This paper is concerned with high moment and pathwise error estimates for both velocity and pressure approximations of the Euler-Maruyama scheme for time discretization and its two fully discrete mixed finite element discretizations. The main idea for deriving the high moment error estimates for the velocity approximation is to use a bootstrap technique starting from the second moment error estimate. The pathwise error estimate, which is sub-optimal in the energy norm, is obtained by using Kolmogorov's theorem based on the high moment error estimates. Unlike for the velocity error estimate, the higher moment and pathwise error estimates for the pressure approximation are derived in a time-averaged norm. In addition, the impact of noise types on the rates of convergence for both velocity and pressure approximations is also addressed.

翻译：本文涉及时间离散和两个完全离散的混合有限元素离散的尤勒-马鲁山方案速度和压力近似值的高时值和路由错误估计。 计算速度近似值的高时值估计的主要想法是从第二时刻误差估计开始采用靴套技术。 在能源规范中,路径错误估计值低于最佳值,通过使用高时刻误差估计值获得。 与速度误差估计不同的是,压力近似值的较高时值和路由错误估计值是在一个平均时间的规范中得出的。 此外,还涉及噪音类型对速度和压力近似速率的趋同率的影响。