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

The closest point method (CPM) is an embedding method that represents surfaces using a function that maps points in a neighbourhood of the surface to their closest points on the surface. This mapping brings intrinsic data onto the embedding space, allowing us to numerically approximate surface intrinsic partial differential equations by standard numerical methods in a tubular neighbourhood of the surface. Solving the surface positive Helmholtz equation by the CPM paired with finite differences yields a large, sparse, and non-symmetric linear system. Here Schwarz domain decomposition methods (DDMs) provide partitioned, parallel algorithms to solve these linear systems. We theoretically investigate the convergence of optimized Schwarz-CPM methods for 1-manifolds in \(\mathbb {R}^d\) ${\mathbb{R}}^d$ . The analysis is followed by numerical experiments for verification.