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**Domain decomposition methods – algorithms and theory.**
*(English)*
Zbl 1069.65138

Springer Series in Computational Mathematics 34. Berlin: Springer (ISBN 3-540-20696-5/hbk). xv, 450 p. (2005).

This book unifies the results from a number of papers by the authors and their coworkers over the past two decades, and complements them by new insights and some background. The distinguishing feature of this book is a comprehensive and rigorous treatment of convergence bounds based on the theory of finite elements. Most of the book deals with linear, symmetric positive definite problems. The bibliography is quite complete for the fields covered until early 2004. The book belongs on the desk of all specialists involved in domain decomposition and substructuring, particulary those more on the theoretical side.

Chapter 1 presents the basic ideas of domain decomposition, based on a pragmatic approach to discretized elliptic problems and the concept of traces and fluxes. This chapter also previews the key results later in the book and provides motivating examples. In Chapter 2, the authors treat the abstract theory of Schwarz methods by methods of linear algebra in abstract normed spaces. Chapter 3 deals with overlapping methods and a variety of coarse spaces, including smoothed aggregation and partition of unity approaches. Chapter 4 presents substructuring methods, where the problem is reduced to interfaces between the subdomains, including a treatment of discrete harmonic extensions and Schur complements.

Chapter 5 then deals with the solution of the problem reduced to the interfaces by additive Schwarz type edge, face, and vertex based algorithms, and a variety of coarse spaces. Methods based on Neumann-Neumann and Lagrange multiplier approaches are studied in Chapter 6; these matrix-based methods treat the substructures as a whole. Chapter 7 is devoted to high-order methods, and Chapter 8 deals with special aspects arizing in elasticity problems. Chapter 9 contains a survey of modified methods for saddle point problems, particularly the Stokes problems. Chapter 10 is devoted to detailed treatment of domain decomposition in \(H(\text{div})\) and \(H(\text{curl})\) spaces. Nonsymmetric, indefinite, and nonlinear problems are the subject of Chapter 10. Finally, the appendices contain background material from Sobolev spaces, finite elements, and numerical linear algebra.

Chapter 1 presents the basic ideas of domain decomposition, based on a pragmatic approach to discretized elliptic problems and the concept of traces and fluxes. This chapter also previews the key results later in the book and provides motivating examples. In Chapter 2, the authors treat the abstract theory of Schwarz methods by methods of linear algebra in abstract normed spaces. Chapter 3 deals with overlapping methods and a variety of coarse spaces, including smoothed aggregation and partition of unity approaches. Chapter 4 presents substructuring methods, where the problem is reduced to interfaces between the subdomains, including a treatment of discrete harmonic extensions and Schur complements.

Chapter 5 then deals with the solution of the problem reduced to the interfaces by additive Schwarz type edge, face, and vertex based algorithms, and a variety of coarse spaces. Methods based on Neumann-Neumann and Lagrange multiplier approaches are studied in Chapter 6; these matrix-based methods treat the substructures as a whole. Chapter 7 is devoted to high-order methods, and Chapter 8 deals with special aspects arizing in elasticity problems. Chapter 9 contains a survey of modified methods for saddle point problems, particularly the Stokes problems. Chapter 10 is devoted to detailed treatment of domain decomposition in \(H(\text{div})\) and \(H(\text{curl})\) spaces. Nonsymmetric, indefinite, and nonlinear problems are the subject of Chapter 10. Finally, the appendices contain background material from Sobolev spaces, finite elements, and numerical linear algebra.

Reviewer: Jan Mandel (Denver)

### MSC:

65N55 | Multigrid methods; domain decomposition 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 |

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

74B05 | Classical linear elasticity |

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

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

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

35Q30 | Navier-Stokes equations |