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

04/15/2020

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by   Hakima Bessaih, et al.

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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(Ω), 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(Ω)-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.