{
  "abstract": "We develop and analyze HDG methods for two central problems in finite element exterior calculus, the Hodge-Dirac problem and the Hodge-Laplace problem, in arbitrary dimension $n$. Our analysis allows for equal-order polynomial spaces with $\\mathcal{O}(1)$ penalty parameters, by contrast with previous work on the Hodge-Laplace problem requiring an underlying conforming complex and $\\mathcal{O}(h)$ or $\\mathcal{O}(h^{-1})$ penalties. For the Hodge-Dirac problem with degree-$r$ polynomials, our error estimates give optimal order-$(r+1)$ convergence in all form degrees under suitable regularity hypotheses. For the Hodge-Laplace problem, we prove optimal order-$(r+1)$ convergence for the $k$-form solution and order-$(r+\\frac{1}{2})$ convergence for the $(k \\pm 1)$-form auxiliary variables. We obtain improved order-$(r+1)$ auxiliary-variable estimates when $k=1$ and/or $k=n-1$, which in particular sharpens some recent HDG error estimates for the two-dimensional vector Poisson equation. Our analysis also encompasses cases of lower regularity, including solutions with reentrant-corner singularities on non-convex domains, and requires only mild mesh regularity conditions. The results are illustrated by numerical experiments in dimensions two and three.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28995",
  "author_total": 3,
  "authors": [
    {
      "name": "Calvin Reedy"
    },
    {
      "name": "Ari Stern"
    },
    {
      "name": "Yunhui Xue"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b3htrvqcym5nk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28995",
  "fulltext_url": null,
  "id": "b3htrvqcym5nk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28995",
  "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": "W7214337560",
  "pdf_url": "https://arxiv.org/pdf/2609.28995",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b3htrvqcym5nk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.28995",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28995"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "HDG methods in finite element exterior calculus",
  "title_text": "HDG methods in finite element exterior calculus",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/b3htrvqcym5nk",
  "venue": "arXiv",
  "versions": []
}