Optimal Coarse Spaces for FETI and Their Approximation

Abstract

One-level iterative domain decomposition methods share only information between neighboring subdomains, and are thus not scalable in general. For scalability, a coarse space is thus needed. This coarse space can however do more than just make the method scalable: there exists an optimal coarse space in the sense that we have convergence after exactly one coarse correction, and thus the method becomes a direct solver. We introduce and analyze here a new such optimal coarse space for the FETI method for the positive definite Helmholtz equation in one and two space dimensions for strip domain decompositions. We then show how one can approximate the optimal coarse space using optimization techniques. Computational results illustrating the performance and effectiveness of this new coarse space and its approximations are also presented.