Nonlinear Two-Level Schwarz Methods: A Parallel Implementation in FROSch

TL;DR

A novel parallel implementation of a two-level nonlinear Schwarz solver based on the FROSch (Fast and Robust Overlapping Schwarz) solver framework, part of Sandia's Trilinos library is introduced.

Abstract

Owing to the ability of nonlinear domain decomposition methods to improve the nonlinear convergence behavior of Newton's method, they have experienced a rise in popularity recently in the context of problems for which Newton's method converges slowly or not at all. This article introduces a novel parallel implementation of a two-level nonlinear Schwarz solver based on the FROSch (Fast and Robust Overlapping Schwarz) solver framework, part of Sandia's Trilinos library. First, an introduction to the key concepts underlying two-level nonlinear Schwarz methods is given, including a brief overview of the coarse space used to build the second level. Next, the parallel implementation is discussed, followed by preliminary parallel results for a scalar nonlinear diffusion problem and a 2D nonlinear plane-stress Neo-Hooke elasticity problem with large deformations.

Figures (2)

• Figure 1: Time to solution of the one-level and two-level additive and hybrid nonlinear Schwarz variants for the nonlinear diffusion problem \ref{['eq:nonlindiff']}. The outer Newton iterations and total number of GMRES iterations summed over all outer iterations is shown above each column.
• Figure 2: Time to solution of a classical Newton-Krylov-Schwarz (NKS) solver, two-level additive and two-level hybrid nonlinear Schwarz variants for the nonlinear elasticity problem \ref{['eq:nonlinelas']} solved using $576$ subdomains. The outer Newton iterations and total number of GMRES iterations summed over all outer iterations is shown above each column. A red cross indicates that the solver failed to converge.