Synchronous and asynchronous optimized Schwarz methods for Poisson's equation in rectangular domains

Abstract

Convergence results for optimized Schwarz methods (OSM) applied as solvers for Poisson's equation in a bounded rectangular domain with Dirichlet (physical) boundary conditions and zeroth-order (Robin) artificial transmission conditions between subdomains are presented. The analysis presented applies to a continuous formulation on an arbitrary number of subdomains with cross points. Both synchronous and asynchronous versions of OSM are discussed. Convergence theorems are presented, and it is shown numerically that the hypotheses of these theorems are satisfied for certain configurations of the subdomains. Additional numerical experiments illustrate the practical behavior of the methods discussed.