Optimized Schwarz Methods for Model Problems with Continuously Variable Coefficients

Abstract. Optimized Schwarz methods perform better than classical Schwarz methods because 6 they use more e↵ective transmission conditions between subdomains. These transmission conditions 7 are determined by optimizing the convergence factor, which is obtained by Fourier analysis for simple 8 two subdomain model problems. Such optimizations have been performed for many di↵erent types of 9 partial di↵erential equations, but almost exclusively based on the assumption of constant coe cients, 10 because only then Fourier analysis can be applied. We use in this paper the technique of separation 11 of variables to study optimized Schwarz methods for a model problem with a continuously variable 12 reaction term, and a similar analysis could be performed as well for many other problems with 13 variable coe cients. We obtain several new interesting results: first, we show that the technique of 14 separation of variables can successfully decouple the spatial variables and give the convergence factor 15 of subdomain iterations as a function of the eigenvalues of a certain Sturm–Liouville problem that 16 contains the variable coe cient. Second, we introduce a new natural transmission condition involving 17 second order derivatives along the interface, which turns the corresponding optimization problem into 18 a well-studied problem, from which the optimized transmission parameters follow. Finally, we find 19 that for variable coe cient problems, the most important information that enters into the optimized 20 transmission conditions is described by the smallest eigenvalue of the corresponding Sturm–Liouville 21 problem. We illustrate our results with extensive numerical experiments. 22