Best Robin Parameters for Optimized Schwarz Methods at Cross Points

Abstract

Optimized Schwarz methods are domain decomposition methods in which a large-scale PDE problem is solved by subdividing it into smaller subdomain problems, solving the subproblems in parallel, and iterating until one obtains a global solution that is consistent across subdomain boundaries. Fast convergence can be obtained if Robin conditions are used along subdomain boundaries, provided that the Robin parameters $p$ are chosen correctly. In the case of second-order elliptic problems such as the Poisson equation, it is well known for two-subdomain problems without overlap that the optimal choice is $p = O(h^{-1/2})$ (where $h$ is the mesh size), with the resulting method having a convergence factor of $\rho = 1 - O(h^{1/2})$. However, when cross points are present, i.e., when several subdomains meet at a single point, this choice leads to a divergent method. In this article, we show for a model problem that convergence can occur only if $p = O(h^{-1})$ at the cross point; thus, a different scaling of the Robin parameter is needed to ensure convergence. In addition, this choice of $p$ allows us to recover the $1 - O(h^{1/2})$ convergence factor in the resulting method.