{
  "abstract": "We develop an arbitrary-degree geometry-conforming finite element (GC-FE) framework for two-dimensional elliptic boundary value and interface problems on curved domains. Using the Frenet--Serret transformation, curved-boundary and interface-fitted segments are represented exactly, while polynomials in Frenet coordinates generate generally nonpolynomial local shape functions in physical coordinates. For interface-unfitted meshes, GC-FE spaces on curved-boundary elements are coupled with geometry-conforming immersed finite element (GC-IFE) spaces on interface-cut elements, with standard polynomial spaces used elsewhere. We establish optimal approximation, inverse, and trace estimates for the GC-FE spaces. For fitted meshes, we prove well-posedness and optimal error estimates in energy and $L^2$ norms for a symmetric interior penalty discontinuous Galerkin discretization. By retaining the prescribed curves exactly, the method avoids the geometric variational crime associated with curved-geometry approximation and requires no corresponding geometric consistency estimates. Numerical experiments confirm the predicted rates, show global accuracy comparable to nodal isoparametric finite elements and smaller true-interface trace errors in the reported tests, and demonstrate the coupled GC-FE-GC-IFE method on interface-unfitted meshes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.14484",
  "author_total": 4,
  "authors": [
    {
      "name": "Yuanhui Lin"
    },
    {
      "name": "Tao Lin"
    },
    {
      "name": "Xu Zhang"
    },
    {
      "name": "Minfu Feng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0hrdtr6xhbffx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.14484",
  "fulltext_url": null,
  "id": "0hrdtr6xhbffx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.14484",
  "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": "W7203604851",
  "pdf_url": "https://arxiv.org/pdf/2608.14484",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0hrdtr6xhbffx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2608.14484",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.14484"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Geometry-Conforming Finite Element Methods for Interface Problems on Fitted and Unfitted Meshes",
  "title_text": "Geometry-Conforming Finite Element Methods for Interface Problems on Fitted and Unfitted Meshes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/0hrdtr6xhbffx",
  "venue": "arXiv",
  "versions": []
}