Optimal convergence properties of the FETI domain decomposition method

Abstract

Abstract The Finite Element Tearing and Interconnecting (FETI) method is a practical and efficient domain decomposition (DD) algorithm for the solution of self-adjoint elliptic partial differential equations. For large-scale structural problems discretized with shell and beam elements, this method was found to outperform popular iterative algorithms and direct solvers on both serial and parallel computers, and to compare favorably with leading DD methods. In this paper, we discuss some numerical properties of the FETI method that were not addressed before. In particular, we show that the mathematical treatment of the floating subdomains and the specific conjugate projected gradient algorithm that characterize the FETI method are equivalent to the construction and solution of a coarse problem that propagates the error globally, accelerates convergence, and ensures a performance that is independent of the number of subdomains. We also show that when the interface problem is optimally preconditioned and the mesh is partitioned into well structured subdomains with good aspect ratios, the performance of the FETI method is also independent of the mesh size. However, we also argue that the FETI and other leading DD methods for unstructured problems lose in practice these scalability properties when the mesh contains junctures with rotational degrees of freedom, or the decomposition is irregular and characterized by arbitrary subdomain aspect ratios. Finally, we report that for realistic problems, optimal preconditioners are not necessarily computationally efficient and can be outperformed by non-optimal ones.