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Convergence analysis of Schwarz algorithm and multilevel decomposition iterative methods. I: Selfadjoint and positive definite elliptic problems. (English) Zbl 0785.65115
Beauwens, R. (ed.) et al., Iterative methods in linear algebra. Proceedings of the IMACS international symposium, Brussels, Belgium, 2-4 April, 1991. Amsterdam: North-Holland. 93-110 (1992).
Summary: New convergence estimates for the Schwarz alternating algorithm and its generalization are derived for self-adjoint and positive definite second- order elliptic equations. We show that this algorithm converges with rate \(1-CJ^{-1}\) for some constant \(C\) independent of \(J\), the number of overlapping subdomains. The analysis is put into a general framework of product algorithms for projection operators onto Hilbert spaces.
Applications of the abstract theory to finite element approximation yield multilevel decomposition methods. These methods are related to the standard and the hierarchical basis multigrid methods and converge with rate \(1-Cj^{-2}\), where \(j\) is the number of levels of the decomposition. No assumption beyond \(H^ 1\) regularity necessary to define the weak form is used in the analysis.
For the entire collection see [Zbl 0778.00018].

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
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations