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

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.

Optimized Schwarz Domain Decomposition Algorithms for the Closest Point Method on Closed Manifolds

“…The Schwarz methods have also been used as a preconditioner for Krylov space iterative methods [27] to accelerate convergence. In spite of the promising numerical results in [27] and the developed software [28], a robust convergence analysis is missing. The remainder of this article provides a crucial first step towards a convergence analysis of such methods.…”