Optimal analysis of finite element methods for the stochastic Stokes equations

TL;DR

This paper establishes optimal strong convergence rates for finite element methods applied to stochastic Stokes equations, improving error estimates and confirming them through numerical experiments.

Contribution

It provides the first optimal order error estimates for finite element schemes solving stochastic Stokes equations, using a novel semigroup analysis approach.

Findings

Optimal convergence rate of $O( au^{1/2}+ h^2)$ for velocity approximation.

Optimal convergence rate of $O( au^{1/2}+ h)$ for pressure time integral.

Numerical experiments confirm the theoretical error estimates.

Abstract

Numerical analysis for the stochastic Stokes equations is still challenging even though it has been well done for the corresponding deterministic equations. In particular, the pre-existing error estimates of finite element methods for the stochastic Stokes equations { in the $L^\infty(0, T; L^2(\Omega; L^2))$ norm} all suffer from the order reduction with respect to the spatial discretizations. The best convergence result obtained for these fully discrete schemes is only half-order in time and first-order in space, which is not optimal in space in the traditional sense. The objective of this article is to establish strong convergence of $O(\tau^{1/2}+ h^2)$ in the $L^\infty(0, T; L^2(\Omega; L^2))$ norm for approximating the velocity, and strong convergence of $O(\tau^{1/2}+ h)$ in the $L^{\infty}(0, T;L^2(\Omega;L^2))$ norm for approximating the time integral of pressure, where $\tau$ and $h$ denote the temporal step size and spatial mesh size, respectively. The error estimates are of optimal order for the spatial discretization considered in this article (with MINI element), and consistent with the numerical experiments. The analysis is based on the fully discrete Stokes semigroup technique and the corresponding new estimates.