In this work we consider the two dimensional instationary Navier-Stokes equations with homogeneous Dirichlet/no-slip boundary conditions. We show error estimates for the fully discrete problem, where a discontinuous Galerkin method in time and inf-sup stable finite elements in space are used. Recently, best approximation type error estimates for the Stokes problem in the $L^\infty(I;L^2(\Omega))$, $L^2(I;H^1(\Omega))$ and $L^2(I;L^2(\Omega))$ norms have been shown. The main result of the present work extends the error estimate in the $L^\infty(I;L^2(\Omega))$ norm to the Navier-Stokes equations, by pursuing an error splitting approach and an appropriate duality argument. In order to discuss the stability of solutions to the discrete primal and dual equations, a specially tailored discrete Gronwall lemma is presented. The techniques developed towards showing the $L^\infty(I;L^2(\Omega))$ error estimate, also allow us to show best approximation type error estimates in the $L^2(I;H^1(\Omega))$ and $L^2(I;L^2(\Omega))$ norms, which complement this work.

翻译：本文考虑具有齐次Dirichlet/无滑移边界条件的二维非定常Navier-Stokes方程。我们证明了全离散问题的误差估计，其中时间方向采用间断Galerkin方法，空间方向采用inf-sup稳定的有限元。近期已有研究给出了Stokes问题在$L^\infty(I;L^2(\Omega))$、$L^2(I;H^1(\Omega))$和$L^2(I;L^2(\Omega))$范数下的最佳逼近型误差估计。本文的主要结果通过采用误差分裂方法和适当的对偶论证，将$L^\infty(I;L^2(\Omega))$范数下的误差估计推广至Navier-Stokes方程。为讨论离散原始方程与对偶方程解的稳定性，本文提出了一个专门设计的离散Gronwall引理。本文发展的用于建立$L^\infty(I;L^2(\Omega))$范数误差估计的技巧，还使我们能够证明$L^2(I;H^1(\Omega))$和$L^2(I;L^2(\Omega))$范数下的最佳逼近型误差估计，从而完善了本文的工作。