{
  "abstract": "We develop a unified isoparametric moving mesh partial differential equation (MMPDE) method for $P_1$ simplicial and $Q_1$ tensor-product elements, including elements on embedded surfaces. Mesh movement follows a prescribed symmetric positive-definite metric through equidistribution and alignment. For $Q_1$ elements, we prove that bounded, exactly integrated adaptation energy prevents local Jacobian rank loss on a fixed finite mesh at sharp critical exponents. The result also gives local full-rank preservation for embedded bilinear surface elements. A complete physical-coordinate gradient accounts for metric variation at moving quadrature points. For the discrete MMPDE, we use whole-element validity tests and develop sufficient Bernstein certificates for trilinear hexahedra. An explicit positive-margin bound on each fixed stage is provided by an optional certificate barrier. Numerical examples illustrate metric-driven movement on one-dimensional, planar, volume, and graph-surface meshes. Limitations of pointwise hexahedral Jacobian sampling are also presented.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.05026",
  "author_total": 1,
  "authors": [
    {
      "name": "Xianping Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9pnxkhk27h3yx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.05026",
  "fulltext_url": null,
  "id": "9pnxkhk27h3yx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.05026",
  "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/2611.05026",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9pnxkhk27h3yx.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:29Z",
      "native_id": "2611.05026",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.05026"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "A Unified Isoparametric Moving Mesh PDE Method for P1 and Q1 Elements with Certified Local Nonsingularity",
  "title_text": "A Unified Isoparametric Moving Mesh PDE Method for P1 and Q1 Elements with Certified Local Nonsingularity",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/9pnxkhk27h3yx",
  "venue": "arXiv",
  "versions": []
}