Highly scalable parallel domain decomposition methods with an application to biomechanics

Highly scalable parallel domain decomposition methods for elliptic partial differential equations are considered with a special emphasis on problems arising in elasticity. The focus of this survey article is on Finite Element Tearing and Interconnecting (FETI) methods, a family of nonoverlapping domain decomposition methods where the continuity between the subdomains, in principle, is enforced by the use of Lagrange multipliers. Exact onelevel and dual‐primal FETI methods as well as related inexact dual‐primal variants are described and theoretical convergence estimates are presented together with numerical results confirming the parallel scalability properties of these methods. New aspects such as a hybrid onelevel FETI/FETI‐DP approach and the behavior of FETI‐DP for anisotropic elasticity problems are presented. Parallel and numerical scalability of the methods for more than 65 000 processor cores of the JUGENE supercomputer is shown. An application of a dual‐primal FETI method to a nontrivial biomechanical problem from nonlinear elasticity, modeling arterial wall stress, is given, showing the robustness of our domain decomposition methods for such problems.