus matrix notation
11
12 BASIC ITERATIVE SCHEMES
12
13 A FIRST DISCUSSION OF FOURIER COMPONENTS
13
522 Redblack coarsening and quadrupling
104
53 SIMPLE RELAXATION METHODS
105
531 Jacobi relaxation
107
532 Lexicographic GaussSeidel relaxation
108
533 A first definition of the smoothing factor
110
54 PATTERN RELAXATIONS
113
541 Redblack Jacobi RBJAC relaxations
114
542 Spaces of 2hharmonics
115

132 Convergence analysis for the Jacobi method
14
133 Smoothing properties of Jacobi relaxation
16
14 FROM RESIDUAL CORRECTION TO COARSEGRID CORRECTION
19
15 MULTIGRID PRINCIPLE AND COMPONENTS
20
16 A FIRST LOOK AT THE GRAPHICAL USER INTERFACE
22
MAIN FEATURES OF LOCAL FOURIER ANALYSIS FOR MULTIGRID
29
22 BASIC IDEAS
30
222 Necessary simplifications for the discrete problem
31
23 APPLICABILITY OF THE ANALYSIS
32
231 Type of partial differential equation
33
233 Type of discretization
34
MULTIGRID AND ITS COMPONENTS IN LFA
35
312 Aliasing of Fourier components
36
313 Correction scheme
37
32 FULL MULTIGRID
40
33 xlfa FUNCTIONALITYAN OVERVIEW
42
332 Button bar
43
334 Problem display
44
341 Discretization and grid structure
45
342 Coarsening strategies
46
344 Multigrid cycling
48
345 Restriction
49
346 Prolongation
50
35 IMPLEMENTED RELAXATIONS
51
352 Relaxation methods for systems
54
353 Multistage MS relaxations
55
USING THE FOURIER ANALYSIS SOFTWARE
57
41 CASE STUDIES FOR 2D SCALAR PROBLEMS
59
fourthorder discretization
65
Mehrstellen discretization
67
414 Helmholtz equation
69
416 Rotated anisotropic diffusion equation
70
firstorder upwind discretization
73
higherorder upwind discretization
76
42 CASE STUDIES FOR 3D SCALAR PROBLEMS
77
fourthorder discretization
82
424 Helmholtz equation
83
43 CASE STUDIES FOR 2D SYSTEMS OF EQUATIONS
84
432 Stokes equations
86
434 Higherorder discretization of the Oseen equations
91
435 Elasticity system
93
44 CREATING NEW APPLICATIONS
94
The Theory behind LFA
97
FOURIER ONEGRID OR SMOOTHING ANALYSIS
99
51 ELEMENTS OF LOCAL FOURIER ANALYSIS
100
512 Generalization to systems of PDEs
102
52 HIGH AND LOW FOURIER FREQUENCIES
103
543 Auxiliary definitions and relations
118
544 Fourier representation for RBJAC point relaxation
120
545 General definition of the smoothing factor
123
546 Redblack GaussSeidel RBGS relaxations
127
547 Multicolor relaxations
128
55 SMOOTHING ANALYSIS FOR SYSTEMS
129
552 Distributive relaxation
132
56 MULTISTAGE MS RELAXATIONS
134
57 FURTHER RELAXATION METHODS
138
58 THE MEASURE OF hELLIPTICITY
139
anisotropic diffusion equation
141
convection diffusion equation
143
Oseen equations
145
FOURIER TWO AND THREEGRID ANALYSIS
147
61 BASIC ASSUMPTIONS
148
62 TWOGRID ANALYSIS FOR 2D SCALAR PROBLEMS
149
622 Fourier representation of finegrid discretization
151
624 Fourier representation of prolongation
152
625 Fourier representation of coarsegrid discretization
158
626 Invariance property of the twogrid operator
160
627 Definition of the twogrid convergence factor
161
628 Semicoarsening
163
63 TWOGRID ANALYSIS FOR 3D SCALAR PROBLEMS
169
632 Semicoarsening
171
64 TWOGRID ANALYSIS FOR SYSTEMS
173
65 THREEGRID ANALYSIS
176
651 Spaces of 4hharmonics
177
652 Invariance property of the threegrid operator
179
653 Definition of threegrid convergence factor
180
654 Generalizations
181
FURTHER APPLICATIONS OF LOCAL FOURIER ANALYSIS
183
71 ORDERS OF TRANSFER OPERATORS
184
712 High and lowfrequency order
185
72 SIMPLIFIED FOURIER kGRID ANALYSIS
187
73 CELLCENTERED MULTIGRID
189
731 Transfer operators
191
732 Fourier two and threegrid analysis
192
733 Orders of transfer operators
194
734 Numerical experiments
195
74 FOURIER ANALYSIS FOR MULTIGRID PRECONDITIONED BY GMRES
197
741 Analysis based on the GMRESmpolynomial
199
742 Analysis based on the spectrum of the residual transformation matrix
200
FOURIER REPRESENTATION OF RELAXATION
203
A1 Twodimensional case
204
REFERENCES
207
Index
213
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