Finite element approximation of the Hardy constant

Della Pietra F, Fantuzzi G, Ignat LI, Masiello AL, Paoli G, Zuazua Iriondo E (2024)

Publication Language: English

Publication Status: Accepted

Publication Type: Unpublished / Preprint

Future Publication Type: Journal article

Publication year: 2024

Publisher: Journal of Convex Analysis

Edition: Special Issue Giuseppe Buttazzo 70

DOI: 10.48550/arXiv.2308.01580

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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.

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Della Pietra, F., Fantuzzi, G., Ignat, L.I., Masiello, A.L., Paoli, G., & Zuazua Iriondo, E. (2024). Finite element approximation of the Hardy constant. (Unpublished, Accepted).


Della Pietra, Francesco, et al. Finite element approximation of the Hardy constant. Unpublished, Accepted. 2024.

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