Toward an Optimized Global-in-Time Schwarz Algorithm for Diffusion Equations with Discontinuous and Spatially Variable Coefficients, Part 1: The Constant Coefficients Case

In this paper we present a global-in-time non-overlapping Schwarz method applied to the one dimensional unsteady diffusion equation. We address specifically the problem with discontinuous diffusion coefficients, our approach is therefore especially designed for subdomains with heterogeneous properties. We derive efficient interface conditions by solving analytically the minmax problem associated with the search for optimized conditions in a Robin-Neumann case and in a two-sided Robin-Robin case. The performance of the proposed schemes are illustrated by numerical experiments.