{
  "id": "1610.01786",
  "version": "v1",
  "published": "2016-10-06T09:26:38.000Z",
  "updated": "2016-10-06T09:26:38.000Z",
  "title": "Discrete $p$-robust $\\mathbf{H}(\\mathrm{div})$-liftings and a posteriori estimates for elliptic problems with $H^{-1}$ source terms",
  "authors": [
    "Alexandre Ern",
    "Iain Smears",
    "Martin Vohralík"
  ],
  "categories": [
    "math.NA"
  ],
  "abstract": "We establish the existence of liftings into discrete subspaces of $\\mathbf{H}(\\mathrm{div})$ of piecewise polynomial data on locally refined simplicial partitions of polygonal/polyhedral domains. Our liftings are robust with respect to the polynomial degree. This result has important applications in the a posteriori error analysis of parabolic problems, where it permits the removal of so-called transition conditions that link two consecutive meshes. It can also be used in a the posteriori error analysis of elliptic problems, where it allows the treatment of meshes with arbitrary numbers of hanging nodes between elements. We present a constructive proof based on the a posteriori error analysis of an auxiliary elliptic problem with $H^{-1}$ source terms, thereby yielding results of independent interest. In particular, for such problems, we obtain guaranteed upper bounds on the error along with polynomial-degree robust local efficiency of the estimators.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-06T09:26:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65N30"
    ],
    "keywords": [
      "source terms",
      "posteriori error analysis",
      "posteriori estimates",
      "polynomial-degree robust local efficiency",
      "auxiliary elliptic problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}