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

TL;DR

This paper introduces a nonlinear preconditioning technique combining Anderson accelerated Picard steps with Newton's method to improve convergence and stability when solving steady Navier-Stokes equations, especially at high Reynolds numbers.

Contribution

The paper proposes a simple nonlinear preconditioning method that enhances Newton's method for Navier-Stokes equations by adding Anderson accelerated Picard steps, ensuring global stability and quadratic convergence.

Findings

Method remains quadratically convergent.

Enlarges the convergence domain of Newton's method.

Numerical tests confirm theoretical predictions.

Abstract

The Newton's method for solving stationary Navier-Stokes equations (NSE) is known to convergent fast, however, may fail due to a bad initial guess. This work presents a simple-to-implement nonlinear preconditioning of Newton's iteration, that remains the quadratic convergence and enlarges the domain of convergence. The proposed AAPicard-Newton method adds the Anderson accelerated Picard step at each iteration of Newton's method for solving NSE, which has been shown globally stable for the relaxation parameter $\beta_{k+1}\equiv1$ in the Anderson acceleration optimization step, convergent quadratically, and converges faster with a smaller convergence rate for large Reynolds number. Several benchmark numerical tests have been tested and are well-aligned with the theoretical results.