OPTIMIZED SCHWARZ METHODS FOR THE DIFFUSION-REACTION PROBLEM WITH CYLINDRICAL INTERFACES~?

摘要

In this work we consider the Optimized Schwarz Method for the three-dimensional (3D) diffusion-reaction problem. In particular, we treat the case of cylindrical interfaces between the subdomains, and we provide a convergence analysis of the Schwarz method, both in the case of Dirichlet interface conditions and in that of general transmission conditions. This allows us to recover, for the latter case, optimal symbols for the interface conditions, which are supposed to work well for geometries which feature cylindrical interfaces. Moreover, starting from these optimal symbols, we propose effective and easily computable constant interface parameters, derived both from Taylor expansions and from an optimization procedure. We finally present several 3D numerical results aiming at validating the theoretical findings.