A closest point method library for PDEs on surfaces with parallel domain decomposition solvers and preconditioners

TL;DR

This paper introduces a software library for solving PDEs on surfaces using the closest point method, featuring domain decomposition preconditioners, parallel scalability, and compatibility with distributed computing frameworks.

Contribution

The library integrates CPM-based PDE solvers with custom domain decomposition preconditioners and demonstrates scalable parallel performance on surface PDE problems.

Findings

Effective parallel scalability demonstrated

Compatibility with MPI enables distributed computing

Sample problems validate solver accuracy and efficiency

Abstract

The DD-CPM software library provides a set of tools for the discretization and solution of problems arising from the closest point method (CPM) for partial differential equations on surfaces. The solvers are built on top of the well-known PETSc framework, and are supplemented by custom domain decomposition (DD) preconditioners specific to the CPM. These solvers are fully compatible with distributed memory parallelism through MPI. This library is particularly well suited to the solution of elliptic and parabolic equations, including many reaction-diffusion equations. The software is detailed herein, and a number of sample problems and benchmarks are demonstrated. Finally, the parallel scalability is measured.