Optimized Schwarz method with two-sided transmission conditions in an unsymmetric domain decomposition

Domain decomposition (DD) methods are important techniques for designing parallel algorithms for solving partial differential equations. Since the decomposition is often performed using automatic mesh partitioning tools, one can in general not make any assumptions on the shape or physical size of the subdomains, especially if local mesh refinement is used. In many of the popular domain decomposition methods, neighboring subdomains are not using the same type of boundary conditions, e.g. the Dirichlet-Neumann methods invented by [2], or the two-sided optimized Schwarz methods proposed in [3], and one has to decide which subdomain uses which boundary condition. A similar question also arises in mortar methods, see [1], where one has to decide on the master and slave side at the interfaces. In [4], it was found that for optimized Schwarz methods, the subdomain geometry and problem boundary conditions influence the optimized Robin parameters for symmetrical finite domain decompositions, and in [5], it was observed numerically that swapping the optimized two-sided Robin parameters can accelerate the convergence for a circular domain decomposition. We study in this paper two-sided optimized Schwarz methods for a model decomposition into a larger and a smaller subomain, to investigate which Robin parameter should be used on which subdomain in order to get fast convergence. We consider the model problem