Optimal Error Estimates of a Discontinuous Galerkin Method for the Navier-Stokes Equations

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.