Efficient solution of the steady-state Navier-Stokes equations using a multigrid preconditioned Newton-Krylov method

An inexact Newtonʹs method is used to solve the steady-state incompressible Navier-Stokes equations. The equations are discretized using a mixed finite element approximation. A new efficient preconditioning methodology introduced by Kay et al. (SIAM J. Sci. Comput., 2002; 24: 237-256) is applied and its effectiveness in the context of a Newton linearization is investigated. The original strategy was introduced as a preconditioning methodology for discrete Oseen equations that arise from Picard linearization. Our new variant of the preconditioning strategy is constructed from building blocks consisting of two component multigrid cycles; a multigrid V-cycle for a scalar convection-diffusion operator; and a multigrid V-cycle for a pressure Poisson operator. We present numerical experiments showing that the convergence rate of the preconditioned GMRES is independent of the grid size and relatively insensitive to the Reynolds number.