Highly Accurate Nyström Volume Integral Equation Method for the Maxwell equations for 3-D Scatters

Abstract: In this paper, we develop highly accurate Nystr\"{o}m methods for the volume integral equation (VIE) of the Maxwell equation for 3-D scatters. The method is based on a formulation of the VIE equation where the Cauchy principal value of the dyadic Green's function can be computed accurately for a finite size exclusion volume with some explicit corrective integrals of removable singularities. Then, an effective interpolated quadrature formula for tensor product Gauss quadrature nodes in a cube is proposed to handle the hyper-singularity of integrals of the dyadic Green's function. The proposed high order Nystr\"{o}m VIE method is shown to have high accuracy and demonstrates $p$-convergence for computing the electromagnetic scattering of cubes in $R^3$.