A Smooth Counterexample to Second-Difference Stability on Graded Meshes
Manuscript 26 October 2026 · Online 26 October 2026
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.
Record
- Affiliation
- Mercury Software GmbH
- Contact
- [email protected] · GitHub
- Result
- Complete counterexample
- Categories
- math.NA · math.AP
- Manuscript
- 26 October 2026
- Online release
- 26 October 2026
- Version
- 1.0
- License
- Creative Commons Attribution 4.0 International
Files and verification
The PDF is the canonical reading copy. The source archive contains the LaTeX manuscript, bibliography, reproducibility material, and audit documents without build artefacts.
Citation
Alper Ferudun, “A Smooth Counterexample to Second-Difference Stability on Graded Meshes,” EulerSolve Research Papers, OWR-1058-002, 2026. https://doi.org/10.5281/zenodo.23675263.
BibTeX
@misc{ferudun2026gradedcorner,
author = {Ferudun, Alper},
title = {A Smooth Counterexample to Second-Difference Stability on Graded Meshes},
year = {2026},
howpublished = {EulerSolve Research Papers},
url = {https://eulersolve.org/papers/owr-1058-002/},
doi = {10.5281/zenodo.23675263},
note = {OWR-1058-002; unrefereed preprint}
}