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

TL;DR

This paper introduces a new class of space-time discretizations for the stochastic p-Stokes system, demonstrating stability, convergence, and optimal approximation properties through theoretical analysis and numerical validation.

Contribution

It develops a novel discretization framework for the stochastic p-Stokes system, establishing stability, convergence rates, and best-approximation properties under certain conditions.

Findings

Velocity approximation converges with rate 1/2 in time.

Spatial convergence rate is 1.

Numerical experiments support theoretical results.

Abstract

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. The theory is corroborated by numerical experiments.