Neumann problem on a torus

TL;DR

This paper develops a method to solve the Neumann problem on a torus using series expansions in toroidal harmonics, providing a numerical approach and illustrations for interior and exterior domains.

Contribution

It introduces a novel approach to compute solutions for the Neumann problem on a torus via algebraic manipulation of harmonic series coefficients.

Findings

Series expansion method effectively reduces the problem to algebraic computations.

Numerical solutions are successfully computed and illustrated.

The approach extends to toroidal shells combining interior and exterior solutions.

Abstract

We consider the Dirichlet-to-Neumann mapping and the Neumann problem for the Laplace operator on a torus, given in toroidal coordinates. The Dirichlet-to-Neumann mapping is expressed with respect to series expansions in toroidal harmonics and thereby reduced to algebraic manipulations on the coefficients. A method for computing the numerical solutions of the corresponding Neumann problem is presented, and numerical illustrations are provided. We combine the results for interior and exterior domains to solve the Neumann problem for a toroidal shell.