# Practical Fourier Analysis for Multigrid Methods (Google eBook)

CRC Press, Oct 28, 2004 - Mathematics - 240 pages
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.

### What people are saying -Write a review

We haven't found any reviews in the usual places.

### 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 Copyright