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.
翻译:在本文中, 我们对基于时间的 $2D 无法压缩的 Navier- Stokes 模型应用不连续限制元素 Galerkin 方法。 我们用 $L infty (\ textbf{L ⁇ 2) 和$L infty (L ⁇ 2) 来得出速度和 $L infty (L}2) $-norm 的最佳误差估计值。 这些估计值是在新 $L2$- project- profty (L2) 和 $L infty (textbf{H}}2\ capt\ capt\ textbf{f}$ 和源代码 $$L\\ textrotherfty 模型帮助下设定的, 初始数据假设值下估算值一致。 然后, 分析一个基于 后向 Euler 方法的完全不公开的 方案, 和完全不公开的错误估算值是 $rentleflex- drolate road rodustryal- rodudealtial- dest rolevleb roislevle 。 我们想要在这里显示最高级L_ 和 ral- dexlexlexlexxxlal_ 。