Multigrid techniques. 1984 guide with applications to fluid dynamics.
Revised ed.

*(English)*Zbl 1227.65121
Classics in Applied Mathematics 67. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) (ISBN 978-1-611970-74-6/pbk; 978-1-611970-75-3/ebook). xxi, 218 p. (2011).

“The amount of computational work should be proportional to the amount of real physical changes in the computer system. Stalling numerical processes must be wrong.” This golden rule is the starting point of this survey on principles in multigrid methods. The first part “Stages in developing fast solvers” reports on different ways of smoothing processes, prolongations and restrictions. The full multigrid-scheme is described. The principle that the convergence rate should only depend on the equation in the interior of the domain and not on the boundaries leads to the definition of the smoothing rate and to the local mode analysis. Part II: “Advanced techniques and insights” is concerned with the full approximation scheme, local refinements and grid-adaption, the dealgebraization of multigrid and the practical role of rigorous analysis. Part III: “Applications to fluid dynamics” has multigrid algorithms with distributive relaxation for staggered finite-difference approximations in its center. The book is written for researchers who have already some experience with multigrid algorithms and who want to hear from an expert’s insight rather than to have a repetition of a systematic introduction into multigrid algorithms and its variants.

The authors state that only a few modifications of the 1984 guide by the first author were made, and they caution the reader that this edition of the guide falls short of representing later multigrid developments. Therefore, the reviewer adds only a few short remarks to his review of A. Brandt [Multigrid techniques: 1984 guide with applications to fluid dynamics. GMD-Stud. 85 (1984; Zbl 0581.76033)]. The authors distinguish the period before the 1984 guide and the period from 1984 to present. They emphasize that multigrid analysis of many papers in the second period is restricted to the verification of a \(Cn\) bound for the computation effort and that such a bound with an unknown constant \(C\) does not provide information on the quality of the multigrid concept. They suggest to add numerical experience to their favorite, i.e., local mode analysis. Of course, one has to take care that the numerical experience is based on an appropriate realization and not on a poor one. Now a Matlab code for a test cycle is provided in an appendix.

The authors state that only a few modifications of the 1984 guide by the first author were made, and they caution the reader that this edition of the guide falls short of representing later multigrid developments. Therefore, the reviewer adds only a few short remarks to his review of A. Brandt [Multigrid techniques: 1984 guide with applications to fluid dynamics. GMD-Stud. 85 (1984; Zbl 0581.76033)]. The authors distinguish the period before the 1984 guide and the period from 1984 to present. They emphasize that multigrid analysis of many papers in the second period is restricted to the verification of a \(Cn\) bound for the computation effort and that such a bound with an unknown constant \(C\) does not provide information on the quality of the multigrid concept. They suggest to add numerical experience to their favorite, i.e., local mode analysis. Of course, one has to take care that the numerical experience is based on an appropriate realization and not on a poor one. Now a Matlab code for a test cycle is provided in an appendix.

Reviewer: Dietrich Braess (Bochum)

##### MSC:

65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

65F10 | Iterative numerical methods for linear systems |

76M25 | Other numerical methods (fluid mechanics) (MSC2010) |