Seamless integration of elliptic Dirichlet-to-Neumann boundary condition and high order spectral element method for scattering problem

Abstract

In this paper, we present a semi-analytic approach to enhance the integration of elliptic Dirichlet-to-Neumann (DtN) boundary condition and high order spectral element method in solving scattering problem with slender scatterer. By using appropriate elemental mapping in the spectral element discretization, semi-analytic formulas are obtained for the computation of Mathieu expansion coefficients involved in the global DtN operator. Further, a semi-analytic approach is proposed for the computation of global boundary integral terms in the spectral element discretization. The proposed semi-analytic formulas can also be used to calculate Mathieu expansion coefficients for functions given values on spectral element grids. Numerical examples show that spectral element method with the proposed semi-analytic approach can produce high order numerical solution for scattering problem with slender scatterer.