Numerical schemes for the 2d Stochastic Navier-Stokes equations.

We consider a time discretization scheme of Euler type for the 2d stochastic Navier-Stokes equations on the torus.
We prove a mean square rate of convergence. This refines previous results established with a rate of convergence in probability only.
Using exponential moment estimates of the solution of the Navier-Stokes equations and a convergence of a localized scheme, we can prove strong convergence of fully implicit and semi-implicit time Euler discretization and also a splitting scheme. The speed of convergence depends on the diffusion coefficient and the viscosity parameter.
When the noise is additive, we are able to get strong convergence without localization.