{
  "abstract": "We consider conforming $\\Pone$ and lowest-order Crouzeix--Raviart (CR) approximations of the Dirichlet Poisson problem in two and three dimensions. We obtain computable a priori $L^2$-error bounds on anisotropic simplicial meshes without a global $H^2$-regularity assumption. For piecewise constant data, the Marini relation gives an equilibrated lowest-order Raviart--Thomas (RT) flux from the scalar CR solution. The square of the resulting Marini constant is the largest generalised eigenvalue of a problem involving only scalar conforming and CR matrices. Combined with the elementwise Poincaré constant, it controls both the conforming and CR errors. The CR $L^2$ estimate uses positivity of the discrete CR--conforming gap operator rather than a nonconforming Aubin--Nitsche argument. We also derive an exact decomposition of the Marini defect into the CR--conforming energy gap and an explicit geometric term with no aspect-ratio factor. On the algebraically graded L-shaped mesh family considered here, $q>3/2$ yields $κ_{M,h}=O(h)$ and computable $O(h^2)$ $L^2$-operator bounds.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06057",
  "author_total": 1,
  "authors": [
    {
      "name": "Hiroki Ishizaka"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/83dew0sr1vnjh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06057",
  "fulltext_url": null,
  "id": "83dew0sr1vnjh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06057",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "2206|Advanced Numerical Methods in Computational Mathematics"
    ]
  ],
  "openalex": "W7212068158",
  "pdf_url": "https://arxiv.org/pdf/2609.06057",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/83dew0sr1vnjh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06057",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06057"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Crouzeix--Raviart--Marini realisation of computable a priori $L^2$-error bounds on anisotropic meshes",
  "title_text": "Crouzeix--Raviart--Marini realisation of computable a priori L^2-error bounds on anisotropic meshes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/83dew0sr1vnjh",
  "venue": "arXiv",
  "versions": []
}