{
  "abstract": "We study the boundary corrector $θ_ε$ in the two-scale expansion for periodic elliptic problems with Dirichlet conditions. By results of Gérard-Varet--Masmoudi and Prange, on smooth uniformly convex domains $θ_ε$ converges to the solution $θ^*$ of a homogenized problem with boundary data given by the boundary layer tail $d(n)$, which is determined by a half-space problem for each normal $n$. In the lifted form of Gérard-Varet and Masmoudi, $d(n)$ is given explicitly by the cell correctors and the Dirichlet-to-Neumann operator $N_n$ of this half-space problem. Our main result is that $N_n$ satisfies a nonlinear equation on the period cell alone, so that $d(n)$ and the boundary density that represents it can be computed on the cell. This equation is a Riccati equation in which the derivative in depth is replaced by a commutator with the derivative along $n$. This cell problem can therefore be solved by a Riccati doubling algorithm, and we can compute $d(n)$ at irrational normals without rational approximation. Dropping the commutator term in the cell equation gives an operator square root approximation for $N_n$ that captures the leading term of $d(n)$ for small contrast, for the truncated problem. Finally, we approximate $θ_ε$ at finite $ε$ by the sum of two solutions of constant coefficient problems, one with oscillatory boundary data and the other $θ^*$, and test it on the disk.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.17026",
  "author_total": 1,
  "authors": [
    {
      "name": "Shari Moskow"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/46dr5z6qj147j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.17026",
  "fulltext_url": null,
  "id": "46dr5z6qj147j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.17026",
  "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.AP"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "1703|Advanced Mathematical Modeling in Engineering"
    ]
  ],
  "openalex": "W7221788903",
  "pdf_url": "https://arxiv.org/pdf/2610.17026",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/46dr5z6qj147j.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2610.17026",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.17026"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "A cell problem for the boundary density and approximation of boundary correctors in homogenization theory",
  "title_text": "A cell problem for the boundary density and approximation of boundary correctors in homogenization theory",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/46dr5z6qj147j",
  "venue": "arXiv",
  "versions": []
}