RASHO a restricted additive Schwarz preconditioner with harmonic overlap.pdf

RASHO: A Restricted Additive Schwarz Preconditioner with Harmonic Overlap Xiao-Chuan Cai ? Maksymilian Dryja ? Marcus Sarkis ? 1 Introduction A restricted additive Schwarz (RAS) preconditioning technique was introduced recently for solving general nonsymmetric sparse linear systems [1, 3, 4, 7, 8, 9, 11]. The RAS preconditioner improves the classical additive Schwarz precon- ditioner (AS), [10], in the sense that it reduces the number of iterations of the iterative method, such as GMRES, and also reduces munication cost per iteration when implemented on distributed puters. However, RAS in its original form is a nonsymmetric preconditioner and therefore the cannot be used with the Conjugate Gradient method (CG). In this paper, we provide an extension of RAS for symmetric positive de?nite problems using the so-called harmonic overlaps (RASHO). Both RAS and RASHOoutperformtheir counterparts of the classical additive Schwarz variants. Roughly speaking, the design of RASHO is based on a much deeper understanding of thebehavior of Schwarz type methods in the overlapping regions, and in the construction of the overlap. Under RASHO, the overlap is obtained by extending the nonoverlap- ping subdomains only in the directions that do not cut the boundaries of other subdomains, and all functions are made harmonic in the overlapping regions. As a result, the subdomain problems in RASHO a