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Multigrid. With guest contributions by A. Brandt, P. Oswald, K. Stüben. (English) Zbl 0976.65106
Orlando, FL: Academic Press. xv, 631 p. (2001).
Multigrid methods are now an efficient tool for the solution of large systems of equations that arise from the discretization of partial differential equations. They make it possible that problems can be treated on a workstation of medium size also in cases where the discretization leads to more than a million unknowns.
This book on multigrid method adresses to people who want to apply multigrid methods to their actual problems. It focuses on the construction of smoothing procedures and on the transfer between the grids that the practitioner has to design for his actual partial differential equation. To this end, emphasis is put on the local Fourier analysis. If one has a differential equation with constant coefficients on a rectangular grid and if boundary conditions are ignored, then the Fourier modes are eigenfunctions of the stencil. The same holds for many relaxation procedures, and the eigenvalues provide information on the smoothing properties. In this way, Sobolev spaces and mesh dependent norms, i.e. the typical tools of a rigorous theory, are circumvented here.
The first two chapters give the introduction to partial differential equations and to the treatment by multigrid methods. The concept of the \(W\)-cycle, the \(V\)-cycle and full multigrid (nested iteration) are described. Chapter 3 on elements of multigrid theory is very brief. Local Fourier analysis is introduced in Chapter 4. Chapters 5 and 7 describe how to deal with more complicated equations, when e.g. anisotropies, nonlinearities, higher order derivatives, or convection terms are present. The remaining chapters are concerned with parallelization, systems of equations, adaptivity, and software packages.
Finally there is a detailed introduction of 120 pages to algebraic multigrid by Klaus Stüben, a smaller contribution on subspace iteration by Peter Oswald, and a table of “difficulties and possible solutions” by Achi Brandt, who is one of the pioneers in multigrid methods.
The book contains the material with many details. Many figures and tables with results of convergence rates provide the reader witha picture of multigrid algorithms for difference methods. The book covers more than 600 pages. So it is clear that it is not considered as a quick introduction to the method, but more as a “reference book” for a practitioner who may conclude from the comparisons in the text how involved his partial differential equation is in the view of multigrid and how much effort he must put into its solution.

65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65N06 Finite difference methods for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
35J25 Boundary value problems for second-order elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
65H10 Numerical computation of solutions to systems of equations
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
65Y05 Parallel numerical computation
65Y15 Packaged methods for numerical algorithms
35-04 Software, source code, etc. for problems pertaining to partial differential equations