{
  "id": "math/0410184",
  "version": "v1",
  "published": "2004-10-06T20:29:45.000Z",
  "updated": "2004-10-06T20:29:45.000Z",
  "title": "Interior numerical approximation of boundary value problems with a distributional data",
  "authors": [
    "Ivo Babuska",
    "Victor Nistor"
  ],
  "comment": "23 pages",
  "categories": [
    "math.NA",
    "math.AP"
  ],
  "abstract": "We study the approximation properties of a harmonic function $u \\in H\\sp{1-k}(\\Omega)$, $k > 0$, on relatively compact sub-domain $A$ of $\\Omega$, using the Generalized Finite Element Method. For smooth, bounded domains $\\Omega$, we obtain that the GFEM--approximation $u_S$ satisfies $\\|u - u_S\\|_{H\\sp{1}(A)} \\le C h^{\\gamma}\\|u\\|_{H\\sp{1-k}(\\Omega)}$, where $h$ is the typical size of the ``elements'' defining the GFEM--space $S$ and $\\gamma \\ge 0 $ is such that the local approximation spaces contain all polynomials of degree $k + \\gamma + 1$. The main technical result is an extension of the classical super-approximation results of Nitsche and Schatz \\cite{NitscheSchatz72} and, especially, \\cite{NitscheSchatz74}. It turns out that, in addition to the usual ``energy'' Sobolev spaces $H^1$, one must use also the negative order Sobolev spaces $H\\sp{-l}$, $l \\ge 0$, which are defined by duality and contain the distributional boundary data.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-10-06T20:29:45.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "boundary value problems",
      "interior numerical approximation",
      "distributional data",
      "local approximation spaces contain",
      "negative order sobolev spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math.....10184B"
    }
  }
}