Adaptive Nonlinear Domain Decomposition Methods with an Application to the $p$-Laplacian

Abstract

In this article, different nonlinear domain decomposition methods are applied to nonlinear problems with highly heterogeneous coefficient functions with jumps. In order to obtain a robust solver with respect to nonlinear as well as linear convergence, adaptive coarse spaces are employed. First, as an example for a nonlinearly left-preconditioned domain decomposition method, the two-level restricted nonlinear Schwarz method H1-RASPEN (Hybrid Restricted Additive Schwarz Preconditioned Exact Newton) is combined with an adaptive GDSW (generalized Dryja--Smith--Widlund) coarse space. Second, as an example for a nonlinearly right-preconditioned domain decomposition method, a nonlinear FETI-DP (Finite Element Tearing and Interconnecting--Dual Primal) method is equipped with an edge-based adaptive coarse space. Both approaches are compared with the respective nonlinear domain decomposition methods with classical coarse spaces as well as with the respective Newton--Krylov methods with adaptive coarse spaces. For some two-dimensional $p$-Laplace model problems with different spatial coefficient distributions, it can be observed that the best linear and nonlinear convergence can only be obtained when combining the nonlinear domain decomposition methods with adaptive coarse spaces.