{
  "id": "2307.14217",
  "version": "v1",
  "published": "2023-07-26T14:35:05.000Z",
  "updated": "2023-07-26T14:35:05.000Z",
  "title": "Error estimates for finite element discretizations of the instationary Navier-Stokes equations",
  "authors": [
    "Boris Vexler",
    "Jakob Wagner"
  ],
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "In this work we consider the two dimensional instationary Navier-Stokes equations with homogeneous Dirichlet/no-slip boundary conditions. We show error estimates for the fully discrete problem, where a discontinuous Galerkin method in time and inf-sup stable finite elements in space are used. Recently, best approximation type error estimates for the Stokes problem in the $L^\\infty(I;L^2(\\Omega))$, $L^2(I;H^1(\\Omega))$ and $L^2(I;L^2(\\Omega))$ norms have been shown. The main result of the present work extends the error estimate in the $L^\\infty(I;L^2(\\Omega))$ norm to the Navier-Stokes equations, by pursuing an error splitting approach and an appropriate duality argument. In order to discuss the stability of solutions to the discrete primal and dual equations, a specialized discrete Gronwall lemma is presented. The techniques developed towards showing the $L^\\infty(I;L^2(\\Omega))$ error estimate, also allow us to show best approximation type error estimates in the $L^2(I;H^1(\\Omega))$ and $L^2(I;L^2(\\Omega))$ norms, which complement this work.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-26T14:35:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q30",
      "65M60",
      "65M15",
      "65M22",
      "76D05",
      "76M10"
    ],
    "keywords": [
      "finite element discretizations",
      "best approximation type error estimates",
      "dimensional instationary navier-stokes equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}