This paper is concerned with stochastic incompressible Navier-Stokes equations with multiplicative noise in two dimensions with respect to periodic boundary conditions. Based on the Helmholtz decomposition of the multiplicative noise, semi-discrete and fully discrete time-stepping algorithms are proposed. The convergence rates for mixed finite element methods based time-space approximation with respect to convergence in probability for the velocity and the pressure are obtained. Furthermore, with establishing some stability and using the negative norm technique, the partial expectations of the $H^1$ and $L^2$ norms of the velocity error are proved to converge optimally.

翻译：本文件涉及在定期边界条件下在两个维度上具有多复制性噪音的不切实际压缩纳维-斯托克斯方程式,根据Helmholtz对多复制噪音、半分解和完全离散的时间步调算法的分解,提出了混合有限元素方法基于时间-空间近似率与速度和压力的概率趋同率的趋同率;此外,由于建立了某种稳定性并使用负规范技术,速度误差的1美元和2美元部分预期值被证明最佳地趋同。