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

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 ${\beta }_{k+1}\equiv 1$ 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.