A fast direct solver for periodic transmission problems using multi-trace boundary integral method

Abstract

Wave scattering problems for periodic structures are of interest since they have many applications in engineering and science. In this study, we apply the fast direct solver of the Martinsson-Rokhlin type to one-periodic transmission problems using multi-trace boundary integral equation formulation. We point out that the numerical accuracy of the well-known PMCHWT formulation decreases when the contrast between the interior and the exterior dielectric constants is small. We show that this inaccuracy can be avoided with the help of the multi-trace boundary integral equations. We also apply the Ewald's method to speed up the evaluation of periodic Green's function. Finally, we demonstrate the validity of the proposed method with several numerical examples and compare its performance with that of the periodic fast multipole boundary element method based on the PMCHWT formulation and GMRES.