This paper focuses on deriving optimal-order full moment error estimates in strong norms for both velocity and pressure approximations in the Euler-Maruyama time discretization of the stochastic Navier-Stokes equations with multiplicative noise. Additionally, it introduces a novel approach and framework for the numerical analysis of nonlinear stochastic partial differential equations (SPDEs) with multiplicative noise in general. The main ideas of this approach include establishing exponential stability estimates for the SPDE solution, leveraging a discrete stochastic Gronwall inequality, and employing a bootstrap argument.

翻译：本文主要针对具有乘性噪声的随机Navier-Stokes方程，在Euler-Maruyama时间离散格式下，推导速度和压力近似解在强范数意义下的最优阶全矩误差估计。此外，本文为一般具有乘性噪声的非线性随机偏微分方程（SPDE）的数值分析提出了一种新颖的方法和框架。该方法的核心思想包括：建立SPDE解的指数稳定性估计，利用离散随机Gronwall不等式，以及采用自举论证策略。