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Practical Fourier Analysis for Multigrid Methods [Kõva köide]

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(University of Cologne, Germany), (University of Applied Science, Sankt Augustin, Germany)
  • Formaat: Hardback, 234 pages, kõrgus x laius: 229x152 mm, kaal: 494 g, 27 Tables, black and white; 48 Illustrations, black and white
  • Sari: Numerical Insights
  • Ilmumisaeg: 28-Oct-2004
  • Kirjastus: Chapman & Hall/CRC
  • ISBN-10: 1584884924
  • ISBN-13: 9781584884927
Teised raamatud teemal:
Practical Fourier Analysis for Multigrid MethodsSuurem pilt
  • Formaat: Hardback, 234 pages, kõrgus x laius: 229x152 mm, kaal: 494 g, 27 Tables, black and white; 48 Illustrations, black and white
  • Sari: Numerical Insights
  • Ilmumisaeg: 28-Oct-2004
  • Kirjastus: Chapman & Hall/CRC
  • ISBN-10: 1584884924
  • ISBN-13: 9781584884927
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781584884927.html
Märksõnad:
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.

Arvustused

" a rather unique distinguished feature is the accompanying LFA software, which is not often found in other multigrid books. This book presents a thorough and systematic description of [ the subject]. Two main features of this book are an extensive selection of problems of different kinds and an accompanying user-friendly software that can perform rather complex local Fourier analysis by just a few mouse clicks. It was a joy reading this book, and I am happy to have it as a valuable addition to my multigrid book collection." -Mathematics of Computation, April 2007

"This book enables understanding of basic principles of multigrid and local Fourier analysis, allowing investigation of real multigrid effects." -Wilhelm Heinrichs, Zentralblatt MATH

Symbol Description xv
I Practical Application of LFA and xlfa 1(96)
1 INTRODUCTION
3(26)
1.1 SOME NOTATION
4(8)
1.1.1 Boundary value problems
4(1)
1.1.2 Discrete boundary value problems
5(1)
1.1.3 Stencil notation
6(3)
1.1.4 Systems of partial differential equations
9(2)
1.1.5 Operator versus matrix notation
11(1)
1.2 BASIC ITERATIVE SCHEMES
12(1)
1.3 A FIRST DISCUSSION OF FOURIER COMPONENTS
13(6)
1.3.1 Empirical calculation of convergence factors
13(1)
1.3.2 Convergence analysis for the Jacobi method
14(2)
1.3.3 Smoothing properties of Jacobi relaxation
16(3)
1.4 FROM RESIDUAL CORRECTION TO COARSE-GRID CORRECTION
19(1)
1.5 MULTIGRID PRINCIPLE AND COMPONENTS
20(2)
1.6 A FIRST LOOK AT THE GRAPHICAL USER INTERFACE
22(7)
2 MAIN FEATURES OF LOCAL FOURIER ANALYSIS FOR MULTIGRID
29(6)
2.1 THE POWER OF LOCAL FOURIER ANALYSIS
29(1)
2.2 BASIC IDEAS
30(2)
2.2.1 Main goal
30(1)
2.2.2 Necessary simplifications for the discrete problem
31(1)
2.2.3 Crucial observation
31(1)
2.2.4 Arising questions
31(1)
2.3 APPLICABILITY OF THE ANALYSIS
32(3)
2.3.1 Type of partial differential equation
33(1)
2.3.2 Type of grid
33(1)
2.3.3 Type of discretization
34(1)
3 MULTIGRID AND ITS COMPONENTS IN LFA
35(22)
3.1 MULTIGRID CYCLING
35(5)
3.1.1 Coarse-grid correction operator
35(1)
3.1.2 Aliasing of Fourier components
36(1)
3.1.3 Correction scheme
37(3)
3.2 FULL MULTIGRID
40(2)
3.3 xlfa FUNCTIONALITY- AN OVERVIEW
42(2)
3.3.1 Menu bar
42(1)
3.3.2 Button bar
43(1)
3.3.3 Parameter display
43(1)
3.3.4 Problem display
44(1)
3.4 IMPLEMENTED COARSE-GRID CORRECTION COMPONENTS
44(7)
3.4.1 Discretization and grid structure
45(1)
3.4.2 Coarsening strategies
46(1)
3.4.3 Coarse-grid operator
46(2)
3.4.4 Multigrid cycling
48(1)
3.4.5 Restriction
49(1)
3.4.6 Prolongation
50(1)
3.5 IMPLEMENTED RELAXATIONS
51(6)
3.5.1 Relaxation type and ordering of grid points
51(3)
3.5.2 Relaxation methods for systems
54(1)
3.5.3 Multistage (MS) relaxations
55(2)
4 USING THE FOURIER ANALYSIS SOFTWARE
57(40)
4.1 CASE STUDIES FOR 2D SCALAR PROBLEMS
59(18)
4.1.1 Anisotropic diffusion equation: second-order discretization
59(6)
4.1.2 Anisotropic diffusion equation: fourth-order discretization
65(2)
4.1.3 Anisotropic diffusion equation: Mehrstellen discretization
67(2)
4.1.4 Helmholtz equation
69(1)
4.1.5 Biharmonic equation
69(1)
4.1.6 Rotated anisotropic diffusion equation
70(3)
4.1.7 Convection diffusion equation: first-order upwind discretization
73(3)
4.1.8 Convection diffusion equation: higher-order upwind discretization
76(1)
4.2 CASE STUDIES FOR 3D SCALAR PROBLEMS
77(7)
4.2.1 Ansiotropic diffusion equation: second-order discretization
77(5)
4.2.2 Anisotropic diffusion equation: fourth-order discretization
82(1)
4.2.3 Anisotropic diffusion equation: Mehrstellen discretization
82(1)
4.2.4 Helmholtz equation
83(1)
4.2.5 Biharmonic equation
83(1)
4.2.6 Convection diffusion equation: first-order upwind discretization
83(1)
4.3 CASE STUDIES FOR 2D SYSTEMS OF EQUATIONS
84(10)
4.3.1 Biharmonic system
84(2)
4.3.2 Stokes equations
86(1)
4.3.3 First-order discretization of the Oseen equations
86(5)
4.3.4 Higher-order discretization of the Oseen equations
91(2)
4.3.5 Elasticity system
93(1)
4.3.6 A linear shell problem
93(1)
4.4 CREATING NEW APPLICATIONS
94(3)
II The Theory behind LFA 97(106)
5 FOURIER ONE-GRID OR SMOOTHING ANALYSIS
99(48)
5.1 ELEMENTS OF LOCAL FOURIER ANALYSIS
100(3)
5.1.1 Basic definitions
100(2)
5.1.2 Generalization to systems of PDEs
102(1)
5.2 HIGH AND LOW FOURIER FREQUENCIES
103(2)
5.2.1 Standard and semicoarsening
103(1)
5.2.2 Red-black coarsening and quadrupling
104(1)
5.3 SIMPLE RELAXATION METHODS
105(8)
5.3.1 Jacobi relaxation
107(1)
5.3.2 Lexicographic Gauss-Seidel relaxation
108(2)
5.3.3 A first definition of the smoothing factor
110(3)
5.4 PATTERN RELAXATIONS
113(16)
5.4.1 Red-black Jacobi (RB-JAC) relaxations
114(1)
5.4.2 Spaces of 2h-harmonics
115(3)
5.4.3 Auxiliary definitions and relations
118(2)
5.4.4 Fourier representation for RB-JAC point relaxation
120(3)
5.4.5 General definition of the smoothing factor
123(4)
5.4.6 Red-black Gauss-Seidel (RB-GS) relaxations
127(1)
5.4.7 Multicolor relaxations
128(1)
5.5 SMOOTHING ANALYSIS FOR SYSTEMS
129(5)
5.5.1 Collective versus decoupled smoothing
129(3)
5.5.2 Distributive relaxation
132(2)
5.6 MULTISTAGE (MS) RELAXATIONS
134(4)
5.7 FURTHER RELAXATION METHODS
138(1)
5.8 THE MEASURE OF h-ELLIPTICITY
139(8)
5.8.1 Example 1: anisotropic diffusion equation
141(2)
5.8.2 Example 2: convection diffusion equation
143(2)
5.8.3 Example 3: Oseen equations
145(2)
6 FOURIER TWO- AND THREE-GRID ANALYSIS
147(36)
6.1 BASIC ASSUMPTIONS
148(1)
6.2 TWO-GRID ANALYSIS FOR 2D SCALAR PROBLEMS
149(20)
6.2.1 Spaces of 2h-harmonics
149(2)
6.2.2 Fourier representation of fine-grid discretization
151(1)
6.2.3 Fourier representation of restriction
151(1)
6.2.4 Fourier representation of prolongation
152(6)
6.2.5 Fourier representation of coarse-grid discretization
158(2)
6.2.6 Invariance property of the two-grid operator
160(1)
6.2.7 Definition of the two-grid convergence factor
161(2)
6.2.8 Semicoarsening
163(6)
6.3 TWO-GRID ANALYSIS FOR 3D SCALAR PROBLEMS
169(4)
6.3.1 Standard coarsening
169(2)
6.3.2 Semicoarsening
171(2)
6.4 TWO-GRID ANALYSIS FOR SYSTEMS
173(3)
6.5 THREE-GRID ANALYSIS
176(7)
6.5.1 Spaces of 4h-harmonics
177(2)
6.5.2 Invariance property of the three-grid operator
179(1)
6.5.3 Definition of three-grid convergence factor
180(1)
6.5.4 Generalizations
181(2)
7 FURTHER APPLICATIONS OF LOCAL FOURIER ANALYSIS
183(20)
7.1 ORDERS OF TRANSFER OPERATORS
184(3)
7.1.1 Polynomial order
184(1)
7.1.2 High- and low-frequency order
185(2)
7.2 SIMPLIFIED FOURIER k-GRID ANALYSIS
187(2)
7.3 CELL-CENTERED MULTIGRID
189(8)
7.3.1 Transfer operators
191(1)
7.3.2 Fourier two- and three-grid analysis
192(2)
7.3.3 Orders of transfer operators
194(1)
7.3.4 Numerical experiments
195(2)
7.4 FOURIER ANALYSIS FOR MULTIGRID PRECONDITIONED BY GMRES
197(6)
7.4.1 Analysis based on the GMRES(m)-polynomial
199(1)
7.4.2 Analysis based on the spectrum of the residual transformation matrix
200(3)
A FOURIER REPRESENTATION OF RELAXATION 203(4)
A.1 Two-dimensional case
204(1)
A.2 Three-dimensional case
204(3)
REFERENCES 207(6)
INDEX 213


Roman Wienands, Wolfgang Joppich