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Article Dans Une Revue Stochastics and Partial Differential Equations: Analysis and Computations Année : 2022

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

Résumé

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 $L^2(\Omega)$, 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 $L^2(\Omega)$-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.

Dates et versions

hal-02546901 , version 1 (19-04-2020)

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Hakima Bessaih, Annie Millet. Space-time Euler discretization schemes for the stochastic 2D Navier-Stokes equations. Stochastics and Partial Differential Equations: Analysis and Computations, 2022, 10, pp.1515-1558. ⟨10.1007/s40072-021-00217-7⟩. ⟨hal-02546901⟩
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