Maximal regularity for the Stokes operator plays a crucial role in the theory of the non-stationary Navier--Stokes equations. In this paper, we consider the finite element semi-discretization of the non-stationary Stokes problem and establish the discrete counterpart of maximal regularity in $L^q$ for $q \in \left( \frac{2N}{N+2}, \frac{2N}{N-2} \right)$. For the proof of discrete maximal regularity, we introduce the temporally regularized Green's function. With the aid of this notion, we prove discrete maximal regularity without the Gaussian estimate. As an application, we present $L^p(0,T;L^q(\Omega))$-type error estimates for the approximation of the non-stationary Stokes problem.
翻译：Stokes算子的最大正则性在非定常Navier-Stokes方程的理论中发挥着关键作用。在本文中，我们考虑非定常Stokes问题的有限元半离散化，并建立$L^q$中（$q\in\left(\frac{2N}{N+2},\frac{2N}{N-2}\right)$）最大正则性的离散对应关系。为了证明离散最大正则性，我们引入时间正则化的格林函数。在这个概念的帮助下，我们在没有高斯估计的情况下证明了离散最大正则性。作为一个应用，我们提出了非定常Stokes问题的$L^p(0,T;L^q(\Omega))$型误差估计。
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