Domain Decomposition for the Closest Point Method

The discretization of elliptic PDEs leads to large coupled systems of equations. Domain decomposition methods (DDMs) are one approach to the solution of these systems, and can split the problem in a way that allows for parallel computing. Herein, we extend two DDMs to elliptic PDEs posed intrinsic to surfaces as discretized by the Closest Point Method (CPM) \cite{SJR:CPM,CBM:ICPM}. We consider the positive Helmholtz equation $\left(c-\Delta\mathcal{S}\right)u = f$, where $c\in\mathbb{R}^+$ is a constant and $\Delta\mathcal{S}$ is the Laplace-Beltrami operator associated with the surface $\mathcal{S}\subset\mathbb{R}^d$. The evolution of diffusion equations by implicit time-stepping schemes and Laplace-Beltrami eigenvalue problems \cite{CBM:Eig} both give rise to equations of this form. The creation of efficient, parallel, solvers for this equation would ease the investigation of reaction-diffusion equations on surfaces \cite{CBM:RDonPC}, and speed up shape classification \cite{Reuter:ShapeDNA}, to name a couple applications.