Optimized Schwarz Methods for Model Problems with Continuously Variable Coefficients

Abstract

Optimized Schwarz methods perform better than classical Schwarz methods because they use more effective transmission conditions between subdomains. These transmission conditions are determined by optimizing the convergence factor, which is obtained by Fourier analysis for simple two subdomain model problems. Such optimizations have been performed for many different types of partial differential equations, but almost exclusively based on the assumption of constant coefficients, because only then Fourier analysis can be applied. We use in this paper the technique of separation of variables to study optimized Schwarz methods for a model problem with a continuously variable reaction term, and a similar analysis could be performed as well for many other problems with variable coefficients. We obtain several new interesting results: first, we show that the technique of separation of variables can successfully decouple the spatial variables and give the convergence factor of subdomain iterations as a function of the eigenvalues of a certain Sturm--Liouville problem that contains the variable coefficient. Second, we introduce a new natural transmission condition involving second order derivatives along the interface, which turns the corresponding optimization problem into a well-studied problem, from which the optimized transmission parameters follow. Finally, we find that for variable coefficient problems, the most important information that enters into the optimized transmission conditions is described by the smallest eigenvalue of the corresponding Sturm--Liouville problem. We illustrate our results with extensive numerical experiments.