{
  "id": "1901.06821",
  "version": "v1",
  "published": "2019-01-21T08:25:55.000Z",
  "updated": "2019-01-21T08:25:55.000Z",
  "title": "Explicit $k-$dependency for $P_k$ finite elements in $W^{m,p}$ error estimates: application to probabilistic laws for accuracy analysis",
  "authors": [
    "Joel Chaskalovic",
    "Franck Assous"
  ],
  "comment": "20 pages. arXiv admin note: substantial text overlap with arXiv:1803.09552",
  "categories": [
    "math.NA"
  ],
  "abstract": "An explicit $k-$dependency in $W^{m,p}$ error estimates is proved to derive two probabilistic laws which evaluate the relative accuracy between Lagrange finite elements $P_{k_1}$ and $P_{k_2}, (k_1 < k_2)$. Thanks to these estimates we show a weak asymptotic relation in ${\\cal D}'(\\mathbb{R})$ between these probabilistic laws when $k_2-k_1$ goes to infinity. In this case, we get that $P_{k_2}$ finite element is surely more accurate than $P_{k_1}$, for any value of the mesh size $h$. This brings a complementary perspective regarding the classical way of comparing two finite elements, which is usually limited to the asymptotic rate of convergence, when $h$ goes to zero. Moreover, we also get new insights which highlight cases such that $P_{k_1}$ finite element is more likely accurate than $P_{k_2}$ one, for a range of specific values of $h$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-21T08:25:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65N15",
      "65N30",
      "65N75"
    ],
    "keywords": [
      "probabilistic laws",
      "error estimates",
      "accuracy analysis",
      "dependency",
      "application"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}