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Publication Type: Unpublished / Preprint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
Open Access Link: https://dcn.nat.fau.eu/wp-content/uploads/hardy_revised_new.pdf
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.
APA:
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 (Unpublished, Accepted).
MLA:
Della Pietra, Francesco, et al. Finite element approximation of the Hardy constant. Unpublished, Accepted. 2024.
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