Silvester, DavidElman, HowardKay, DavidWathen, AndrewWe outline a new class of robust and efficient methods for solving subproblems that arise in the linearization and operator splitting of Navier-Stokes equations. We describe a very general strategy for preconditioning that has two basic building blocks; a multigrid V-cycle for the scalar convection-diffusion operator, and a multigrid V-cycle for a pressure Poisson operator. We present numerical experiments illustrating that a simple implementation of our approach leads to an effective and robust solver strategy in that the convergence rate is independent of the grid and the time-step, and only deteriorates very slowly as the Reynolds number is increased. (Also cross-referenced as UMIACS-TR-99-66)en-USTechnical Report