Space-Time Euler Discretization Schemes for the Stochastic 2D Navier-Stokes Equations

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Date
2020-05-06MFO Scientific Program
Research in Pairs 2019Series
Oberwolfach Preprints;2020,12Author
Bessaih, Hakima
Millet, Annie
Metadata
Show full item recordOWP-2020-12
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 $L^2(\Omega)$, 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.