We study a two-grid scheme fully discrete in time and
space for solving the Navier-Stokes system. In the first step, the
fully non-linear problem is discretized in space on a coarse grid
with mesh-size H and time step k. In the second step, the
problem is discretized in space on a fine grid with mesh-size h
and the same time step, and linearized around the velocity uH
computed in the first step. The two-grid strategy is motivated by
the fact that under suitable assumptions, the contribution of
uH to the error in the non-linear term, is measured in the
L2 norm in space and time, and thus has a higher-order than if
it were measured in the H1 norm in space. We present the
following results: if h = H2 = k, then the global error of
the two-grid algorithm is of the order of h, the same as would
have been obtained if the non-linear problem had been solved
directly on the
fine grid.