A novel implicit cell-vertex finite volume method is described for the solution of the Navier-Stokes equations at high Reynolds numbers. The key idea is the elimination of the pressure term from the momentum equation by multiplying the momentum equation with the unit normal vector to a control volume boundary and integrating thereafter around this boundary. The resulting equations are expressed solely in terms of the velocity components. Thus any difficulties with pressure or vorticity boundary conditions are circumvented and the number of primary variables that need to be determined equals the number of space dimensions. The method is applied to both the steady and unsteady two-dimensional lid-driven cavity problem at Reynolds numbers up to 10000. Results are compared with those in the literature and show excellent agreement. Copyright (C) 2003 John Wiley Sons, Ltd.