  In this article, we analyze a two-level finite element method for the two
dimensional time-dependent incompressible Navier-Stokes equations with
non-smooth initial data. It involves solving the non-linear Navier-Stokes
problem on a coarse grid of size $H$ and solving a Stokes problem on a fine
grid of size $h, h<<H$. This method gives optimal convergence for velocity in
$H^1$-norm and for pressure in $L^2$-norm. The analysis takes in to account the
loss of regularity of the solution at $t=0$ of the Navier-Stokes equations.
