Space-time Euler discretization schemes for the stochastic 2D Navier-Stokes equations

We prove that the implicit time Euler scheme coupled with finite elements space discretization for the 2D Navier-Stokes equations on the torus subject to a random perturbation converges in L-2(Omega), and describe the rate of convergence for an H-1-valued initial condition. This refines previous results which only established the convergence in probability of these numerical approximations. Using exponential moment estimates of the solution of the stochastic Navier-Stokes equations and convergence of a localized scheme, we can prove strong convergence of this space-time approximation. The speed of the L-2(Omega)-convergence depends on the diffusion coefficient and on the viscosity parameter. In case of Scott-Vogelius mixed elements and for an additive noise, the convergence is polynomial.