A simple-to-implement nonlinear preconditioning of Newton’s method for solving the steady Navier-Stokes equations

Abstract

Newton’s method for solving stationary Navier-Stokes equations (NSE) is known for its fast convergence; however, it may fail when provided with a poor initial guess. This work presents a simple-to-implement nonlinear preconditioning technique for Newton’s iteration that retains quadratic convergence and expands the domain of convergence. The proposed AAPicard-Newton method adds an Anderson accelerated Picard step at each iteration of Newton’s method for solving NSE. This approach has been shown to be globally stable with a relaxation parameter ${\beta }_{k+1}\equiv 1$ in the Anderson acceleration optimization step, converges quadratically, and achieves faster convergence with a small convergence rate for large Reynolds numbers. Several benchmark numerical tests have been carried out and are well aligned with the theoretical results.