{
  "abstract": "Kellogg's checkerboard interface problem is a classical benchmark for robust adaptive finite element methods. Its successful adaptive meshes are radial: they refine strongly toward the interface crossing but show no angular structure, despite the large contrast and the asymmetric solution. We explain this by proving that the singular solution $u(r,θ)=r^γμ(θ)$ satisfies the exact identities $κ|\\nabla u|^2=Λr^{2γ-2}$ and $κ|\\nabla^2 u|_F^2=2(1-γ)^2Λr^{2γ-4}$, with $Λ=γ^2\\cos^2(πγ/4)$ and the Hessian taken separately in each quadrant. The point is what has disappeared: the right-hand sides depend on $r$ alone, although $κ$ and $u$ each depend on the angle as well. Combined with equal discretization-error distribution, this shows that the target element density is radial, so a correct mesh should display nothing but refinement toward the center, and a Kellogg mesh that is not radial is visible evidence that the computation is not following the coefficient-weighted local difficulty. The reading is specific to this benchmark: on a second interface problem the same estimator correctly produces a strongly material-biased mesh, with a computed element-count ratio of $3.934$ against the predicted $4$. A byproduct gives the benchmark constants in closed form, so the problem data can be generated from $γ$ alone at any precision.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.22991",
  "author_total": 1,
  "authors": [
    {
      "name": "Shun Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/08090q7r466fp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.22991",
  "fulltext_url": null,
  "id": "08090q7r466fp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.22991",
  "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",
      "2206|Advanced Numerical Methods in Computational Mathematics"
    ]
  ],
  "openalex": "W7204199192",
  "pdf_url": "https://arxiv.org/pdf/2608.22991",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/08090q7r466fp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.22991",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.22991"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Why the Kellogg Mesh Is Radial: A Mathematical Explanation of a Classical Computational Benchmark",
  "title_text": "Why the Kellogg Mesh Is Radial: A Mathematical Explanation of a Classical Computational Benchmark",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/08090q7r466fp",
  "venue": "arXiv",
  "versions": []
}