A Spectral Element Method with Transparent Boundary Condition for Periodic Layered Media Scattering

Abstract

We present a high-order spectral element method for solving layered media scattering problems featuring an operator that can be used to transparently enforce the far-field boundary condition. The incorporation of this Dirichlet-to-Neumann (DtN) map into the spectral element framework is a novel aspect of this work, and the resulting method can accommodate plane-wave radiation of arbitrary angle of incidence. In order to achieve this, the governing Helmholtz equations subject to quasi-periodic boundary conditions are rewritten in terms of periodic unknowns. We construct a spectral element operator to approximate the DtN map, thus ensuring nonreflecting outgoing waves on the artificial boundaries introduced to truncate the computational domain. We present an explicit formula that accurately computes the Fourier coefficients of the solution in the spectral element discretization space projected onto the boundary which is required by the DtN map. Our solutions are represented by the tensor product basis of one-dimensional Legendre---Lagrange interpolation polynomials based on the Gauss---Lobatto---Legendre grids. We study the scattered field in singly and doubly layered media with smooth and nonsmooth interfaces. We consider rectangular, triangular, and sawtooth interfaces that are accurately represented by the body-fitted quadrilateral elements. We use GMRES iteration to solve the resulting linear system, and we validate our results by demonstrating spectral convergence in comparison with exact solutions and the results of an alternative computational method.