{
  "id": "2101.08331",
  "version": "v1",
  "published": "2021-01-20T21:33:21.000Z",
  "updated": "2021-01-20T21:33:21.000Z",
  "title": "A posteriori error estimates for hierarchical mixed-dimensional elliptic equations",
  "authors": [
    "Jhabriel Varela",
    "Elyes Ahmed",
    "Eirik Keilegavlen",
    "Jan Martin Nordbotten",
    "Florin Adrian Radu"
  ],
  "comment": "30 pages, 4 tables, 5 figures",
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "In this paper, we derive a posteriori error estimates for mixed-dimensional elliptic equations exhibiting a hierarchical structure. Exploiting the exterior calculus perspective of such equations, we introduce mixed-dimensional variables and operators, which, together with careful construction of the functional spaces, allow us to recast the set of partial differential equations as a regular linear elliptic problem structure-wise. We therefrom apply the well-established theory of functional a posteriori error estimates to our model to derive guaranteed abstract as well as fully computable upper bounds. Our estimators are tested using three different families of locally-mass conservative methods on synthetic problems and verification benchmarks of flow in fractured porous media. The numerical results support our theoretical findings while showcasing satisfactory effectivity indices.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-20T21:33:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "posteriori error estimates",
      "hierarchical mixed-dimensional elliptic equations",
      "partial differential equations",
      "regular linear elliptic problem structure-wise"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}