An algebraic convergence theory for restricted additive Schwarz methods using weighted max norms (Q2719217)

The article is concerned with restricted additive Schwarz methods for systems \(Ax=b\), considered as preconditioners. In such methods, the preconditioning matrix produces in a certain sense ``less overlap'' than in the classical additive Schwarz method. For the restricted as well as for the classical method, the preconditioned matrix can be written as a sum of projections, which however for the restricted method are not \(A\)-orthogonal, in case of an symmetric positive definite matrix \(A\). Therefore, the classical convergence theory cannot be applied to restricted additive Schwarz methods.NEWLINENEWLINENEWLINEThe central result of the paper is a convergence theorem in case of an \(M\)-matrix \(A\); the proof is based on multisplitting theory. Furthermore, the influence of the choice of the overlap, and variants of the restricted additive Schwarz method are investigated using weighted norms.