{
  "abstract": "We develop a one-level restricted additive Schwarz solver with perfectly matched layer transmission conditions (RAS--PML) for high-frequency Helmholtz systems. For problems with ${O}(k^d)$ unknowns, $d=2,3$, the solver uses Cartesian decompositions and ${O}(k^d)$ concurrent local solves for a single right-hand side, where $k$ is the wavenumber. Starting from the convergence theory for fixed decompositions established in~\\texttt{arXiv:2404.02156}, we address the growing cost {as $k$ increases} of local solves through a nested RAS--PML construction. Under suitable stability and contraction assumptions, we prove that sufficiently accurate recursive local solves preserve convergence of the outer iteration. We also prove that a single inner iteration at each non-terminal level yields a correction algebraically equivalent to one-level RAS--PML, provided that the hierarchy-induced finest local problems and composite restriction and weighted extension operators are retained. This equivalence provides a hierarchical interpretation and permits a direct implementation without recursive transfers. We assess the one-level solver using PML and overlap widths of order ${O}(k^{-1}\\log k)$. Constant-coefficient experiments cover $300\\leq k\\leq19200$ in two dimensions and $60\\leq k\\leq780$ in three dimensions, reaching $1.47\\times10^9$ and $4.75\\times10^8$ complex unknowns, respectively. Richardson iteration counts grow approximately linearly with $k$ over these ranges, while total parallel runtime, including setup and solution, exhibits near-linear growth at the larger tested wavenumbers. Further experiments assess the effects of mesh refinement and heterogeneous media.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01075",
  "author_total": 5,
  "authors": [
    {
      "name": "Yan Xie"
    },
    {
      "name": "Yue Yu"
    },
    {
      "name": "Shihua Gong"
    },
    {
      "name": "Ivan G. Graham"
    },
    {
      "name": "Chen-song Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3s25xzxs2n3b8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01075",
  "fulltext_url": null,
  "id": "3s25xzxs2n3b8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01075",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.01075",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3s25xzxs2n3b8.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.01075",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01075"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Massively parallel Schwarz methods for the high-frequency Helmholtz equation with O(k) convergence",
  "title_text": "Massively parallel Schwarz methods for the high-frequency Helmholtz equation with O(k) convergence",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/3s25xzxs2n3b8",
  "venue": "arXiv",
  "versions": []
}