×

On the abstract theory of additive and multiplicative Schwarz algorithms. (English) Zbl 0826.65098

The authors consider additive (ASM) and multiplicative (MSM) Schwarz methods for solving discrete (e.g. finite element) symmetric positive definite problems of the form: \(\text{Find }u \in V\) such that \(a(u,v) = \Phi (v)\) \(\forall v \in V\), where \(V\) is some fixed, finite-dimensional (e.g. finite element) Hilbert space. Starting with a splitting of the space \(V\) into \((J + 1)\) subspaces such that \(V = \sum^J_{j = 0} V_j\) and with \((J + 1)\) preconditioning forms \(b_j (.,.)\) for the original form \(a(.,.)\) restricted to the subspaces \(V_j\) (\(j = 0, \dots, J\)), one can construct the corresponding ASM and MSM.
The authors first review the convergence results for the ASM and give then a quite pretty connection of the ASM and the MSM analysis to the classical matrix analysis of the classical Jacobi-/Richardson- and Gauss- Seidel-/SOR-like iterations. Finally, they present new bounds for the convergence rate \(\rho_{ms}\) of the appropriately scaled MSM in terms of the condition number \(\chi\) of the corresponding ASM-operator. The rigorous bound is given by the estimate \(\rho_{ms} \leq \sqrt {1- 1/(\log_2 (4J) \times \chi)}\).
Reviewer: U.Langer (Linz)

MSC:

65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65J10 Numerical solutions to equations with linear operators
47A50 Equations and inequalities involving linear operators, with vector unknowns
65F10 Iterative numerical methods for linear systems
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
PDFBibTeX XMLCite
Full Text: DOI