  In this paper, we apply discontinuous finite element Galerkin method to the
time-dependent $2D$ incompressible Navier-Stokes model. We derive optimal error
estimates in $L^\infty(\textbf{L}^2)$-norm for the velocity and in
$L^\infty(L^2)$-norm for the pressure with the initial data $\textbf{u}_0\in
\textbf{H}_0^1\cap \textbf{H}^2$ and the source function $\textbf{f}$ in
$L^\infty(\textbf{L}^2)$ space. These estimates are established with the help
of a new $L^2$-projection and modified Stokes operator on appropriate broken
Sobolev space and with standard parabolic or elliptic duality arguments.
Estimates are shown to be uniform under the smallness assumption on data. Then,
a completely discrete scheme based on the backward Euler method is analyzed,
and fully discrete error estimates are derived. We would like to highlight here
that the estiablished semi-discrete error estimates related to the
$L^\infty(\textbf{L}^2)$-norm of velocity and $L^\infty(L^2)$-norm of pressure
are optimal and sharper than those derived in the earlier articles. Finally,
numerical examples validate our theoretical findings.
