Additive and hybrid nonlinear two-level schwarz methods and energy minimizing coarse spaces for unstructured grids

Abstract

Nonlinear domain decomposition (DD) methods, such as ASPIN (additive Schwarz preconditioned inexact Newton), RASPEN (restricted additive Schwarz preconditioned inexact Newton), nonlinear FETI-DP (finite element tearing and interconnecting-dual primal), and nonlinear BDDC (balancing DD by constraints), can be reasonable alternatives to classical Newton-Krylov-DD methods for the solution of sparse nonlinear systems of equations, e.g., arising from a discretization of a nonlinear partial differential equation (PDE). These nonlinear DD approaches are often able to effectively tackle unevenly distributed nonlinearities and outperform Newton's method with respect to convergence speed as well as global convergence behavior. Furthermore, they often improve parallel scalability due to a superior ratio of local to global work. Nonetheless, as for linear DD methods, it is often necessary to incorporate an appropriate coarse space in a second level to obtain numerical scalability for increasing numbers of subdomains. In addition, an appropriate coarse space can also improve the nonlinear convergence of nonlinear DD methods. In this paper, we introduce four variants for integrating coarse spaces in nonlinear Schwarz methods in an additive or multiplicative way using Galerkin projections. These new variants can be interpreted as natural nonlinear equivalents to well-known linear additive and hybrid two-level Schwarz preconditioners. Furthermore, they facilitate the use of various coarse spaces, e.g., coarse spaces based on energy-minimizing extensions, which can easily be used for irregular DDs, such as, e.g., those obtained by graph partitioners. In particular, multiscale finite element method (MsFEM)-Type coarse spaces are considered, and it is shown that they outperform classical approaches for certain heterogeneous nonlinear problems. The new approaches are then compared with classical Newton-Krylov-DD and nonlinear one-level Schwarz approaches for different homogeneous and heterogeneous model problems based on the p-Laplace operator.