{
  "id": "2105.12973",
  "version": "v1",
  "published": "2021-05-27T07:31:11.000Z",
  "updated": "2021-05-27T07:31:11.000Z",
  "title": "$H^m$-Conforming Virtual Elements in Arbitrary Dimension",
  "authors": [
    "Xuehai Huang"
  ],
  "comment": "29 pages",
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "The $H^m$-conforming virtual elements of any degree $k$ on any shape of polytope in $\\mathbb R^n$ with $m, n\\geq1$ and $k\\geq m$ are recursively constructed by gluing conforming virtual elements on faces in a universal way. For the lowest degree case $k=m$, the set of degrees of freedom only involves function values and derivatives up to order $m-1$ at the vertices of the polytope. The inverse inequality and several norm equivalences for the $H^m$-conforming virtual elements are rigorously proved. The $H^m$-conforming virtual elements are then applied to discretize a polyharmonic equation with a lower order term. With the help of the interpolation error estimate and norm equivalences, the optimal error estimates are derived for the $H^m$-conforming virtual element method.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-27T07:31:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65N12",
      "65N15",
      "65N22",
      "65N30"
    ],
    "keywords": [
      "arbitrary dimension",
      "norm equivalences",
      "lowest degree case",
      "optimal error estimates",
      "interpolation error estimate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}