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 measured in the Lt∞Lx2∩Lt2Wx1,2-norm). Our main result provides linear convergence in space and convergence of order (almost) 1/2 in time. This improves earlier results from Carelli and Prohl (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.