  A two-grid scheme based on mixed finite-element approximations to the
incompressible Navier-Stokes equations is introduced and analyzed. In the first
level the standard mixed finite-element approximation over a coarse mesh is
computed. In the second level the approximation is postprocessed by solving a
discrete Oseen-type problem on a finer mesh. The two-level method is optimal in
the sense that, when a suitable value of the coarse mesh diameter is chosen, it
has the rate of convergence of the standard mixed finite-element method over
the fine mesh. Alternatively, it can be seen as a postprocessed method in which
the rate of convergence is increased by one unit with respect to the coarse
mesh. The analysis takes into account the loss of regularity at initial time of
the solution of the Navier-Stokes equations in absence of nonlocal
compatibility conditions. Some numerical experiments are shown.
