{
  "schema_version": 1,
  "problem_number": "OWR-1058-002",
  "title": "A Smooth Counterexample to Second-Difference Stability on Graded Meshes",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We give a smooth counterexample to the mesh-independent second-order difference-quotient estimate in Becker and Rannacher's 2005 Oberwolfach report, equation (5), represented by UnsolvedMath record OWR-1058-002. The fixed polynomial u(x,y)=xy(1-x^2)(1-y^2) on the unit square has homogeneous Dirichlet data. Its conforming piecewise affine Ritz projections are computed on uniformly shape-regular meshes whose maximum diameter tends to zero. They converge in the energy norm, but an interior normal-derivative jump, scaled by the edge length to represent a second-order difference, is unbounded. Exact local stiffness matrices yield a positive tridiagonal recurrence and an explicit geometric lower bound. A separate fixed-outer-mesh family is nested. The five-page English manuscript includes the mesh construction, complete proof, scope qualifications, a vector figure, and reproducible exact-arithmetic and numerical diagnostics. The separate averaging conjecture and specified adaptive refinement rules are not addressed. This is an unrefereed preprint, with no claim of formal verification or absolute priority.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.NA",
    "math.AP"
  ],
  "keywords": [
    "OWR-1058-002",
    "finite elements",
    "Ritz projection",
    "graded meshes",
    "second differences",
    "counterexample",
    "normal derivative jump"
  ],
  "manuscript_version_date": "2026-10-26",
  "publication_date": "2026-10-26",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-26",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-1058-002/",
  "pdf_url": "https://eulersolve.org/papers/owr-1058-002/paper.pdf?v=fc56a9e598fa",
  "doi": "10.5281/zenodo.23675263",
  "zenodo_record_url": "https://zenodo.org/records/23675263",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete counterexample to the uniform second-order difference-quotient estimate in Becker and Rannacher, OWR 37/2005, p.2098, equation (5), represented by OWR-1058-002. On the unit square the single smooth polynomial u=xy(1-x^2)(1-y^2) has conforming piecewise affine Ritz projections on uniformly shape-regular meshes with maximum diameter tending to zero, yet one scaled interior normal-derivative jump diverges. A separate fixed-outer-mesh family is nested. The proof uses exact local stiffness matrices, positive loads, and an explicit geometric lower bound. This closes the unrestricted source estimate represented by OWR-1058-002. The separate averaging conjecture, stronger mesh-grading conditions, and prescribed adaptive refinement algorithms are not addressed. This preprint is AI-assisted, self-audited and unrefereed; no independent or formal verification or certified novelty or absolute-priority claim is made.",
  "files": {
    "paper.pdf": {
      "sha256": "fc56a9e598fa023772873174842a2642e6061c7f10f702cabd2443402292b587"
    },
    "source.zip": {
      "sha256": "1c9de0dee01fbf6c2d44f92df8c7d6520abe4ef5290ae1ac80befce3e53e2328"
    },
    "verification_report.md": {
      "sha256": "2db4192ea8c5fda6a7cb43b85508b77754df09794ecf05a9de227d3127d67f6d"
    }
  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}