{
  "abstract": "We study the singular limit of a family of linear degenerate interface transmission problems arising from a regularization procedure in the newly proposed Two-Parameter Diffuse Domain Method (DDM2p). For $α>0$, the regularized problem admits a strictly convex variational formulation on $H^{1}(Ω)$. In the limit $α\\to0$, the problem degenerates to a weakly coupled interface system with a nonstandard energy structure. To characterize the limit, we introduce a closed Hilbert subspace $\\mathcal{H}\\subset H^{1}(Ω)$, defined through an auxiliary Helmholtz problem on an annular subdomain $Ω_2\\subset Ω$, and identify the limiting energy functional $\\mathcal{E}_{0}$ on $\\mathcal{H}$. We prove that the regularized energies $\\mathcal{E}_α$ $Γ$-converge to $\\mathcal{E}_{0}$ in the strong $L^{2}(Ω)$ topology, using the standard framework. Consequently, minimizers of $\\mathcal{E}_α$ converge to the unique minimizer of $\\mathcal{E}_{0}$, which is shown to be equivalent to the solution of the limiting interface problem. We further prove strong convergence $u_α\\to u_{0}$ in $H^{1}(Ω)$ and establish an $O(α)$ convergence rate. Numerical experiments in one spatial dimension confirm the predicted first-order convergence rate and suggest that this rate is sharp.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.01237",
  "author_total": 5,
  "authors": [
    {
      "name": "Toai Luong"
    },
    {
      "name": "Tadele Mengesha"
    },
    {
      "name": "Kerrek Stinson"
    },
    {
      "name": "Steven M. Wise"
    },
    {
      "name": "Ming Hei Wong"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4f59afb53tn8w.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.01237",
  "fulltext_url": null,
  "id": "4f59afb53tn8w",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.01237",
  "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.AP"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "2206|Advanced Numerical Methods in Computational Mathematics"
    ]
  ],
  "openalex": "W7206161863",
  "pdf_url": "https://arxiv.org/pdf/2609.01237",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4f59afb53tn8w.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2609.01237",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.01237"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "The $α$-Limit Problem: Convergence of a Linear Degenerate Interface Transmission Problem",
  "title_text": "The α-Limit Problem: Convergence of a Linear Degenerate Interface Transmission Problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/4f59afb53tn8w",
  "venue": "arXiv",
  "versions": []
}