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 $u(r,θ)=r^γμ(θ)$ 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.