A Restricted Additive Schwarz Preconditioner with Harmonic Overlap for Symmetric Positive Definite Linear Systems
A restricted additive Schwarz (RAS) preconditioning technique was introduced recently for solving general nonsymmetric sparse linear systems. In this paper, we provide an extension of RAS for symmetric positive definite problems using the so-called harmonic overlaps (RASHO). Both RAS and RASHO outperform their counterparts of the classical additive Schwarz variants (AS). The design of RASHO is based on a much deeper understanding of the behavior of Schwarz type methods in overlapping subregions, and in the construction of the overlap. In RASHO, the overlap is obtained by extending the nonoverlapping 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 are smaller than that of AS, and the communication cost is also smaller when implemented on distributed memory computers, since the right-hand sides of discrete harmonic systems are always zero that do not need to be communicated. We also show numerically that RASHO preconditioned CG takes fewer number of iterations than the corresponding AS preconditioned CG. A nearly optimal theory is included for the convergence of RASHO/CG for solving elliptic problems discretized with a finite element method.