## Multi-Grid Methods and ApplicationsMulti-grid methods are the most efficient tools for solving elliptic boundary value problems. The reader finds here an elementary introduction to multi-grid algorithms as well as a comprehensive convergence analysis. One section describes special applications (convection-diffusion equations, singular perturbation problems, eigenvalue problems, etc.). The book also contains a complete presentation of the multi-grid method of the second kind, which has important applications to integral equations (e.g. the "panel method") and to numerous other problems. Readers with a practical interest in multi-grid methods will benefit from this book as well as readers with a more theoretical interest. |

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### Contents

1 | |

Introductory Model Problem | 17 |

General TwoGrid Method | 41 |

General MultiGrid Iteration | 80 |

Nested Iteration Technique | 98 |

Convergence of the TwoGrid Iteration | 112 |

Fourier Analysis | 169 |

Nonlinear MultiGrid Methods | 181 |

Elliptic Systems | 231 |

Eigenvalue Problems and Singular Equations | 252 |

Continuation Techniques | 270 |

Extrapolation and Defect Correction Techniques | 277 |

Local Techniques | 293 |

The MultiGrid Method of the Second Kind | 305 |

Bibliography | 354 |

375 | |

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analogous analysis applied approximation property Assume boundary conditions boundary value problem bounded Brandt choice coarse-grid correction computed contraction number convergence rate damped Jacobi iteration defect correction defined denoted discrete discretisation error eigenvalue problem eigenvectors elliptic equivalent estimate Euclidean norm Example Exercise finite element method formula Gauß-Seidel iteration grid function grid points Hackbusch Hence Hilbert space holds implies inequality integer integral equations interpolation iteration matrix Lemma Li'l linear linear interpolation M-matrix matrix norm multi-grid algorithm multi-grid convergence multi-grid iteration multi-grid method nested iteration nine-point NMGM nonlinear norm Note O(hi obtain operator piecewise Poisson equation positive definite post-smoothing prolongation Proof Proposition prove replaced restriction satisfies scalar product smoothing iteration smoothing process smoothing property spectral norm spectral radius step symmetric Table Theorem tion two-grid iteration V-cycle variant vector y-line yields