Resumen
This paper derives a linear convergence for the Schwarz overlapping domain decomposition method when applied to constrained minimization problems. The convergence analysis is based on a minimization approach to the corresponding functional over a convex set. A general framework of convergence is established for some multiplicative Schwarz algorithm. The abstract theory is particularly applied to some obstacle problems, which yields a linear convergence for the corresponding Schwarz overlapping domain decomposition method of one and two levels. Numerical experiments are presented to confirm the convergence estimate derived in this paper.

Resumen
A restricted additive Schwarz (RAS) preconditioning technique was introduced recently for solving general nonsymmetric sparse linear systems. In this paper, we provide one-level and two-level extensions 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 those 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 and therefore do not need to be communicated. We also show numerically that RASHO-preconditioned CG takes fewer iterations than the corresponding AS-preconditioned CG. A nearly optimal theory is included for the convergence of RASHO-preconditioned CG for solving elliptic problems discretized with a finite element method.