{
  "abstract": "We study the uniaxial perfectly matched layer (UPML) problem for the Helmholtz equation in two and three dimensions and establish wavenumber-explicit stability estimates at high wavenumbers. For the full-space UPML problem, we prove the optimal stability estimate with the inf-sup constant satisfying $μ_\\infty\\eqsim k^{-1}$ for sufficiently large $k$ if the absorption function $σ\\in W^{1,\\infty}$ is a piecewise-power function of order at least one for which $\\|σ'\\|_{L^\\infty(\\{t:0<σ(t)\\leqδ_*\\})}\\leq \\varepsilon_*$, where $\\varepsilon_*$ and $δ_*$ depend only on the PML parameters. Using this stability estimate, we obtain the optimal inf-sup estimate $μ_L\\eqsim k^{-1}$ for the truncated UPML problem. These estimates further yield preasymptotic error estimates for a linear continuous interior penalty finite element method (CIP-FEM). Numerical simulations confirmed the theoretical predictions and demonstrated the effectiveness of CIP-FEM under high-wavenumber conditions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.06288",
  "author_total": 2,
  "authors": [
    {
      "name": "Silin Wen"
    },
    {
      "name": "Yu Du"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4nanny7yearcp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.06288",
  "fulltext_url": null,
  "id": "4nanny7yearcp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.06288",
  "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/2611.06288",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4nanny7yearcp.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.06288",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.06288"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Optimal stability estimates and preasymptotic error analysis of the CIP-FEM for the Helmholtz equation with uniaxial perfectly matched layers",
  "title_text": "Optimal stability estimates and preasymptotic error analysis of the CIP-FEM for the Helmholtz equation with uniaxial perfectly matched layers",
  "type": "preprint",
  "updated_at": "2026-11-05T15:21:00Z",
  "url": "https://research.draconitic.com/paper/4nanny7yearcp",
  "venue": "arXiv",
  "versions": []
}