Asymptotic analysis of optimized Schwarz methods for maxwell’s equations with discontinuous coefficients

Abstract

Discretized time harmonic Maxwell’s equations are hard to solve by iterative methods, and the best currently available methods are based on domain decomposition and optimized transmission conditions. Optimized Schwarz methods were the first ones to use such transmission conditions, and this approach turned out to be so fundamentally important that it has been rediscovered over the last years under the name sweeping preconditioners, source transfer, single layer potential method and the method of polarized traces. We show here how one can optimize transmission conditions to take benefit from the jumps in the coefficients of the problem, when they are aligned with the subdomain interface, and obtain methods which converge for two subdomains in certain situations independently of the mesh size, which would not be possible without jumps in the coefficients.

A Schwarz Method for the Magnetotelluric Approximation of Maxwell's equations

“…He also introduced a parallel variant of the Schwarz method, where all subdomain problems are solved simultaneously. Schwarz methods have also been introduced and studied for the original Maxwell equations (1), see [2,1,6,5,7,8] and references therein; maximum principle arguments can not be used here for…”

Closed form optimized transmission conditions for complex diffusion with many subdomains

“…We are interested here in designing and analyzing domain decomposition methods for complex diffusion problems of the form (1.1) that use optimized transmission conditions; for an analysis of the performance of the classical Schwarz method, see [12]. Traditionally, such transmission conditions are derived and optimized for simple two subdomain configurations, like in optimized Schwarz methods, see [6,17,3,19,11,21,20,22,1,16,15] and references therein. We investigate here the optimization of transmission conditions directly for the many subdomain case, and also study the optimization problem in the limit when the number of subdomains goes to infinity, using the tool of limiting spectra, see [2] and the references therein.…”