Learning Adaptive Constraints in Nonlinear FETI-DP Methods

Abstract: While linear FETI-DP (Finite Element Tearing and Interconnecting - Dual Primal) is an efficient iterative domain decomposition solver for discretized linear PDEs (partial differential equations), nonlinear FETI-DP is its consequent extension to the nonlinear case. In both methods, the parallel efficiency of the method results from a decomposition of the computational domain into nonoverlapping subdomains and a resulting localization of the computational work. For a fast linear convergence of the linear FETI-DP method, a global coarse problem has to be considered. Adaptive coarse spaces are provably robust variants for many complicated micro-heterogeneous problems, as, for example, stationary diffusion problems with large jumps in the diffusion coefficient. Unfortunately, the set-up and exact computation of adaptive coarse spaces is known to be computationally expensive. Therefore, recently, surrogate models based on neural networks have been trained to directly predict the adaptive coarse constraints. Here, these learned constraints are implemented in nonlinear FETI-DP and it is shown numerically that they are able to improve the nonlinear as well as linear convergence speed of nonlinear FETI-DP.