A Convergence Analysis of the Parallel Schwarz Solution of the Continuous Closest Point Method

TL;DR

This paper analyzes the convergence of the parallel Schwarz method when applied to the Closest Point Method for solving surface PDEs, specifically focusing on 1-manifolds, providing theoretical insights into its effectiveness.

Contribution

It offers a novel convergence analysis of Schwarz methods combined with CPM for surface PDEs, an area with limited prior theoretical work.

Findings

Convergence conditions for Schwarz methods with CPM are established.

The analysis is specific to 1-manifolds in d-dimensional space.

Provides theoretical foundation for iterative solvers in surface PDE discretizations.

Abstract

The discretization of surface intrinsic PDEs has challenges that one might not face in the flat space. The closest point method (CPM) is an embedding method that represents surfaces using a function that maps points in the flat space to their closest points on the surface. This mapping brings intrinsic data onto the embedding space, allowing us to numerically approximate PDEs by the standard methods in the tubular neighborhood of the surface. Here, we solve the surface intrinsic positive Helmholtz equation by the CPM paired with finite differences which usually yields a large, sparse, and non-symmetric system. Domain decomposition methods, especially Schwarz methods, are robust algorithms to solve these linear systems. While there have been substantial works on Schwarz methods, Schwarz methods for solving surface differential equations have not been widely analyzed. In this work, we investigate the convergence of the CPM coupled with Schwarz method on 1-manifolds in d-dimensional space of real numbers.