Strong rates of convergence of space-time discretization schemes for the 2D Navier–Stokes equations with additive noise

We consider the strong solution of the 2D Navier–Stokes equations in a torus subject to an additive noise. We implement a fully implicit time numerical scheme and a finite element method in space. We prove that the space-time rate of convergence is the “optimal” one, namely, [Formula: see text] in time and 1 in space. Let us mention that the coefficient [Formula: see text] is equal to the time regularity of the solution with values in [Formula: see text]. Our method relies on the existence of finite exponential moments for both the solution and its time approximation. Unlike previous results, our main new idea is the use of a discrete Grönwall lemma for the error estimate without any localization.