Finite element approximation of the Hardy constant

F. Della Pietra, G. Fantuzzi, LI Ignat, AL Masiello, G. Paoli, E. Zuazua (2024)Finite element approximation of the Hardy constant J. Convex Anal. Special Issue Giuseppe Buttazzo 70

Abstract. We consider finite element approximations to the optimal constant for the Hardy inequality with exponent $p=2$ in bounded domains of dimension $n=1$ or $n\geq 3$. For finite element spaces of piecewise linear and continuous functions on a mesh of size $h$, we prove that the approximate Hardy constant, $S_h^n$, converges to the optimal Hardy constant $S^n$ no slower than $O(1/\abs{\log h})$. We also show that the convergence is no faster than $O(1/\abs{\log h}^2)$ if $n=1$ or if $n\geq 3$, the domain is the unit ball, and the finite element discretization exploits the rotational symmetry of the problem. Our estimates are compared to exact values for $S_h^n$ obtained computationally.

arxiv: arXiv.2308.01580