{
  "id": "1701.07415",
  "version": "v1",
  "published": "2017-01-25T18:20:29.000Z",
  "updated": "2017-01-25T18:20:29.000Z",
  "title": "On the numerical approximation of $p$-Biharmonic and $\\infty$-Biharmonic functions",
  "authors": [
    "Nikos Katzourakis",
    "Tristan Pryer"
  ],
  "comment": "21 pages, 5 figures",
  "categories": [
    "math.NA",
    "math.AP"
  ],
  "abstract": "In [KP16] (https://arxiv.org/pdf/1605.07880) the authors introduced a second order variational problem in $L^{\\infty}$. The associated equation, coined the $\\infty$-Bilaplacian, is a \\emph{third order} fully nonlinear PDE given by $\\Delta^2_\\infty u\\, := (\\Delta u)^3 | D (\\Delta u) |^2 = 0.$ In this work we build a numerical method aimed at quantifying the nature of solutions to this problem which we call $\\infty$-Biharmonic functions. For fixed $p$ we design a mixed finite element scheme for the pre-limiting equation, the $p$-Bilaplacian $\\Delta^2_p u\\, := \\Delta(| \\Delta u |^{p-2} \\Delta u) = 0.$ We prove convergence of the numerical solution to the weak solution of $\\Delta^2_p u = 0$ and show that we are able to pass to the limit $p\\to\\infty$. We perform various tests aimed at understanding the nature of solutions of $\\Delta^2_\\infty u$ and in 1-$d$ we prove convergence of our discretisation to an appropriate weak solution concept of this problem, that of $\\mathcal D$-solutions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-25T18:20:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "biharmonic functions",
      "numerical approximation",
      "appropriate weak solution concept",
      "second order variational problem",
      "mixed finite element scheme"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}