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

Abstract

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 \(L^2(\varOmega )\), 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 \(L^2(\varOmega )\)-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.