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Abstract: We present an Optimized Schwarz Domain Decomposition Methods applied to Helmholtz transmission problems in periodic layered media. Unlike the classical domain decomposition approach that relies on exchange of Robin data on the subdomain boundaries, we incorporate instead transmission operators that are approximations of Dirichlet-to-Neumann (DtN) operators. The latter approximations, in turn, is obtained via shape perturbation series. The Robin-to-Robin (RtR) operators that are the building blocks of Domain Decomposition Methods are expressed via boundary integral equation formulations that are shown to be robust for all frequencies, including the challenging Wood frequencies. We use Nyström discretizations of quasi-periodic boundary integral operators to construct high-order approximations of RtR. Based on the premise that the quasi-optimal transmission operators should act like perfect transparent boundary conditions, we construct an approximate LU factorization of the tridiagonal QO DD matrix associated with periodic layered media, which is then used as a double sweep preconditioner. We present a variety of numerical results that showcase the effectiveness of the sweeping preconditioners for the iterative solution of Helmholtz transmission problems in periodic layered media.

Joint work with David Nicholls (UIC) and Carlos Perez Arancibia (PUC Chile)