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

whereΔS denotes the Laplace-Beltrami operator associated with the surfaceS ⊂ R , and 2 > 0 is a constant. Discretization of this equation arises in many applications including the time-stepping of reaction-diffusion equations on surfaces [1], the comparison of shapes [2], and the solution of Laplace-Beltrami eigenvalue problems [3]. As a consequence, considerable recent work has taken place to develop efficient, high-speed solvers for this and other related PDEs on surfaces. There are several methods to solve surface intrinsic differential equations (DEs). If a surface parameterization (a mapping from the surface to a parameter space) is known, then the equation can be solved in the parameter domain [4]. For triangulated surfaces, a finite element discretization can be created [5]. Alternatively, we can solve the DE in a neighborhood of the surface using standard PDE methods in the underlying embedding space [6, 7, 8, 9]. Here, we discretize via the closest point method (CPM), which is an embedding method suitable for the discretization of PDEs on surfaces. The closest point method leads to non-symmetric linear systems to solve. On complex geometries or when varying scales arise, iterative solvers can be slow despite the sparsity of the underlying systems. In order to develop an efficient iterative solver which is also capable of parallelism, Parallel Schwarz (PS) and Optimized Parallel Schwarz (OPS) algorithms have been applied to the CPM for