{
  "abstract": "We study the eigenvalues and eigenvectors of the Schur complement $\\textbf{S}$ of the graph Laplacian, defined on vertices sampled from a domain with $C^2$ boundary. We use a thin boundary layer to approximate the continuum boundary as the number of vertices increases and rely on transportation theory to justify the consistency of a discrete Steklov problem. In particular, $\\textbf{S}$ acts on vectors defined over the shrinking boundary layer but behaves asymptotically like a known operator. In this paper, we use a variational argument to determine conditions for which the eigenvalues and eigenvectors of $\\textbf{S}$ converge to the eigenvalues and eigenfunctions of a Dirichlet-to-Neumann map, and we provide evidence of this spectral convergence through a numerical example.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.40920",
  "author_total": 1,
  "authors": [
    {
      "name": "Andrew Lyons"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/41tn39be496pz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.40920",
  "fulltext_url": null,
  "id": "41tn39be496pz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.40920",
  "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"
    ],
    [
      "arxiv",
      "math.SP"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.40920",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/41tn39be496pz.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.40920",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.40920"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Spectral Convergence of the Discrete Steklov Problem",
  "title_text": "Spectral Convergence of the Discrete Steklov Problem",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/41tn39be496pz",
  "venue": "arXiv",
  "versions": []
}