Nonlinear Monolithic Two-Level Schwarz Methods for the Navier-Stokes Equations

TL;DR

This paper introduces a novel nonlinear two-level Schwarz method with monolithic GDSW coarse basis functions for Navier-Stokes equations, demonstrating improved convergence for high Reynolds number lid-driven cavity problems.

Contribution

It is the first to combine monolithic GDSW coarse basis functions with a nonlinear Schwarz approach for Navier-Stokes equations, enhancing nonlinear convergence.

Findings

Improved convergence for high Reynolds number problems.

Comparison shows advantages over classical Newton's method.

Inclusion of local pressure corrections and basis recycling enhances performance.

Abstract

Nonlinear domain decomposition methods became popular in recent years since they can improve the nonlinear convergence behavior of Newton's method significantly for many complex problems. In this article, a nonlinear two-level Schwarz approach is considered and, for the first time, equipped with monolithic GDSW (Generalized Dryja-Smith-Widlund) coarse basis functions for the Navier-Stokes equations. Results for lid-driven cavity problems with high Reynolds numbers are presented and compared with classical global Newton's method equipped with a linear Schwarz preconditioner. Different options, for example, local pressure corrections on the subdomain and recycling of coarse basis functions are discussed in the nonlinear Schwarz approach for the first time.