Optimized Domain Decomposition Methods for the Spherical Laplacian

Abstract

The Schwarz iteration decomposes a boundary value problem over a large domain $\Omega$ into smaller subproblems by iteratively solving Dirichlet problems on a cover $\Omega_{1},\dots,\Omega_{p}$ of $\Omega$. In this paper, we discuss alternate transmission conditions that lead to faster convergence for the Laplacian on the sphere $\Omega$. We look at Robin conditions, second order tangential conditions, and a discretized version of an optimal but nonlocal operator.