{
  "abstract": "This paper develops a piecewise $P_2$ finite element space on rectangular meshes in arbitrary dimensions for the biharmonic equation. The discrete space imposes continuity of vertex sums on $(d-2)$-faces and of averaged normal derivatives across interior $(d-1)$-faces. A local algebraic characterization of $P_2$ is first established through the unisolvence of an enriched polynomial space and the resulting compatibility conditions. For each interior element, a $3^d$-element patch is then introduced, and the associated homogeneous patch space is shown to be one-dimensional. The corresponding patch functions, extended by zero, form a basis of the global homogeneous discrete space. A local quadratic interpolation operator is further constructed in arbitrary dimensions. Its approximation properties, together with a consistency estimate derived from the weak continuity conditions, lead to the optimal broken $H^2$ error estimate for the biharmonic problem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.32048",
  "author_total": 3,
  "authors": [
    {
      "name": "Rui Ma"
    },
    {
      "name": "Yang Xing"
    },
    {
      "name": "Shuo Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8r0qyaryfh62y.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.32048",
  "fulltext_url": null,
  "id": "8r0qyaryfh62y",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.32048",
  "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": "W7223705910",
  "pdf_url": "https://arxiv.org/pdf/2610.32048",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8r0qyaryfh62y.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2610.32048",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.32048"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Piecewise $P_2$ finite element space on rectangular meshes in arbitrary dimensions for the biharmonic equation",
  "title_text": "Piecewise P_2 finite element space on rectangular meshes in arbitrary dimensions for the biharmonic equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/8r0qyaryfh62y",
  "venue": "arXiv",
  "versions": []
}