Optimal analysis of finite element methods for the stochastic Stokes equations

Numerical analysis for the stochastic Stokes/Navier-Stokes equations is still challenging even though it has been well done for the corresponding deterministic equations. In particular, the existing error estimates of finite element methods for the stochastic equations 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 purpose of this article is to establish the strong convergence of $O(\tau^{1/2}+ h^2)$ and $O(\tau^{1/2}+ h)$ in the $L^2$ norm for the inf-sup stable velocity-pressure finite element approximations, where $\tau$ and $h$ denote the temporal stepsize 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.