{
  "id": "2004.06932",
  "version": "v1",
  "published": "2020-04-15T08:05:55.000Z",
  "updated": "2020-04-15T08:05:55.000Z",
  "title": "Space-time Euler discretization schemes for the stochastic 2D Navier-Stokes equations",
  "authors": [
    "Hakima Bessaih",
    "Annie Millet"
  ],
  "categories": [
    "math.PR",
    "cs.NA",
    "math.NA"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-15T08:05:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H15",
      "60H35",
      "76D06",
      "76M35"
    ],
    "keywords": [
      "stochastic 2d navier-stokes equations",
      "space-time euler discretization schemes",
      "convergence",
      "implicit time euler scheme"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}