Fast rapidly convergent penetrable scattering computations

Abstract

We present a fast high-order scheme for the numerical solution of a volume-surface integro-differential equation. Such equations arise in problems of scattering of time-harmonic acoustic and electromagnetic waves by inhomogeneous media with variable density wherein the material properties jump across the medium interface. The method uses a partition of unity to segregate the interior and the boundary regions of the scattering obstacle, enabling us to make use of specially designed quadratures to deal with the material discontinuities in a high-order manner. In particular, the method uses suitable changes of variables to resolve the singularities present in the integrals in conjunction with a decomposition of Green’s function via the addition theorem. To achieve a reduced computational cost, the method employs a Fast Fourier Transform (FFT) based acceleration strategy to compute the integrals over the boundary region. Moreover, the necessary offgrid evaluation of the density and the inter-grid transfer of data is achieved by applying an FFT-based refined-grid interpolation strategy. We validate the performance of the method through multiple scattering simulations. In particular, the numerical experiments demonstrate that the proposed method can handle high-contrast material properties without any adverse effect on the number of GMRES iterations.