Toward an Optimized Global-in-Time Schwarz Algorithm for Diffusion Equations with Discontinuous and Spatially Variable Coefficients, Part 2: the Variable Coefficients Case

This paper is the second part of a study dealing with the application of a global-in-time Schwarz method to a one dimensional diffusion problem defined on two non-overlapping subdomains. In the first part, we considered that the diffusion coefficients were constant and possibly discontinuous. In the present study, we address the problem for spatially variable coefficients with a discontinuity at the interface between subdomains. For this particular case, we derive a new approach to determine analytically the convergence factor of the associated algorithm. The theoretical results are illustrated by numerical experiments withDirichlet-Neumann and Robin-Robin interface conditions. In the Robin-Robin case, thanks to the convergence factor found at the analytical level, we can optimize the convergence speed of the Schwarz algorithm.