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**Domain decomposition. Parallel multilevel methods for elliptic partial differential equations.**
*(English)*
Zbl 0857.65126

Cambridge: Cambridge University Press. xii, 224 p. (1996).

The book presents domain decomposition algorithms, their implementation and analysis. Multigrid and multilevel algorithms are treated as particular examples of the broad class of domain decomposition methods. Discussions of the implementation of domain decomposition methods on parallel computers are also included. Many numerical examples are included to demonstrate the behaviour of the numerical methods. A mathematical framework for the analysis and complete understanding of the methods is carefully developed.

Each chapter is divided in three parts. The first one contains a detailed description of the algorithms. This is followed by a discussion of implementation issues. The third section contains more advanced mathematical ideas which are useful to understand why the algorithms work. The software used for numerical calculations is the PETSc (Portable Extensionable Toolkit for Scientific computation) developed by two of the authors.

Approximately the first half of the book is devoted to Schwarz methods, including the classical multigrid methods. The first chapter is devoted to a thorough discussion of one-level overlapping Schwarz methods and introduces the notations used in the book. The second chapter presents two-level algorithms. The third chapter demonstrates the intertwining relationship between multigrid and domain decomposition algorithms. In chapter 4 a large class of methods that use nonoverlapping subdomains, i.e. substructuring methods is introduced. Chapter 5 contains an introduction to the mathematical convergence analysis of domain decomposition algorithms. Appendix 1 contains a brief introduction to the concepts of preconditioners and Krylov subspace methods. The used software is introduced in Appendix 2.

The book is a valuable resource for all people interested in numerical computational methods and parallel computing. It is intended for a general audience of engineers, scientists, mathematicians, and computer scientists. The mathematics needed to understand the material are multivariable calculus and linear algebra plus a basic understanding of finite discretizations and the iterative solution of linear systems. Since the text is intended for a wide audience, each section does not contain an abstract theory, but develops the motivation and the description of the algorithms, and has notes and references that direct the reader to more detailed information on topics of particular interest.

Each chapter is divided in three parts. The first one contains a detailed description of the algorithms. This is followed by a discussion of implementation issues. The third section contains more advanced mathematical ideas which are useful to understand why the algorithms work. The software used for numerical calculations is the PETSc (Portable Extensionable Toolkit for Scientific computation) developed by two of the authors.

Approximately the first half of the book is devoted to Schwarz methods, including the classical multigrid methods. The first chapter is devoted to a thorough discussion of one-level overlapping Schwarz methods and introduces the notations used in the book. The second chapter presents two-level algorithms. The third chapter demonstrates the intertwining relationship between multigrid and domain decomposition algorithms. In chapter 4 a large class of methods that use nonoverlapping subdomains, i.e. substructuring methods is introduced. Chapter 5 contains an introduction to the mathematical convergence analysis of domain decomposition algorithms. Appendix 1 contains a brief introduction to the concepts of preconditioners and Krylov subspace methods. The used software is introduced in Appendix 2.

The book is a valuable resource for all people interested in numerical computational methods and parallel computing. It is intended for a general audience of engineers, scientists, mathematicians, and computer scientists. The mathematics needed to understand the material are multivariable calculus and linear algebra plus a basic understanding of finite discretizations and the iterative solution of linear systems. Since the text is intended for a wide audience, each section does not contain an abstract theory, but develops the motivation and the description of the algorithms, and has notes and references that direct the reader to more detailed information on topics of particular interest.

Reviewer: D.Petcu (Timişoara)

### MSC:

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

65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |

65F10 | Iterative numerical methods for linear systems |

35J25 | Boundary value problems for second-order elliptic equations |

65N06 | Finite difference methods for boundary value problems involving PDEs |

00A06 | Mathematics for nonmathematicians (engineering, social sciences, etc.) |

65F35 | Numerical computation of matrix norms, conditioning, scaling |

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

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