# Domain Decomposition for the Closest Point Method

## TL;DR

This paper extends domain decomposition methods to solve elliptic PDEs on surfaces using the Closest Point Method, enabling parallel computation for applications like reaction-diffusion systems and shape analysis.

## Contribution

It introduces novel domain decomposition techniques tailored for surface PDEs discretized by the Closest Point Method, enhancing parallel solver efficiency.

## Key findings

- Extended two DDMs to surface PDEs with CPM
- Demonstrated potential for parallel computation in surface PDEs
- Facilitated faster solutions for applications like shape classification

## Abstract

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.