{
  "abstract": "We establish Sobolev--Poincaré inequalities for piecewise $W^{2,p}$ functions in two and three dimensions over families of general polytopic (including shape-regular simplicial and Cartesian) meshes, allowing for degenerate facets. The proposed inequalities feature control by either the broken Hessian or, more naturally for general fourth-order problems, the broken Laplacian. Important tools in their derivation are a Helmholtz decomposition in Banach--Sobolev spaces and Sobolev--Poincaré inequalities for piecewise $W^{1,p}$ functions; we avoid the use of any inverse inequality. We provide estimates with fairly general Lebesgue exponents. Applications to the stability analysis of nonconforming discretizations for second- and fourth-order, linear and nonlinear problems are exhibited.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.06646",
  "author_total": 2,
  "authors": [
    {
      "name": "Michele Botti"
    },
    {
      "name": "Lorenzo Mascotto"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6gt87xvykwtw7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.06646",
  "fulltext_url": null,
  "id": "6gt87xvykwtw7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.06646",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.06646",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6gt87xvykwtw7.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.06646",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.06646"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Sobolev--Poincaré inequalities for piecewise $W^{2,p}$ functions over general polytopic meshes",
  "title_text": "Sobolev--Poincaré inequalities for piecewise W^2,p functions over general polytopic meshes",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:46Z",
  "url": "https://research.draconitic.com/paper/6gt87xvykwtw7",
  "venue": "arXiv",
  "versions": []
}