## A multigrid tutorial. 2nd ed.(English)Zbl 0958.65128

Philadelphia, PA: SIAM, Society for Industrial and Applied Mathematics. 193 p. (2000).
The second edition of “A multigrid tutorial” provides an extension of the first author’s tutorial (1987; Zbl 0659.65095) in order to incorporate some advanced material about multigrid algorithms, and two experts have been won for this purpose. Chapters 6 to 10 are completely new. Nevertheless, the lively and clear style of the original tutorial is also found in the new part of the book.
The basic multigrid principles for solving linear equations arising from the discretization of elliptic equations are presented in Chapters 1-5. The discussion focuses on one- and two-dimensional Poisson equation although numerical results for different problems are included. The very slow convergence of the classical relaxation methods is illustrated, before the remedy by the multigrid idea and the interplay between algebraic arguments and the Fourier analysis is presented. The standard choices of the smoothing procedures and transfer operators are very efficient for most problems. The chapter on implementation provides also hints how to recognize which side has to be improved in case of difficult problems.
Chapter 6 is concerned with nonlinear problems. In the nonlinear case the transfer between the grids is more involved. Selected Applications in Chapter 7 show for example that the compatibility condition of the data of Neumann problems are inherited to coarser grids in a simple way. Chapter 8 treats Algebraic Multigrid and Chapter 9 multigrid methods with local refinements. The book concludes with a brief introduction of finite elements.
This “Tutorial” is a good introduction into multigrid methods, and advanced knowledges on the theory of partial differential equations are not required. On the other hand, the authors say clearly that it is not their intention to present theoretical results.
In this context, the “Tutorial” is a useful book that profits from the authors’ lively style, and it will find its audience.
Reviewer: D.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 35J25 Boundary value problems for second-order elliptic equations 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs

Zbl 0659.65095
Full Text: