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, in the Dirichlet-case estimates 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.

中文翻译：

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有界域中二维随机 Navier--Stokes 方程的误差分析

我们研究了在补充有无滑移边界条件的有界域中二维随机 Navier-Stokes 方程的基于有限元的时空离散化。我们证明了能量范数中关于概率收敛的最佳收敛率，即时间上（几乎）1/2 级和空间上 1 级的收敛。这以前仅在空间周期情况下才知道，在这种情况下，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，在 Dirichlet 情况下，估计仅因（可能很大）停止时间而闻名。我们通过引入一种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）。