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