Schwarz domain decomposition algorithms for the closest point method on closed manifolds

The discretization of surface intrinsic PDEs has challenges that one might not face in flat spaces. 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 standard methods in a tubular neighbourhood 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 linear system. Domain decomposition methods, especially Schwarz methods, are robust algorithms to solve these linear systems. In this work, we investigate the convergence of four Schwarz-CPM methods for 1-manifolds in ℝᵈ . The analysis is followed by numerical experiments for verification