A class of space-time discretizations for the stochastic $p$-Stokes system

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 $1\mathrm{/}2$ in time and $1$ in space towards the (unknown) target velocity with respect to the natural distance. The theory is corroborated by numerical experiments.