{
  "abstract": "We prove sharp pointwise error estimates for finite element approximations of the clamped biharmonic equation on convex polygonal domains. For $C^1$-conforming elements, we establish $W^{2,p}$-stability for all $2\\le p\\le \\infty$, together with best-approximation type estimates in $W^{1,\\infty}$ and $L^\\infty$. For the nonconforming Morley element, we derive optimal estimates in $W^{2,p}$, $W^{1,\\infty}$, and $L^\\infty$; in particular, the $L^\\infty$-error is of order $O(h^2)$ without an additional logarithmic factor. The analysis is based on three ingredients: local energy estimates for conforming and nonconforming plate elements, a removal of the local $L^2$-error term in these estimates, and Hölder estimates for derivatives of the Green's function. These tools are combined with a dyadic decomposition argument for regularized Green's functions. The results remove logarithmic losses present in earlier pointwise estimates for fourth-order problems and yield sharp rates for classical plate elements such as the Argyris, Bogner--Fox--Schmit (BFS), Hsieh--Clough--Tocher (HCT), and Morley elements.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.25115",
  "author_total": 4,
  "authors": [
    {
      "name": "Buyang Li"
    },
    {
      "name": "Zhenglei Li"
    },
    {
      "name": "Qigang Liang"
    },
    {
      "name": "Xuejun Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cbrp2z4t3ffh6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.25115",
  "fulltext_url": null,
  "id": "cbrp2z4t3ffh6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.25115",
  "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": "W7223049224",
  "pdf_url": "https://arxiv.org/pdf/2610.25115",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cbrp2z4t3ffh6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.25115",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.25115"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Sharp Pointwise Error Estimates of Finite Element Methods for the Biharmonic Equation",
  "title_text": "Sharp Pointwise Error Estimates of Finite Element Methods for the Biharmonic Equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/cbrp2z4t3ffh6",
  "venue": "arXiv",
  "versions": []
}