Boundary Integral Equations for the Laplace-Beltrami Operator

TL;DR

This paper develops a boundary integral and boundary element method for solving boundary-value problems involving the Laplace-Beltrami operator on the sphere, focusing on a boundary curve dividing the sphere into two regions.

Contribution

It introduces a novel boundary integral formulation and discretization technique for the Laplace-Beltrami operator on spherical surfaces with boundary curves.

Findings

Effective boundary integral equations derived for the Laplace-Beltrami problem.

Discretization method implemented for the sphere's surface.

Potential for accurate solutions on spherical domains with boundaries.

Abstract

We present a boundary integral method, and an accompanying boundary element discretization, for solving boundary-value problems for the Laplace-Beltrami operator on the surface of the unit sphere $\S$ in $\mathbb{R}^3$. We consider a closed curve ${\cal C}$ on ${\cal S}$ which divides ${\cal S}$ into two parts ${\cal S}_1$ and ${\cal S}_2$. In particular, ${\cal C} = \partial {\cal S}_1$ is the boundary curve of ${\cal S}_1$. We are interested in solving a boundary value problem for the Laplace-Beltrami operator in $\S_2$, with boundary data prescribed on $\C$.