A domain decomposition approach for heterogeneous time-harmonic Maxwell equations

Abstract

The time-harmonic Maxwell equations for a composite medium which behaves like a conductor in one part and like a perfect insulator in the other one are considered. An existence and uniqueness theorem is proven for this degenerate problem in the case of Dirichlet boundary conditions. A finite element domain decomposition approach is then proposed for the numerical approximation of the exact solution. This leads to an iteration-by-subdomain procedure, where at each step a non-degenerate boundary value problem has to be solved in each subdomain. The convergence of these iterations is proven, and the rate of convergence turns out to be independent of the mesh size h, showing that the preconditioner implicitly defined by the iteration procedure is optimal.