Convergence rates for the numerical approximation of the 2D stochastic Navier-Stokes equations

We study stochastic Navier-Stokes equations in two dimensions with respect to periodic boundary conditions. The equations are perturbed by a nonlinear multiplicative stochastic forcing with linear growth (in the velocity) driven by a cylindrical Wiener process. We establish convergence rates for a finite-element based space-time approximation with respect to convergence in probability (where the error is measure in the $L^{\inftytL2x\cap} L^{2tW{1,2}x$-norm).} Our main result provides linear convergence in space and convergence of order (almost) 1/2 in time. This improves earlier results from [E. Carelli, A. Prohl: Rates of convergence for discretizations of the stochastic incompressible Navier-Stokes equations. SIAM J. Numer. Anal. 50(5), 2467-2496. (2012)] where the convergence rate in time is only (almost) 1/4. Our approach is based on a careful analysis of the pressure function using a stochastic pressure decomposition.