{
  "id": "1807.01161",
  "version": "v1",
  "published": "2018-06-30T22:17:51.000Z",
  "updated": "2018-06-30T22:17:51.000Z",
  "title": "Duality in finite element exterior calculus",
  "authors": [
    "Yakov Berchenko-Kogan"
  ],
  "comment": "33 pages",
  "categories": [
    "math.NA",
    "math.DG"
  ],
  "abstract": "In order to generalize finite element methods to differential forms, Arnold, Falk, and Winther constructed two families of spaces of polynomial differential forms on a simplex $T$, the $\\mathcal P_r\\Lambda^k(T)$ spaces and the $\\mathcal P_r^-\\Lambda^k(T)$ spaces, where $k$ is the degree of the form and $r$ is the degree of its coefficients. The geometric decomposition for these finite element spaces hinges on a duality relationship between the $\\mathcal P$ and $\\mathcal P^-$ spaces proved by Arnold, Falk, and Winther. In this article, we give a natural alternate construction of the $\\mathcal P_r\\Lambda^k(T)$ and $\\mathcal P_r^-\\Lambda^k(T)$ spaces, leading to a new basis-free proof of this duality relationship using a modified Hodge star operator.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-30T22:17:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65N30",
      "58A10"
    ],
    "keywords": [
      "finite element exterior calculus",
      "finite element spaces hinges",
      "duality relationship",
      "polynomial differential forms",
      "generalize finite element methods"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}