Matrix representation of the Neumann-Poincar\'e operator for a torus

Abstract

We represent a matrix representation of the Neumann-Poincar\'e operator defined on the boundaries of a torus. A torus is a doubly connected domain in three dimensions. There is a well-known parametrization for the shape of the torus, the toroidal coordinate system. Based on the coordinate system, we use toroidal harmonics to get an expansion of the NP operator for the torus. Along with proper bases, the Neumann-Poincar\'e operator can be explicitly represented by an infinite matrix.