Comparison of the Dirichlet-Neumann and Optimal Schwarz Method on the Sphere

We investigate the performance of domain decomposition methods for solving the Poisson equation on the surface of the sphere. This equation arises in a global weather model as a consequence of an implicit time discretization. We consider two different types of algorithms: the Dirichlet-Neumann algorithm and the optimal Schwarz method. We show that both algorithms applied to a simple two subdomain decomposition of the surface of the sphere converge in two iterations. While the Dirichlet-Neumann algorithm achieves this with local transmission conditions, the optimal Schwarz algorithm needs non-local transmission conditions. This seems to be a disadvantage of the optimal Schwarz method. We then show however that for more than two subdomains or overlapping subdomains, both the optimal Schwarz algorithm and the Dirichlet Neumann algorithm need non-local interface conditions to converge in a finite number of steps. Hence the apparent advantage of DirichletNeumann over optimal Schwarz is only an artifact of the special two subdomain decomposition.