Optimized Schwarz Methods with Elliptical Domain Decompositions

Abstract

Over the past decade, partial differential equation models in elliptical geometries have become a focus of interest in several scientific and engineering applications: the classical studies of flow past a cylinder, the spherical particles in nano-fluids and spherical water filled domains are replaced by elliptical geometries which more accurately describe a wider class of physical problems of interest. Optimized Schwarz methods (OSMs) are among the best parallel methods for such models. We study here for the first time OSMs with elliptical domain decompositions, i.e. decompositions into an ellipse and elliptical rings. Using the technique of separation of variables, we decouple the spatial variables and reduce the subdomain problems to radial Mathieu like equations defined on finite intervals, which allows us to derive and study a new family of OSMs. Our analysis reveals that the optimized transmission parameters are not constants any more along the elliptical interfaces. We can prove however also that using the constant optimized parameters from the straight interface analysis in the literature scaled locally by the interface curvature is still efficient in an asymptotic sense, which leads to the important discovery of a unique factor in the optimized parameters and asymptotic performance determined by the geometry of the decomposition. We use numerical examples to illustrate our analysis and findings.