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.
翻译:我们研究一个以2D 随机导航-斯托克斯等式为基点的基于空间时间的有限分解,该等式在封闭的域内,加上无滑动边界条件;我们证明能源规范在概率趋同方面的最佳趋同率,即顺序(几乎)1/2在时间上和1在空间上的趋同率;这在以往仅发生在空间周期中才为人所知,即任何给定(确定性)时间的能量估计都具有较高的顺序;与此相反,在Drichlet个案中,估计的能量只知道一个(可能很大)停留时间;我们通过采用以离散停止时间为基础的方法克服了这一问题;这取代了先前贡献中(关于抽样空间)的本地化估计。