We study stochastic Navier–Stokes equations in two dimensions with respect to periodic boundary conditions. The equations are perturbed by a nonlinear multiplicative stochastic forcing with linear growth (in the velocity) driven by a cylindrical Wiener process. We establish convergence rates for a finite-element based space-time approximation with respect to convergence in probability (where the error is measured in the \(L^\infty _tL^2_x\cap L^2_tW^{1,2}_x\)-norm). Our main result provides linear convergence in space and convergence of order (almost) 1/2 in time. This improves earlier results from Carelli and Prohl (SIAM J Numer Anal 50(5):2467–2496, 2012) where the convergence rate in time is only (almost) 1/4. Our approach is based on a careful analysis of the pressure function using a stochastic pressure decomposition.

我们研究关于周期边界条件的二维随机Navier-Stokes方程。方程由圆柱维纳过程驱动的线性增长（速度）的非线性乘法随机强迫扰动。我们针对概率收敛建立了基于有限元的时空近似的收敛速率（其中误差以\（L ^ \ infty _tL ^ 2_x \ cap L ^ 2_tW ^ {1,2} _x \ ）-规范）。我们的主要结果提供了空间上的线性收敛和时间上（几乎）1/2阶的收敛。这改善了Carelli和Prohl的早期结果（SIAM J Numer Anal 50（5）：2467-2496，2012），其中时间的收敛速度仅为（几乎）1/4。我们的方法是基于使用随机压力分解对压力函数的仔细分析。