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Towards a unified theory of domain decomposition algorithms for elliptic problems. (English) Zbl 0719.65084
Domain decomposition methods for partial differential equations, Proc. 3rd Int. Symp. Houston/TX (USA) 1989, 3-21 (1990).
[For the entire collection see Zbl 0695.00026.]
The aim of the paper is to introduce an abstract method and to develop a simple framework for the analysis of its rate of convergence.
After introducing two elliptic model problems and certain finite element methods the paper reviews Schwarz’s alternating algorithm in its classical setting and indicates how this algorithm can be expressed in a variational form.
An additive variant of Schwarz’s method is also introduced and presented in a general form. A lower bound of the eigenvalues is obtained in terms of an upper bound of a Rayleigh quotient which measures the extent by which the subspaces are linearly independent. The rest part of the paper presents a series of applications.

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
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations