{
  "abstract": "I present a matrix-free symmetric interior penalty discontinuous Galerkin method for adaptively refined Cartesian meshes on GPUs. At non-matching interfaces, auxiliary \\emph{shadow cells} represent the adjacent coarse polynomial on the fine level. This approach converts non-matching interfaces into matching faces, permitting uniform face evaluation throughout the mesh. I prove that this construction is equivalent to the standard non-matching formulation. Furthermore, this representation yields a local geometric multigrid method in which frozen shadows provide inter-level boundary data and carry residual contributions to coarser levels. A primal--dual pairing eliminates shadow assembly from the Krylov iteration. Numerical experiments with cubic elements ($p=3$) on an NVIDIA A100 show stable multigrid convergence under increasing refinement depth and efficient GPU execution.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.04534",
  "author_total": 1,
  "authors": [
    {
      "name": "Michał Wichrowski"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0mdmpyaam3ms5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.04534",
  "fulltext_url": null,
  "id": "0mdmpyaam3ms5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.04534",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "2206|Advanced Numerical Methods in Computational Mathematics"
    ]
  ],
  "openalex": "W7211908653",
  "pdf_url": "https://arxiv.org/pdf/2609.04534",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0mdmpyaam3ms5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:18Z",
      "native_id": "2609.04534",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.04534"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Adaptive Multilevel Discontinuous Galerkin Methods on GPUs",
  "title_text": "Adaptive Multilevel Discontinuous Galerkin Methods on GPUs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:18Z",
  "url": "https://research.draconitic.com/paper/0mdmpyaam3ms5",
  "venue": "arXiv",
  "versions": []
}