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**Discretization methods and iterative solvers based on domain decomposition.**
*(English)*
Zbl 0966.65097

Lecture Notes in Computational Science and Engineering. 17. Berlin: Springer. x, 199 p. (2001).

The author presents both discretization techniques and iterative solvers based on the domain decomposition approach. An abstract framework for domain decomposition (DD) methods is given and an analysis is carried out for new techniques of special interest. Some optimal estimates for the methods considered are established. The presented results from numerical experiments confirm the theoretical predictions. The book consists of two main chapters a quite full bibliography of 160 items.

Chapter 1 concerns special discretization methods based on DD techniques. The decomposition of geometrical complex structures into subdomains of simple shape, optimal discretizations and mortar finite element methods are of special interest. Abstract conditions on the Lagrange multiplier spaces such that the obtained nonconforming discretization schemes yield optimal a priori results are derived. In the new approach the locality of the support of the nodal basis functions of the constrained space can be presented. Examples with several crosspoints, a corner singularity, discontinuous coefficients, a rotating geometry and a linear elasticity problems are considered and the corresponding numerical results are presented and discussed..

Chapter 2 concerns iterative solution techniques based on DD. A brief overview on general Schwarz methods and multigrid techniques is presented. Examples for the standard \(H^1\)-case illustrate overlapping, non-overlapping and hierarchical decomposition techniques. A polylogarithmical bound independent of the jumps of the coefficients across the subdomain boundaries of the author’s substracting method is established. The author presents a combination of dual basis functions for the Lagrange multiplier space with standard multigrid techniques for symmetric positive definite systems. Dirichlet-Neumann type algorithm for the mortar method and a multigrid method for the saddle point formulation are studied. A block diagonal smoother and a smoother reflecting the saddle point structure are discussed. Numerical results which confirm the theoretical predictions are presented.

Chapter 1 concerns special discretization methods based on DD techniques. The decomposition of geometrical complex structures into subdomains of simple shape, optimal discretizations and mortar finite element methods are of special interest. Abstract conditions on the Lagrange multiplier spaces such that the obtained nonconforming discretization schemes yield optimal a priori results are derived. In the new approach the locality of the support of the nodal basis functions of the constrained space can be presented. Examples with several crosspoints, a corner singularity, discontinuous coefficients, a rotating geometry and a linear elasticity problems are considered and the corresponding numerical results are presented and discussed..

Chapter 2 concerns iterative solution techniques based on DD. A brief overview on general Schwarz methods and multigrid techniques is presented. Examples for the standard \(H^1\)-case illustrate overlapping, non-overlapping and hierarchical decomposition techniques. A polylogarithmical bound independent of the jumps of the coefficients across the subdomain boundaries of the author’s substracting method is established. The author presents a combination of dual basis functions for the Lagrange multiplier space with standard multigrid techniques for symmetric positive definite systems. Dirichlet-Neumann type algorithm for the mortar method and a multigrid method for the saddle point formulation are studied. A block diagonal smoother and a smoother reflecting the saddle point structure are discussed. Numerical results which confirm the theoretical predictions are presented.

Reviewer: Krassimir Georgiev (Sofia)

### MSC:

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

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

35R05 | PDEs with low regular coefficients and/or low regular data |

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 |