Heterogeneous Optimized Schwarz Methods for Second Order Elliptic PDEs

Due to their property of convergence in the absence of overlap, optimized Schwarz methods are the natural domain decomposition framework for heterogeneous problems, where the spatial decomposition is provided by the multiphysics of the phenomena. We study here heterogeneous problems which arise from the coupling of second order elliptic PDEs. Theoretical results and asymptotic formulas are proposed solving the corresponding min-max problems both for single and double sided optimizations, while numerical results confirm the effectiveness of our approach even when analytical conclusions are not available. Our analysis shows that optimized Schwarz methods do not suffer the heterogeneity, it is the opposite, they are faster the stronger the heterogeneity is. It is even possible to have h independent convergence choosing two independent Robin parameters. This property was proved for a Laplace equation with discontinuous coefficients, but only conjectured for more general couplings in [M. J. Gander and O. Dubois, Numer. Algorithms, 69 (2015), pp. 109--144]. Our study is completed by an application to a contaminant transport problem.