  We consider two-grid mixed-finite element schemes for the spatial
discretization of the incompressible Navier-Stokes equations. A standard
mixed-finite element method is applied over the coarse grid to approximate the
nonlinear Navier-Stokes equations while a linear evolutionary problem is solved
over the fine grid. The previously computed Galerkin approximation to the
velocity is used to linearize the convective term. For the analysis we take
into account the lack of regularity of the solutions of the Navier-Stokes
equations at the initial time in the absence of nonlocal compatibility
conditions of the data. Optimal error bounds are obtained.
