{
  "abstract": "We analyze the geometric variational crime of nonlinear finite element heterogeneous multiscale methods (FE--HMM) for $p$-growth monotone laws on curved surfaces, $p\\ge2$. The practical method projects discrete gradients onto the exact tangent planes; we compare it with the exactly lifted method on the same discrete space. Without further hypotheses, the difference is $O(h_g^{p/(p-1)})$ in the metric induced by the cell correctors. The optimal order $O(h_g^2)$ holds with a constant given by a weighted negative moment of the corrected gradients, evaluated along the practical solution only; no bounds on discrete gradients in $L^\\infty$ are used. The negative moment is a sufficient condition for the lower shifted $p$-structure of the discrete homogenized flux, which for $p>2$ is not available in general. For the model law and a fixed cell mesh, its uniform boundedness is equivalent to a finite-dimensional directional criterion on degenerate correctors, and for laminates it holds with an explicit bound, uniform in the cell mesh. With macro--micro quasi-optimality and an estimate for coefficient sampling, the reconstructed microscopic field converges as $O(h+H^{\\min{s,1}}+h_g^2)$, where $H^s$ is the cell approximation rate. Experiments on a torus with an exact closest-point lift and a transported anisotropic $p=4$ material show second-order geometric convergence up to $98,304$ macroscopic quadrature points, each with a fully solved nonlinear cell problem, with bounded a posteriori constants for both load treatments.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.07761",
  "author_total": 1,
  "authors": [
    {
      "name": "Marcial Nguemfouo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f0z0fkcyykg12.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.07761",
  "fulltext_url": null,
  "id": "f0z0fkcyykg12",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.07761",
  "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",
      "math.AP"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.07761",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f0z0fkcyykg12.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.07761",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.07761"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Second-order geometric consistency of nonlinear FE-HMM on surfaces via corrector non-degeneracy",
  "title_text": "Second-order geometric consistency of nonlinear FE-HMM on surfaces via corrector non-degeneracy",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:48Z",
  "url": "https://research.draconitic.com/paper/f0z0fkcyykg12",
  "venue": "arXiv",
  "versions": []
}