We consider a two-level method for resolving the nonlinearity in finite element approximations of the equilibrium Navier-Stokes equations. The method yields L 2 and H 1 optimal velocity approximations and an L 2 optimal pressure approximation. The two-level method involves solving one small, nonlinear coarse mesh system, one Oseen problem (hence, linear with positive definite symmetric part) on the fine mesh, and one linear correction problem on the coarse mesh. The algorithm we study produces an approximate solution with the optimal, asymptotic in h, accuracy for any fixed Reynolds number. We do not consider the behavior of the error for fixed h as Re → ∞, i.e., for flows in transition to turbulence.