Multi-Domain Transmission Conditions for Domain Decomposition Methods Applied to Scattering Problems

The choice of the transmission conditions (TCs) across the interfaces between subdomains and its free parameters has a significant impact on the convergence of domain decomposition methods (DDM), especially when applied to propagation and scattering problems. In this paper, we study the convergence behavior of the optimized Schwarz DDM applied to electromagnetic scattering problem in two dimensions for different TCs based on a polynomial approximation of the Dirichlet-to-Neumann map on the Fourier domain. We then propose a methodology to obtain the free parameters of the TC based on the numerical optimization of the spectral radius of the Schwarz iteration matrix on the Fourier domain, taking into account the number of subdomains and some geometrical characteristics such as length and overlap size. To confirm the improvements in convergence, we perform numerical experiments and compare the iteration count for different TCs.