{
  "abstract": "Rounding errors in finite element computations can lead to a complete loss of accuracy, stalled convergence, and incorrect results. Moreover, the effects of rounding errors accumulated within automatically generated and compiled kernels are difficult to analyze a priori. We present the first software framework for automated rounding error estimation within finite element kernels. The proposed methodology is based on an a posteriori technique called Running Error Analysis (REA), where a forward error estimate is automatically computed concurrently with the value. An open-source implementation is provided for the FEniCS Form Compiler (FFCx), based on a C++ backend for generating type-generic templated kernels over a custom arithmetic type that tracks both the value and its error estimate. We demonstrate REA on two examples. First, we use it to detect catastrophic cancellation in the assembly of a Neo-Hooke hyperelastic model in the small deformation regime. A series expansion circumvents the cancellation problem and the computed error estimates show this. Second, we study the assembly of the Laplace operator on a near-degenerate mesh. We demonstrate that our REA implementation typically incurs only 2-4x performance overhead. Applications of this work include robust reduced-precision computations in embedded systems, numerical debugging of new, possibly ill-conditioned or unstable PDE formulations, and guiding the design of mixed-precision kernels.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.37844",
  "author_total": 4,
  "authors": [
    {
      "name": "Michal Habera"
    },
    {
      "name": "Paul T. Kühner"
    },
    {
      "name": "Matteo Croci"
    },
    {
      "name": "Andreas Zilian"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2y3d584gdm4dt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.37844",
  "fulltext_url": null,
  "id": "2y3d584gdm4dt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.37844",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.MS"
    ],
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.37844",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2y3d584gdm4dt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.37844",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.37844"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Running error bounds in finite element kernels",
  "title_text": "Running error bounds in finite element kernels",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/2y3d584gdm4dt",
  "venue": "arXiv",
  "versions": []
}