{
  "abstract": "We analyse the Cartesian perfectly matched layer (PML) approximation of the two-dimensional Helmholtz equation $-c^2Δu-ω^2 u=f$, in which the propagation speed $c$ is variable inside the region of interest and equal to a constant $c_\\infty$ outside a disc $B_R$. Existing analyses of the Cartesian PML assume a homogeneous, or piecewise homogeneous, background. We assume of $c$ only a two-sided bound, so that discontinuous speeds -- layered media and compact inclusions -- are covered. This is possible because the speed enters the PML sesquilinear form only through its zeroth order coefficient, the principal part being built from the stretching profiles alone. Well-posedness of the physical problem at this regularity follows from a Lippmann-Schwinger equation, uniqueness from Rellich's lemma and unique continuation. A Morawetz multiplier yields a sharp non-trapping condition on $c$, namely $\\nabla c(x)\\cdot(x-x_0)<c(x)$ for some $x_0$, which reduces in the radial case to the Herglotz condition $\\frac{d}{dρ}(ρ/c(ρ))>0$; its failure produces a trapped bicharacteristic. The variable speed is invisible to the essential spectrum: after factoring out $c^2$, the two operators differ by a compactly supported multiplication, so the spectral analysis of the homogeneous Cartesian layer applies verbatim. This yields well-posedness of the truncated problem for every $ω>0$, with a stability constant independent of the layer width, and convergence to the physical solution in $H^1$ at a rate exponential in $ωγ_Mδ_M$, where $γ_M$ and $δ_M$ are the absorption strength and the layer thickness. Numerical experiments with a low-velocity inclusion confirm the exponential decay in both $δ$ and $ω$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20992",
  "author_total": 2,
  "authors": [
    {
      "name": "César Ortiz-Saavedra"
    },
    {
      "name": "Mauricio A. Londoño-Arboleda"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/czdpzvgve6qx0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20992",
  "fulltext_url": null,
  "id": "czdpzvgve6qx0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20992",
  "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",
      "math.AP"
    ],
    [
      "openalex",
      "2208|Electromagnetic Simulation and Numerical Methods"
    ]
  ],
  "openalex": "W7213871413",
  "pdf_url": "https://arxiv.org/pdf/2609.20992",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/czdpzvgve6qx0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:07Z",
      "native_id": "2609.20992",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20992"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Cartesian perfectly matched layers for the Helmholtz equation with variable propagation speed: well-posedness and exponential convergence",
  "title_text": "Cartesian perfectly matched layers for the Helmholtz equation with variable propagation speed: well-posedness and exponential convergence",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:07Z",
  "url": "https://research.draconitic.com/paper/czdpzvgve6qx0",
  "venue": "arXiv",
  "versions": []
}