Best Robin Parameters for Optimized Schwarz Methods at Cross Points

Optimized Schwarz methods are domain decomposition methods in which a largescale 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 ρ = 1− O(h1/2). 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 only occur if p = O(h) 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(h1/2) convergence factor in the resulting method.