Some nonoverlapping domain decomposition methods. (English) Zbl 0913.65115

Xu, Jinchao; Zou, Jun

SIAM Rev. 40, No. 4, 857-914 (1998).

The rough contents of the present paper are as follows: 1. Introduction. 2. Algebraic aspects of preconditioning techniques. 3. A model problem and outline. 4. Preliminaries of Sobolev spaces and finite element spaces. 5. Substructuring methods. 6. Neumann-Neumann methods. 7. Some other interface preconditioners. 8. Methods with inexact subdomain solvers. 9. Implementation issues. The paper contains a list of 87 references.
The authors give an overview upon several well-known nonoverlapping domain decomposition methods for solving large sparse systems of linear equations which arise from finite element discretizations of second-order selfadjoint elliptic problems defined on a bounded Lipschitz domain.
They focus on the \(h\)-versions of finite element methods and carry out a unified and coherent presentation of the theoretical aspects of these methods.

Reviewer: C.I.Gheorghiu (Cluj-Napoca)

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
65F35	Numerical computation of matrix norms, conditioning, scaling
35J25	Boundary value problems for second-order elliptic equations
65F10	Iterative numerical methods for linear systems

Keywords:

nonoverlapping domain decomposition; Schur complement; local-global and global-local techniques; jumps in coefficients; substructuring; balancing methods; preconditioning; Neumann-Neumann methods; large sparse systems; finite element; \(h\)-versions