Optimized Schwarz Methods for Heterogeneous Helmholtz and Maxwell's Equations

In [14, 15], it was discovered that heterogeneous media can actually im- prove the convergence of optimized Schwarz methods, provided that the coefficient jumps are aligned with the interfaces, and the jumps are taken into account in an appropriate way in the transmission conditions. Similar results were found for Maxwell's equations in [9] and [10]; it is even possible to obtain convergence independently of the mesh size in certain situations. We present and study here transmission conditions for the Helmholtz equation with heterogeneous media, and establish a relation to the results of [9, 10] written for Maxwell's equations. We then study improved convergence behavior for specific choices of the discretization parameters related to the pollution effect [2].