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 $u(r,\theta )=r^{\gamma }\mu (\theta )$ satisfies the exact identities $\kappa |\nabla u{|}^2=\Lambda r^{2\gamma -2}$ and $\kappa |{\nabla }^{2}u{|}_F^2=2(1-\gamma {)}^2\Lambda r^{2\gamma -4}$, with $\Lambda ={\gamma }^2{\cos }^2(\pi \gamma /4)$ and the Hessian taken separately in each quadrant. The point is what has disappeared: the right-hand sides depend on $r$ alone, although $\kappa$ 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 $\gamma$ alone at any precision.