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算子的极大正则性在非定常Navier--Stokes方程理论中起着关键作用。本文考虑非定常Stokes问题的有限元半离散化，建立$L^q$空间中$q \in \left( \frac{2N}{N+2}, \frac{2N}{N-2} \right)$情形下离散极大正则性的对应结论。为证明离散极大正则性，我们引入了时间正则化Green函数。借助这一概念，我们在无需高斯估计的情况下证明了离散极大正则性。作为应用，我们给出了非定常Stokes问题逼近的$L^p(0,T;L^q(\Omega))$型误差估计。