Optimized Schwarz Methods for Maxwell Equations with Discontinuous Coefficients

After the development of optimized Schwarz methods for the Helmholtz equation [2, 3, 4, 12, 14], extensions to the more difficult case of Maxwell's equations were developed: for curl-curl formulations, see [1]. For first order formulations without conductivity, see [7], and with conductivity, see [5, 11]. For DG discretizations of Maxwell's equations, optimized Schwarz methods can be found in [6, 8, 9], and for scattering problems with applications, see [15, 16]. We present here optimized Schwarz methods for Maxwell's equations in hetero- geneous media with discontinuous coefficients, and show that the discontinuities need to be taken into account in the transmission conditions in order to obtain effec- tive Schwarz methods. For diffusive problems, it was shown in [10] that jumps in the coefficients can actually lead to faster iterations, when they are taken into account correctly in the transmission conditions. We show here that for the case of Maxwell's equations with jumps along the interfaces, one can obtain a non-overlapping opti- mized Schwarz method that converges independently of the mesh parameter; this is not possible without coefficient jumps.