Practical Fourier Analysis for Multigrid Methods

Practical Fourier Analysis for Multigrid Methods

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Description

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.show more

Product details

  • Hardback | 240 pages
  • 160 x 236.2 x 20.3mm | 476.28g
  • Taylor & Francis Inc
  • Chapman & Hall/CRC
  • Boca Raton, FL, United States
  • English
  • New.
  • 48 black & white illustrations, 27 black & white tables
  • 1584884924
  • 9781584884927

Table of contents

PRACTICAL APPLICATION OF LFA AND xlfa Introduction Some Notation Basic Iterative Schemes A First Discussion of Fourier Components From Residual Correction to Coarse-Grid Correction Multigrid Principle and Components A First Look at the Graphical User Interface Main Features of Local Fourier Analysis for Multigrid The Power of Local Fourier Analysis Basic Ideas Applicability of the Analysis Multigrid and Its Components in LFA Multigrid Cycling Full Multigrid xlfa Functionality-An Overview Implemented Coarse-Grid Correction Components Implemented Relaxations Using the Fourier Analysis Software Case Studies for 2D Scalar Problems Case Studies for 3D Scalar Problems Case Studies for 2D SYSTEMS of Equations Creating New Applications THE THEORY BEHIND LFA Fourier One-Grid or Smoothing Analysis Elements of Local Fourier Analysis High and Low Fourier Frequencies Simple Relaxation Methods Pattern Relaxations Smoothing Analysis for Systems Multistage (MS) Relaxations Further Relaxation Methods The Measure of h-Ellipticity Fourier Two- and Three-Grid Analysis Basic Assumptions Two-Grid Analysis for 2D Scalar Problems Two-Grid Analysis for 3D Scalar Problems Two-Grid Analysis for Systems Three-Grid Analysis Further Applications of Local Fourier Analysis Orders of Transfer Operators Simplified Fourier k-Grid Analysis Cell-Centered Multigrid Fourier Analysis for Multigrid Preconditioned by GMRES APPENDIX Fourier Representation of Relaxation Two-Dimensional Case Three-Dimensional Caseshow more