  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 $L^\infty_tL^2_x\cap
L^2_tW^{1,2}_x$-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.
