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

TL;DR

This paper presents the first parallel implementation of a two-level nonlinear Schwarz method using GDSW-type coarse spaces, demonstrating excellent scalability and robustness for large-scale nonlinear PDE problems like high Reynolds number flows and nonlinear elasticity.

Contribution

The paper introduces a novel parallel implementation of a two-level nonlinear Schwarz method with GDSW coarse spaces, enhancing robustness and scalability for complex nonlinear PDEs.

Findings

Scales effectively up to 9,000 subdomains.

More robust than standard Newton-Krylov-Schwarz methods for high Reynolds number Navier-Stokes.

Demonstrates practical viability for large-scale nonlinear simulations.

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.