{
  "abstract": "We construct and analyse a fully discrete scheme for a Kirchhoff-type nonlinear integro-differential equation in two spatial dimensions with time-dependent coefficients. The scheme combines a symmetric three-layer time discretization of Crank--Nicolson type with a conforming finite element method of arbitrary polynomial degree $p$ and a Picard iteration for the nonlinearity. We prove that the Picard iteration is well defined and, that the iterates converge to the Ritz projection of the exact solution measured in fractional-power norms of the discrete Laplacian. Running the same energy argument in the unshifted norms, and under the same hypotheses, we then obtain the optimal order $\\mathcal{O} ( h^{p+1} + τ^2 )$ in $L^2$ and $\\mathcal{O} ( h^{p} + τ^2 )$ in $H^1$ for the full error. For the homogeneous problem with constant coefficients, the exact solution of the scheme moreover preserves a discrete energy identically. Numerical experiments with manufactured solutions reproduce the predicted second order in time and the full spatial order $h^{p+1}$ in the $L^2$-norm.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.31487",
  "author_total": 5,
  "authors": [
    {
      "name": "Benjamin Dörich"
    },
    {
      "name": "Felix Krumbiegel"
    },
    {
      "name": "Jemal Rogava"
    },
    {
      "name": "Andreas Rupp"
    },
    {
      "name": "Zurab Vashakidze"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1pyq3psj8p2g5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.31487",
  "fulltext_url": null,
  "id": "1pyq3psj8p2g5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.31487",
  "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": "W7223387347",
  "pdf_url": "https://arxiv.org/pdf/2610.31487",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1pyq3psj8p2g5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.31487",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.31487"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Conservative Finite Element Methods for Kirchhoff-Type Equations",
  "title_text": "Conservative Finite Element Methods for Kirchhoff-Type Equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/1pyq3psj8p2g5",
  "venue": "arXiv",
  "versions": []
}