{
  "id": "1809.03790",
  "version": "v1",
  "published": "2018-09-11T10:59:28.000Z",
  "updated": "2018-09-11T10:59:28.000Z",
  "title": "Laplacian preconditioning of elliptic PDEs: Localization of the eigenvalues of the discretized operator",
  "authors": [
    "Tomáš Gergelits",
    "Kent-André Mardal",
    "Bjørn Fredrik Nielsen",
    "Zdeněk Strakoš"
  ],
  "categories": [
    "math.NA"
  ],
  "abstract": "In the paper \\textit{Preconditioning by inverting the {L}aplacian; an analysis of the eigenvalues. IMA Journal of Numerical Analysis 29, 1 (2009), 24--42}, Nielsen, Hackbusch and Tveito study the operator generated by using the inverse of the Laplacian as preconditioner for second order elliptic PDEs $\\nabla \\cdot (k(x) \\nabla u) = f$. They prove that the range of $k(x)$ is contained in the spectrum of the preconditioned operator, provided that $k$ is continuous. Their rigorous analysis only addresses mappings defined on infinite dimensional spaces, but the numerical experiments in the paper suggest that a similar property holds in the discrete case. % Motivated by this investigation, we analyze the eigenvalues of the matrix $\\bf{L}^{-1}\\bf{A}$, where $\\bf{L}$ and ${\\bf{A}}$ are the stiffness matrices associated with the Laplace operator and general second order elliptic operators, respectively. Without any assumption about the continuity of $k(x)$, we prove the existence of a one-to-one pairing between the eigenvalues of $\\bf{L}^{-1}\\bf{A}$ and the intervals determined by the images under $k(x)$ of the supports of the FE nodal basis functions. As a consequence, we can show that the nodal values of $k(x)$ yield accurate approximations of the eigenvalues of $\\bf{L}^{-1}\\bf{A}$. Our theoretical results are illuminated by several numerical experiments.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-09-11T10:59:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65F08",
      "65F15",
      "65N12",
      "35J99"
    ],
    "keywords": [
      "eigenvalues",
      "discretized operator",
      "laplacian preconditioning",
      "general second order elliptic operators",
      "second order elliptic pdes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}