{
  "id": "2308.01580",
  "version": "v1",
  "published": "2023-08-03T07:35:58.000Z",
  "updated": "2023-08-03T07:35:58.000Z",
  "title": "Finite element approximation of the Hardy constant",
  "authors": [
    "Francesco Della Pietra",
    "Giovanni Fantuzzi",
    "Liviu I. Ignat",
    "Alba Lia Masiello",
    "Gloria Paoli",
    "Enrique Zuazua"
  ],
  "comment": "18 pages, 6 figures",
  "categories": [
    "math.NA",
    "cs.NA",
    "math.AP"
  ],
  "abstract": "We consider finite element approximations to the optimal constant for the Hardy inequality with exponent $p=2$ in bounded domains of dimension $n=1$ or $n\\geq 3$. For finite element spaces of piecewise linear and continuous functions on a mesh of size $h$, we prove that the approximate Hardy constant, $S_h^n$, converges to the optimal Hardy constant $S^n$ no slower than $O(1/\\vert \\log h \\vert)$. We also show that the convergence is no faster than $O(1/\\vert \\log h \\vert^2)$ if $n=1$ or if $n\\geq 3$, the domain is the unit ball, and the finite element discretization exploits the rotational symmetry of the problem. Our estimates are compared to exact values for $S_h^n$ obtained computationally.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-03T07:35:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65N30",
      "46E35"
    ],
    "keywords": [
      "finite element approximation",
      "finite element discretization exploits",
      "approximate hardy constant",
      "optimal hardy constant",
      "finite element spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}