{
  "id": "1801.05879",
  "version": "v1",
  "published": "2018-01-17T22:38:31.000Z",
  "updated": "2018-01-17T22:38:31.000Z",
  "title": "Analysis of the Vanishing Moment Method and its Finite Element Approximations for Second-order Linear Elliptic PDEs in Non-divergence Form",
  "authors": [
    "Xiaobing Feng",
    "Thomas Lewis",
    "Stefan Schnake"
  ],
  "categories": [
    "math.NA"
  ],
  "abstract": "This paper is concerned with continuous and discrete approximations of $W^{2,p}$ strong solutions of second-order linear elliptic partial differential equations (PDEs) in non-divergence form. The continuous approximation of these equations is achieved through the Vanishing Moment Method (VMM) which adds a small biharmonic term to the PDE. The structure of the new fourth-order PDE is a natural fit for Galerkin-type methods unlike the original second order equation since the highest order term is in divergence form. The well-posedness of the weak form of the perturbed fourth order equation is shown as well as error estimates for approximating the strong solution of the original second-order PDE. A $C^1$ finite element method is then proposed for the fourth order equation, and its existence and uniqueness of solutions as well as optimal error estimates in the $H^2$ norm are shown. Lastly, numerical tests are given to show the validity of the method.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-17T22:38:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65N12",
      "65N15",
      "65N30"
    ],
    "keywords": [
      "second-order linear elliptic pdes",
      "vanishing moment method",
      "finite element approximations",
      "non-divergence form",
      "elliptic partial differential equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}