Numerical analysis for the stochastic Stokes equations is still challenging even though it has been well done for the corresponding deterministic equations. In particular, the pre-existing error estimates of finite element methods for the stochastic Stokes equations { in the $L^\infty(0, T; L^2(\Omega; L^2))$ norm} all suffer from the order reduction with respect to the spatial discretizations. The best convergence result obtained for these fully discrete schemes is only half-order in time and first-order in space, which is not optimal in space in the traditional sense. The objective of this article is to establish strong convergence of $O(\tau^{1/2}+ h^2)$ in the $L^\infty(0, T; L^2(\Omega; L^2))$ norm for approximating the velocity, and strong convergence of $O(\tau^{1/2}+ h)$ in the $L^{\infty}(0, T;L^2(\Omega;L^2))$ norm for approximating the time integral of pressure, where $\tau$ and $h$ denote the temporal step size and spatial mesh size, respectively. The error estimates are of optimal order for the spatial discretization considered in this article (with MINI element), and consistent with the numerical experiments. The analysis is based on the fully discrete Stokes semigroup technique and the corresponding new estimates.

翻译：随机Stokes方程的数值分析仍具有挑战性，尽管其对应确定性方程的数值分析已较为完善。特别地，现有有限元方法求解随机Stokes方程在$L^\infty(0, T; L^2(\Omega; L^2))$范数下的误差估计均存在空间离散阶数降低问题。这些全离散格式目前的最佳收敛结果仅为时间半阶与空间一阶，这在传统意义上并非空间最优。本文旨在建立速度逼近在$L^\infty(0, T; L^2(\Omega; L^2))$范数下的$O(\tau^{1/2}+ h^2)$强收敛性，以及压力的时间积分在$L^{\infty}(0, T;L^2(\Omega;L^2))$范数下的$O(\tau^{1/2}+ h)$强收敛性，其中$\tau$和$h$分别表示时间步长与空间网格尺寸。对于本文考虑的MINI单元空间离散，该误差估计达到最优阶，并与数值实验相符。分析基于全离散Stokes半群技术及相应的新估计方法。