{
  "abstract": "This paper presents an optimal interior penalty discontinuous Galerkin (IPDG) scheme for the planar biharmonic equation using piecewise polynomials of degree $k=3$ or $4$. In standard IPDG methods, large penalty parameters force the discrete solution into an overconstrained space, severely degrading accuracy---phenomenon known as numerical locking. To overcome this, our method enforces only vertex continuity and projects the jumps of the function and its normal derivative onto $\\mathcal{P}^{k-3}$ and $\\mathcal{P}^{k-2}$, respectively. Consequently, as the penalty parameters tend to infinity, the discrete solution is forced to lie in a constrained subspace $V_{h,\\infty}^k$, which we identify as the optimal nonconforming finite element space $B_h^k$. This intrinsic connection fundamentally eliminates numerical locking. We prove optimal error estimates of $\\mathcal{O}(h^{k-1})$ in a mesh-dependent energy norm and $\\mathcal{O}(h^{k+1})$ in the $L^2$ norm. Numerical experiments on both convex and L-shaped domains confirm that the proposed scheme is robust and entirely locking-free, even for extremely large penalty parameters.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12675",
  "author_total": 3,
  "authors": [
    {
      "name": "Bohua Zhang"
    },
    {
      "name": "Xia Ji"
    },
    {
      "name": "Shuo Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9qy7qb9fg4s1x.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12675",
  "fulltext_url": null,
  "id": "9qy7qb9fg4s1x",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12675",
  "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": "W7212639068",
  "pdf_url": "https://arxiv.org/pdf/2609.12675",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9qy7qb9fg4s1x.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.12675",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12675"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "An Optimal IPDG Scheme for the Biharmonic Equation",
  "title_text": "An Optimal IPDG Scheme for the Biharmonic Equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/9qy7qb9fg4s1x",
  "venue": "arXiv",
  "versions": []
}