Comparison of a One and Two Parameter Family of Transmission Conditions for Maxwell's Equations with Damping

Transmission conditions between subdomains have a substantial influence on the convergence of iterative domain decomposition algorithms. For Maxwell’s equations, transmission conditions which lead to rapidly converging algorithms have been developed both for the curl-curl formulation of Maxwell’s equation, see [2, 3, 1], and also for first order formulations, see [7, 6]. These methods have well found their way into applications, see for example [9] and the references therein. It turns out that good transmission conditions are approximations of transparent boundary conditions. For each form of approximation chosen, one can try to find the best remaining free parameters in the approximation by solving a min-max problem. Usually allowing more free parameters leads to a substantially better solution of the min-max problem, and thus to a much better algorithm. For a particular one parameter family of transmission conditions analyzed in [4], we investigate in this paper a two parameter counterpart. The analysis, which is substantially more complicated than in the one parameter case, reveals that in one particular asymptotic regime there is only negligible improvement possible using two parameters, compared to the one parameter results. This analysis settles an important open question for this family of transmission conditions, and also suggests a direction for systematically reducing the number of parameters in other optimized transmission conditions.