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Multigrid methods for process simulation. (English) Zbl 0801.65095

Computational Microelectronics. Wien: Springer-Verlag. xvii, 309 p. (1993).
There are numerous difficulties in the application of multigrid methods to real-life problems, since practical problems usually do not resemble much the Poisson equation on a square grid. This book contains a comprehensive treatment of practical multigrid techniques for nonlinear elliptic and parabolic equations and systems on uniform rectangular grids. The main motivation and application are taken from semiconductor modeling.
The book starts with a comprehensive overview of standard multigrid techniques for difference equations on uniform grids. The authors treat in detail the basic algorithm development, full multigrid, refined grids, MLAT, and FAC, block approaches to systems of partial differential equations, and mode analysis for model problems and as a heuristic technique to estimate multigrid performance. The first part of the book is concluded by mesh partitioning and parallelization.
The bulk of the book is then devoted to highly nonlinear evolution problems such as arise in semiconductor modeling. The difference schemes and the algorithms are developed in full detail with special attention to the selection of the various components and parameters. A large number of well documented experimental results illustrates the methods and provides practical guidance for applications. There are no attempts at analysis besides the standard heuristics of difference equations and truncation errors, and no theorems or proofs.
Although the book does explain the basic principles well, it is not meant to be a tutorial for the novice. It is a very good monograph and reference for the specialist who needs to apply multigrid methods to difficult practical problems on rectangular grids with refinements.
Reviewer: J.Mandel (Denver)

MSC:

65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65N06 Finite difference methods for boundary value problems involving PDEs
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
35J65 Nonlinear boundary value problems for linear elliptic equations
35Q60 PDEs in connection with optics and electromagnetic theory
35K55 Nonlinear parabolic equations
78A55 Technical applications of optics and electromagnetic theory

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