{
  "abstract": "We study the selection and stable realization of local approximation spaces in Trefftz discontinuous Galerkin discretizations of the Helmholtz equation. A scaled Cauchy-trace inner product places plane waves and Fourier-Bessel functions in a common geometry: Fourier-Bessel modes are orthogonal with explicit weights, while the same weights determine the circulant spectrum of an equispaced plane-wave trace Gram matrix. This separates amplitude scaling from genuine trace-rank loss and yields an exact best-approximation identity for mixed plane-wave--Fourier--Bessel spaces. With complex plane-wave angles, the unresolved modal tail is an exponential sequence, so propagating and evanescent components can be identified by the same ESPRIT/variable-projection procedure. We prove exact recovery and a perturbation estimate for the recovered angles and, under the standard PWDG quasi-optimality bound, transfer these perturbations to the DG error. Trace-Riesz orthonormalization is then separated from a graph--Riesz normalization of the assembled operator. Numerical experiments verify the identities, recover sparse ray fields to roundoff, and resolve a propagating-to-evanescent transition without a prescribed critical angle.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12282",
  "author_total": 1,
  "authors": [
    {
      "name": "Shelvean Kapita"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2fdh06fd1s37e.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12282",
  "fulltext_url": null,
  "id": "2fdh06fd1s37e",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12282",
  "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",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "2206|Advanced Numerical Methods in Computational Mathematics"
    ]
  ],
  "openalex": "W7213064973",
  "pdf_url": "https://arxiv.org/pdf/2609.12282",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2fdh06fd1s37e.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.12282",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12282"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Adaptive local representations for Helmholtz Trefftz discontinuous Galerkin methods",
  "title_text": "Adaptive local representations for Helmholtz Trefftz discontinuous Galerkin methods",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/2fdh06fd1s37e",
  "venue": "arXiv",
  "versions": []
}