On strong $L^2$ convergence of time numerical schemes for the stochastic 2D Navier-Stokes equations
Résumé
We prove that some time discretization schemes for the 2D Navier-Stokes equations on the torus subject to a random perturbation converge in $L^2(\Omega)$. 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 fully implicit and semi-implicit time Euler discretizations, and of a splitting scheme. The speed of the $L^2(\Omega)$-convergence depends on the diffusion coefficient and on the viscosity parameter.
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