  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.
