An Algebraic Convergence Theory for Restricted Additive and Multiplicative Schwarz Methods

In this contribution we use the algebraic representation recently developed for the classical additive and multiplicative Schwarz methods in [FS99, BFNS01] to analyze the restrictive additive Schwarz (RAS) and restrictive multiplicative Schwarz (RMS) methods; see [CS96, CFS98, CS99, QV99]. RAS was introduced in [CS99] as an efficient alternative to the classical additive Schwarz preconditioner. Practical experiments have proven RAS to be particularly attractive, because it reduces communication time while maintaining the most desirable properties of the classical Schwarz methods [CFS98, CS99]. RAS preconditioners are widely used in practice and are the default preconditioner in the PETSc software package [BGMS97]. Similar savings in communication time can be expected in the case of RMS; see [CS96]. In fact, we announce here that we can prove that RMS is better than RAS, in the sense that the corresponding iteration matrix has a smaller norm, for a certain weighted max norm. Our results provide the theoretical underpinnings for the behavior of the RAS preconditioners as observed in [CS99]. The theory we develop is not complete in the sense that we do not get quantitative results (like mesh independence in the presence of a coarse grid, for example). However, such results can be obtained indirectly by using some of the comparison results of [FS01] and classical results for the usual Schwarz method. Our approach is purely algebraic, and therefore our results apply to discretization of differential equations as well as to algebraic additive Schwarz. We believe that the algebraic tools used here and in [FS99, BFNS01] complement the usual analytic tools used for the analysis of Schwarz methods; see, e.g., the books [SBG96, QV99] and the extensive bibliography therein. One of the reasons why the algebraic approach presented here is a good alternative to the classical approach is that the operators defining RAS and RMS are not orthogonal projections (see [FS01]), and thus the usual theory as described, e.g., in [BM91] does not apply. This paper is organized as follows. We start by giving algebraic representations of the usual and the restricted additive Schwarz methods and we introduce the splittings associated with each of the methods. Then, our convergence theorem for RAS, as well as results on the effect of overlap on the quality of the preconditioner are presented. Finally, convergence of RMS is shown, together with the comparison between RMS and RAS. We note that using the same formulation described in this paper, several variants of RAS