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

TL;DR

This paper develops an explicit matrix representation of the Neumann-Poincaré operator for a torus using toroidal harmonics and coordinate systems, facilitating analysis of boundary integral operators on doubly connected domains.

Contribution

It introduces a novel matrix representation of the NP operator on a torus using toroidal harmonics and parametrization, enabling explicit analysis of the operator.

Findings

Explicit infinite matrix representation derived

Utilizes toroidal harmonics for expansion

Provides a basis for further spectral analysis

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.