Preconditioners for FETI-DP formulations with mortar methods

Abstract

In this dissertation, we consider FETI methods which are known as the most efficient domain decomposition method especially for solving large scale problems. In FETI methods, Lagrange multipliers are introduced to enforce the continuity of solutions across subdomain interfaces. This gives a mixed problem with the continuity condition as constraints. After eliminating unknowns other than the Lagrange multipliers, the resulting linear system is solved using the preconditioned conjugate gradient method. There are three variants of FETI methods, FETI, two-level FETI and dual-primal FETI(FETI-DP) method. Until now, FETI methods have been developed for the problems discretized with conforming finite elements. Among them, we extend FETI-DP methods to the problems with nonconforming discretizations, that arise from nonmatching triangulations across subdomain interfaces. The nonmatching triangulations are important for problems with corner singularities, contact problems as well as multi-physics problems. Moreover, the generation of meshes can be done independently in each subdomain. To resolve the nonconformity of the approximation, we consider mortar methods, which gives the same order of accuracy as conforming finite elements. In the mortar methods, the Lagrange multiplier space is introduced to enforce the continuity of solutions across the subdomain interfaces. The saddle point formulation of mortar methods gives a similar linear system to the mixed formulation of the FETI methods. The linear system is ill-conditioned. Moreover, it is difficult to find a good preconditioner for this system. We apply the FETI-DP method to solving this linear system efficiently and to finding a good preconditioner easily. This dissertation concerns elliptic problems both in 2D and 3D, and Stokes problem in 2D. Especially, redundant continuity constraints are introduced for 3D elliptic problems and Stokes problem. The Lagrange multipliers to the redundant constraints are treated as t...