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 $L^2(\Omega )$, 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.

References in corpus (5)

• Stochastic 2D hydrodynamical type systems: Well posedness and large deviations
• Galerkin approximations for the stochastic Burgers equation
• Splitting up method for the 2D stochastic Navier-Stokes equations
• On strong $L^2$ convergence of time numerical schemes for the stochastic 2D Navier-Stokes equations
• Strong convergence for explicit space-time discrete numerical approximation methods for stochastic Burgers equations

Cited by in corpus (5)

• Strong rates of convergence of space-time discretization schemes for the 2D Navier-Stokes equations with additive noise
• Speed of convergence of time Euler schemes for a stochastic 2D Boussinesq model
• Strong L2 convergence of time Euler schemes for stochastic 3D Brinkman-Forchheimer-Navier-Stokes equations
• Numerical analysis of 2D Navier--Stokes equations with additive stochastic forcing
• A splitting semi-implicit method for stochastic incompressible Euler equations on ${\mathbb{T}}^2$