Nonlinear FETI-DP and BDDC Methods: A Unified Framework and Parallel Results

Parallel Newton--Krylov FETI-DP (Finite Element Tearing and Interconnecting---Dual-Primal) domain decomposition methods are fast and robust solvers, e.g., for nonlinear implicit problems in structural mechanics. In these methods, the nonlinear problem is first linearized and then decomposed into loosely coupled (linear) problems, which can be solved in parallel. By changing the order of the operations, new parallel communication can be constructed, where the loosely coupled local problems are nonlinear. We discuss different nonlinear FETI-DP methods which are equivalent when applied to linear problems but which show a different performance for nonlinear problems. Moreover, a new unified framework is introduced which casts all nonlinear FETI-DP domain decomposition approaches discussed in the literature into a single algorithm. Furthermore, the equivalence of nonlinear FETI-DP methods to specific nonlinearly right-preconditioned Newton--Krylov methods is shown. For the methods using nested Newton iterations, a strategy is presented to stop the inner Newton iteration early, resulting in an approximate local nonlinear elimination. Additionally, the nonlinear BDDC (Balancing Domain Decomposition by Constraint) method is presented as a right-preconditioned Newton approach. Finally, for the first time, parallel weak scaling results for four different nonlinear FETI-DP approaches are compared to standard Newton--Krylov FETI-DP in two and three dimensions, using both exact as well as highly scalable inexact linear FETI-DP preconditioners and up to 131,072 message passing interface (MPI) ranks on the JUQUEEN supercomputer at Forschungszentrum Jülich. For a model problem with nonlocal nonlinearities, nonlinear FETI-DP methods are shown to be up to five times faster than the standard Newton--Krylov FETI-DP approach.