## Practical Fourier Analysis for Multigrid MethodsBefore 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 |

MAIN FEATURES OF LOCAL FOURIER ANALYSIS FOR MULTIGRID | 29 |

MULTIGRID AND ITS COMPONENTS IN LFA | 35 |

USING THE FOURIER ANALYSIS SOFTWARE | 57 |

The Theory behind LFA | 97 |

FOURIER ONEGRID OR SMOOTHING ANALYSIS | 99 |

FOURIER TWO AND THREEGRID ANALYSIS | 147 |

FURTHER APPLICATIONS OF LOCAL FOURIER ANALYSIS | 183 |

FOURIER REPRESENTATION OF RELAXATION | 203 |

207 | |

213 | |

### Common terms and phrases

accompanying software anisotropic diffusion equation applied approximation biharmonic biharmonic equation bilinear interpolation boundary conditions cell-centered Chapter coarse coarsening strategy compare with Section convection diffusion equation corresponding d-dimensional Definition Dirichlet boundary conditions discrete operators Eh Lh eigenvalues error example Figure fine-grid Fourier analysis Fourier components Fourier frequencies Fourier representation Fourier symbols given grid points Helmholtz equation high-frequency infinite grid iteration Jacobi Laplacian linear low-frequency matrix measure of h-ellipticity mesh size h multigrid components multigrid convergence multigrid methods multistage norm optimal parameters pattern relaxations point relaxation Poisson equation polynomial order problem RB-JAC relaxation red-black relaxation methods residual restriction scalar second-order semicoarsening Smax smoothing factor smoothing properties spaces of 2h-harmonics spectral norm spectral radius standard coarsening stencil subspace systems of equations systems of PDEs three-dimensional tion transfer operators two-dimensional two-grid analysis two-grid Operator upwind discretization V-cycle variant xlfa yields