Boundary Integral Equations for the Laplace-Beltrami Operator

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 $\mathcal{C}$ on $\mathcal{S}$ which divides $\mathcal{S}$ into two parts ${\mathcal{S}}_1$ and ${\mathcal{S}}_2$. In particular, $\mathcal{C}=\partial {\mathcal{S}}_1$ is the boundary curve of ${\mathcal{S}}_1$. We are interested in solving a boundary value problem for the Laplace-Beltrami operator in ${\S }_2$, with boundary data prescribed on $\C$.

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