We study a finite-element based space-time discretisation for the 2D stochastic Navier–Stokes equations in a bounded domain supplemented with no-slip boundary conditions. We prove optimal convergence rates in the energy norm with respect to convergence in probability, that is convergence of order (almost) 1/2 in time and 1 in space. This was previously only known in the space-periodic case, where higher order energy estimates for any given (deterministic) time are available. In contrast to this, estimates in the Dirichlet-case are only known for a (possibly large) stopping time. We overcome this problem by introducing an approach based on discrete stopping times. This replaces the localised estimates (with respect to the sample space) from earlier contributions.

我们研究了有界域中二维随机纳维-斯托克斯方程的基于有限元的时空离散化，并补充了无滑移边界条件。我们证明了能量范数中相对于概率收敛的最佳收敛率，即时间上（几乎）1/2 阶和空间上 1 阶的收敛。此前，这仅在空间周期情况下才为人所知，其中任何给定（确定性）时间的高阶能量估计都是可用的。与此相反，狄利克雷情况下的估计仅已知（可能很大）停止时间。我们通过引入基于离散停止时间的方法来克服这个问题。这取代了早期贡献的局部估计（相对于样本空间）。