{
  "abstract": "We develop a framework for equilibrated residual a posteriori error analysis of finite element methods, based on decomposing vector- and tensor-valued Sobolev spaces into $H^2$ and $H^1$ spaces. To estimate the residual functional acting on $H^2$ component, the method reconstructs symmetric and piecewise polynomial tensors on vertex patches in broken $H(\\operatorname{div}\\operatorname{div})$ spaces. We apply the proposed framework to conforming finite element discretizations of biharmonic equation, Poisson's equation, and linear elasticity in mixed form. As a result, we obtain several novel equilibrated residual a posteriori error estimates, providing guaranteed error control for energy error of biharmonic equation, $L^2$ norm error of Poisson's equation, and stress error of mixed linear elasticity. They also serve as the first polynomial-degree-robust a posteriori lower bounds of discretization errors. Numerical experiments illustrate the effectiveness of the $H^2$-based error estimators.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39210",
  "author_total": 2,
  "authors": [
    {
      "name": "Kexin Ding"
    },
    {
      "name": "Yuwen Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3j437kq2ph8y3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39210",
  "fulltext_url": null,
  "id": "3j437kq2ph8y3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39210",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.39210",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3j437kq2ph8y3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.39210",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39210"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "$H^2$-based equilibrated residual error estimates",
  "title_text": "H^2-based equilibrated residual error estimates",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/3j437kq2ph8y3",
  "venue": "arXiv",
  "versions": []
}