Griebel, M.; Oswald, P. On the abstract theory of additive and multiplicative Schwarz algorithms. (English) Zbl 0826.65098 Numer. Math. 70, No. 2, 163-180 (1995). 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) Cited in 87 Documents 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 Keywords:Jacobi-Richardson method; error bound; additive Schwarz methods; multiplicative Schwarz methods; finite element; symmetric positive definite problems; Hilbert space; preconditioning; convergence; Gauss- Seidel-/SOR-like iterations; condition number PDFBibTeX XMLCite \textit{M. Griebel} and \textit{P. Oswald}, Numer. Math. 70, No. 2, 163--180 (1995; Zbl 0826.65098) Full Text: DOI