High-order Nyström implementation of an augmented volume integral equation

Integral equation methods provide an effective approach for solving electromagnetic radiation and scattering problems. For linear homogeneous materials, surface integral equations are commonly applied. However, for nonlinear, high contrast and/or inhomogeneous materials, volume integral equations (VIEs) often provide a more useful alternative. The locally corrected Nyström (LCN) method provides a scheme for discretizing the VIE formulation. The main advantages of the LCN method over other discretization methods include the ability to easily handle different mesh element types at the same time, and the ability to easily incorporate high-order representations. Moreover, by applying the LCN discretization to the VIE formulation, the material properties can be confined to the diagonal of the system matrix, which significantly reduces the computational costs in cases for which a given geometry with different material properties is of interest. Unfortunately, the regular VIE formulation has a number of limitations. These limitations include poor matrix condition numbers for problems with high contrast materials, and deteriorating performance for problems with complex, multi-scale meshes.