{
  "id": "2301.02077",
  "version": "v1",
  "published": "2023-01-05T14:23:24.000Z",
  "updated": "2023-01-05T14:23:24.000Z",
  "title": "On the $L^\\infty(0,T;L^2(Ω)^d)$-stability of Discontinuous Galerkin schemes for incompressible flows",
  "authors": [
    "Pablo Alexei Gazca-Orozco",
    "Alex Kaltenbach"
  ],
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "The property that the velocity $\\boldsymbol{u}$ belongs to $L^\\infty(0,T;L^2(\\Omega)^d)$ is an essential requirement in the definition of energy solutions of models for incompressible fluids; it is, therefore, highly desirable that the solutions produced by discretisation methods are uniformly stable in the $L^\\infty(0,T;L^2(\\Omega)^d)$-norm. In this work, we establish that this is indeed the case for Discontinuous Galerkin (DG) discretisations (in time and space) of non-Newtonian implicitly constituted models with $p$-structure, in general, assuming that $p\\geq \\frac{3d+2}{d+2}$; the time discretisation is equivalent to a RadauIIA Implicit Runge-Kutta method. To aid in the proof, we derive Gagliardo-Nirenberg-type inequalities on DG spaces, which might be of independent interest",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-05T14:23:24.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "discontinuous galerkin schemes",
      "incompressible flows",
      "radauiia implicit runge-kutta method",
      "essential requirement",
      "energy solutions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}