The main objective of the present paper is to construct a new class of space-time discretizations for the stochastic $p$-Stokes system and analyze its stability and convergence properties. We derive regularity results for the approximation that are similar to the natural regularity of solutions. One of the key arguments relies on discrete extrapolation that allows to relate lower moments of discrete maximal processes. We show that, if the generic spatial discretization is constraint conforming, then the velocity approximation satisfies a best-approximation property in the natural distance. Moreover, we present an example such that the resulting velocity approximation converges with rate $1/2$ in time and $1$ in space towards the (unknown) target velocity with respect to the natural distance.

翻译：本文的主要目标是构造随机$p$-Stokes系统的一类新型时空离散化方法，并分析其稳定性和收敛性。我们推导了近似解的与自然解的正则性相似的正则性结果。关键论证之一依赖于离散外推方法，该方法能够关联离散极大过程低阶矩。研究表明，若一般空间离散化满足约束相容性，则速度逼近解在自然距离下具有最佳逼近性质。此外，我们给出一个算例：在该算例中，所得速度逼近解在自然距离下以时间方向$1/2$阶和空间方向$1$阶的收敛率逼近（未知）目标速度。