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

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.