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

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.

Figures (4)

• Figure 1: Brief algorithmic overview of additive or hybrid nonlinear two-level Schwarz.
• Figure 2: Convergence behavior of nonlinear two-level Schwarz and Newton-Krylov-Schwarz for $Re=1000$(top) and $Re=1500$(bottom) for the lid-driven cavity flow comparing the two different RGDSW coarse spaces.
• Figure 3: From left to right: Evolution of the coarse correction $T_0(u^{(k)})$; shown are the first, second, and fifth (last) iterations. The point-wise Euclidian norm of the velocity is plotted. Top row: Type A coarse space. Bottom row: Type B coarse space.
• Figure 4: From left to right: Evolution of the Newton iteration; shown are the first, second, and sixth iterations for nonlinear Schwarz (top row) and Newton-Krylov-Schwarz (bottom row). The point-wise Euclidian norm of the velocity is plotted.