OPTIMAL CONVERGENCE PROPERTIES OF THE FETI DOMAIN DECOMPOSITION METHOD

The Finite Element Tearing and Interconnecting (FETI) method is a prac tical and efficient domain decomposition (DD) algorithm for the soluti on of self-adjoint elliptic partial differential equations. For large- scale structural problems discretized with shell and beam elements, th is method was found to outperform popular iterative algorithms and dir ect solvers on both serial and parallel computers, and to compare favo rably with leading DD methods. In this paper, we discuss some numerica l properties of the FETI method that were not addressed before. In par ticular, we show that the mathematical treatment of the floating subdo mains and the specific conjugate projected gradient algorithm that cha racterize the FETI method are equivalent to the construction and solut ion of a coarse problem that propagates the error globally, accelerate s convergence, and ensures a performance that is independent of the nu mber of subdomains. We also show that when the interface problem is op timally preconditioned and the mesh is partitioned into well structure d subdomains with good aspect ratios, the performance of the FETI meth od is also independent of the mesh size. However, we also argue that t he FETI and other leading DD methods for unstructured problems lose in practice these scalability properties when the mesh contains juncture s with rotational degrees of freedom, or the decomposition is irregula r and characterized by arbitrary subdomain aspect ratios. Finally, we report that for realistic problems, optimal preconditioners are not ne cessarily computationally efficient and can be outperformed by non-opt imal ones.