{
  "abstract": "We develop a convergence theory for geometric multigrid with vertex-patch smoothers applied to cut finite element discretizations of the Poisson problem. The framework addresses non-inherited level forms and the mismatch between the physical and active domains. Using the discrete extension property, we prove two-level convergence bounds uniform in the mesh size and the cut geometry, and W-cycle bounds under an additional smallness assumption on the two-level rate. The numerical experiments intentionally use the stronger V-cycle, for which no convergence bound is claimed here. The convergence constants degrade with the degree $p$. Lowering the ghost penalty improves iteration counts. An aligned two-cell model exhibits a semidefiniteness threshold of order $p^{-2}$, whereas the visibility scale of a degree-$p$ cut mode decreases exponentially. Experiments at the model threshold reduce the iteration counts, but do not establish an assembled-operator threshold.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.04067",
  "author_total": 1,
  "authors": [
    {
      "name": "Michal Wichrowski"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4d4xc9m0yrxcr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.04067",
  "fulltext_url": null,
  "id": "4d4xc9m0yrxcr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.04067",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "2206|Advanced Numerical Methods in Computational Mathematics"
    ]
  ],
  "openalex": "W7208722855",
  "pdf_url": "https://arxiv.org/pdf/2609.04067",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4d4xc9m0yrxcr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2609.04067",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.04067"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "A Multigrid Method for CutFEM and its Convergence",
  "title_text": "A Multigrid Method for CutFEM and its Convergence",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/4d4xc9m0yrxcr",
  "venue": "arXiv",
  "versions": []
}