L. Ignat, E. Zuazua (2025) Sharp numerical approximation of the Hardy constant
Abstract. We study the P1 finite element approximation of the best constant in the classical Hardy inequality over bounded domains containing the origin in ℝN, for N≥3. Despite the fact that this constant is not attained in the associated Sobolev space H1, our main result establishes an explicit, sharp, and dimension-independent rate of convergence proportional to 1/|logh|2.
The analysis carefully combines an improved Hardy inequality involving a reminder term with logarithmic weights, approximation estimates for Hardy-type singular radial functions constituting minimizing sequences, properties of piecewise linear and continuous finite elements, and weighted Sobolev space techniques.
We also consider other closely related spectral problems involving the Laplacian with singular quadratic potentials obtaining sharp convergence rates.
arXiv: 2506.19422