  In this paper, both semidiscrete and fully discrete finite element methods
are analyzed for the penalized two-dimensional unsteady Navier-Stokes equations
with nonsmooth initial data. First order backward Euler method is applied for
the time discretization, whereas conforming finite element method is used for
the spatial discretization. Optimal $L^2$ error estimates for the semidiscrete
as well as the fully discrete approximations of the velocity and of the
pressure are derived for realistically assumed conditions on the data. The main
ingredient in the proof is the appropriate exploitation of the inverse of the
penalized Stokes operator, negative norm estimates and time weighted estimates.
Numerical examples are discussed at the end which conform our theoretical
results.
