{
  "id": "0712.1241",
  "version": "v1",
  "published": "2007-12-07T22:52:34.000Z",
  "updated": "2007-12-07T22:52:34.000Z",
  "title": "Mixed finite element methods for the fully nonlinear Monge-Ampère equation based on the vanishing moment method",
  "authors": [
    "Xiaobing Feng",
    "Michael Neilan"
  ],
  "comment": "31 pages and 8 figures",
  "categories": [
    "math.NA",
    "math.AP"
  ],
  "abstract": "This paper studies mixed finite element approximations of the viscosity solution to the Dirichlet problem for the fully nonlinear Monge-Amp\\`ere equation $\\det(D^2u^0)=f$ based on the vanishing moment method which was proposed recently by the authors in \\cite{Feng2}. In this approach, the second order fully nonlinear Monge-Amp\\`ere equation is approximated by the fourth order quasilinear equation $-\\epsilon\\Delta^2 u^\\epsilon + \\det{D^2u^\\epsilon} =f$. It was proved in \\cite{Feng1} that the solution $u^\\epsilon$ converges to the unique convex viscosity solution $u^0$ of the Dirichlet problem for the Monge-Amp\\`ere equation. This result then opens a door for constructing convergent finite element methods for the fully nonlinear second order equations, a task which has been impracticable before. The goal of this paper is threefold. First, we develop a family of Hermann-Miyoshi type mixed finite element methods for approximating the solution $u^\\epsilon$ of the regularized fourth order problem, which computes simultaneously $u^\\vepsi$ and the moment tensor $\\sigma^\\vepsi:=D^2u^\\epsilon$. Second, we derive error estimates, which track explicitly the dependence of the error constants on the parameter $\\vepsi$, for the errors $u^\\epsilon-u^\\epsilon_h$ and $\\sigma^\\vepsi-\\sigma_h^\\vepsi$. Finally, we present a detailed numerical study on the rates of convergence in terms of powers of $\\vepsi$ for the error $u^0-u_h^\\vepsi$ and $\\sigma^\\vepsi-\\sigma_h^\\vepsi$, and numerically examine what is the \"best\" mesh size $h$ in relation to $\\vepsi$ in order to achieve these rates.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-12-07T22:52:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65N30",
      "65M60",
      "35J60",
      "53C45"
    ],
    "keywords": [
      "mixed finite element methods",
      "fully nonlinear monge-ampère equation",
      "vanishing moment method",
      "convergent finite element methods",
      "nonlinear second order equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0712.1241F"
    }
  }
}