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

An inexact Newton's 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.