Why the Kellogg Mesh Is Radial: A Mathematical Explanation of a Classical Computational Benchmark
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)=rγμ(θ) satisfies the exact identities κ|∇ u|2=Λr2γ−2 and κ|∇2 u|F2=2(1−γ)2Λr2γ−4, with Λ=γ2cos2(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.
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BibTeX
@misc{zhang2026kellogg,
title = {{Why the Kellogg Mesh Is Radial: A Mathematical Explanation of a Classical Computational Benchmark}},
author = {Shun Zhang},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.22991},
eprint = {2608.22991},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.22991},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Why the Kellogg Mesh Is Radial: A Mathematical Explanation of a Classical Computational Benchmark
AU - Shun Zhang
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.22991
UR - https://arxiv.org/abs/2608.22991
AB - 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|∇ u|^2=Λr^2γ−2 and κ|∇^2 u|_F^2=2(1−γ)^2Λr^2γ−4, with Λ=γ^2cos^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.
ER -