An Algebraic Convergence Theory for Restricted Additive Schwarz Methods Using Weighted Max Norms

Convergence results for the restrictive additive Schwarz (RAS) method of Cai and Sarkis [SIAM J. Sci. Comput.}, 21 (1999), pp. 792--797] for the solution of linear systems of the form Ax = b are provided using an algebraic view of additive Schwarz methods and the theory of multisplittings. The linear systems studied are usually discretizations of partial differential equations in two or three dimensions. It is shown that in the case of A symmetric positive definite, the projections defined by the methods are not orthogonal with respect to the inner product defined by A, and therefore the standard analysis cannot be used here. The convergence results presented are for the class of M-matrices (and more generally for H-matrices) using weighted max norms. Comparison between different versions of the RAS method are given in terms of these norms. A comparison theorem with respect to the classical additive Schwarz method makes it possible to indirectly get quantitative results on rates of convergence which otherwise cannot be obtained by the theory. Several RAS variants are considered, including new ones and two-level schemes.