Towards Extremely Scalable Nonlinear Domain Decomposition Methods for Elliptic Partial Differential Equation

The solution of nonlinear problems, e.g., in material science requires fast and highly scalable parallel solvers. FETI-DP (Finite Element Tearing and Interconnecting) domain decomposition methods are parallel solution methods for implicit problems discretized by finite elements. Recently, nonlinear versions of the well-known FETI-DP methods for linear problems have been introduced. In these methods, the nonlinear problem is decomposed before linearization. This approach can be viewed as a strategy to further localize computational work and to extend the parallel scalability of FETI-DP methods towards extreme-scale supercomputers. Here, a recent nonlinear FETI-DP method is combined with an approach that allows an inexact solution of the FETI-DP coarse problem. We combine the nonlinear FETI-DP domain decomposition method with an AMG (Algebraic Multigrid) method and thus obtain a hybrid nonlinear domain decomposition/multigrid method. We consider scalar nonlinear problems as well as nonlinear hyperelasticity problems in two and three space dimensions. For the first time weak parallel scalability can be shown for a FETI-DP domain decomposition method beyond 100 000 cores. We can show weak parallel scalability for up to 262 144 cores on the Mira BlueGene/Q supercomputer for our new implementation. Our analysis then reveals that scalability beyond 262 144 cores depends critically on, both, the efficient construction and solution of the coarse problem. Indeed, for the current implementation it was not expected that the construction of the coarse problem is a limiting factor.