Two-level nonlinear Schwarz methods - a parallel implementation with application to nonlinear elasticity and incompressible flow problems

Abstract

Nonlinear Schwarz methods are a type of nonlinear domain decomposition method used as an alternative to Newton's method for solving discretized nonlinear partial differential equations. In this article, the first parallel implementation of a two-level nonlinear Schwarz method leveraging the GDSW-type coarse spaces from the Fast and Robust Overlapping Schwarz (FROSch) framework in Trilinos is presented. This framework supports both additive and hybrid two-level nonlinear Schwarz methods and makes use of modifications to the coarse spaces constructed by FROSch to further enhance the robustness and convergence speed of the methods. Efficiency and excellent parallel performance of the software framework are demonstrated by applying it to two challenging nonlinear problems: the two-dimensional lid-driven cavity problem at high Reynolds numbers, and a Neo-Hookean beam deformation problem. The results show that two-level nonlinear Schwarz methods scale exceptionally well up to 9\,000 subdomains and are more robust than standard Newton-Krylov-Schwarz solvers for the considered Navier-Stokes problems with high Reynolds numbers or, respectively, for the nonlinear elasticity problems and large deformations. The new parallel implementation provides a foundation for future research in scalable nonlinear domain decomposition methods and demonstrates the practical viability of nonlinear Schwarz techniques for large-scale simulations.

Figures (17)

• Figure 1: RGDSW, MsFEM and GDSW interface vertex functions for the vertex $\nu$ at the interface between four subdomains. The interface is shown in black, the original mesh in grey and the interface vertex function in green. Figure \ref{['fig:gdsw-edge-function']} shows a GDSW edge function corresponding to the edge marked in red.
• Figure 1: Abstract classes that form the base of our nonlinear Schwarz implementation.
• Figure 1: Time-to-solution as a function of the number of subdomains for the two-level nonlinear Schwarz solver with Reynolds numbers $Re \in \{1000, 2000, 4000\}$. Runtimes are split into contributions from the inner solver, coarse solver, GMRES solver, and remaining overhead. Above each bar, the total number of GMRES and Newton iterations are shown as "GMRES iters. (Newton iters.)".
• Figure 2: Interface values for a monolithic coarse basis function for the velocity component in $x$-direction.
• Figure 2: Class hierarchy of our one-level nonlinear Schwarz implementation.