Finite Element Approximation of the Hardy Constant

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

Publication Type: Journal article

Publication year: 2024

Journal

Book Volume: 31

Pages Range: 497-523

Journal Issue: 2

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 ≥ 3. For finite element spaces of piecewise linear and continuous functions on a mesh of size h, we prove that the approximate Hardy constant converges to the optimal Hardy constant at a rate proportional to 1/|log h|². This result holds in dimension n = 1, in any dimension n ≥ 3 if the domain is the unit ball and the finite element discretization exploits the rotational symmetry of the problem, and in dimension n = 3 for general finite element discretizations of the unit ball. In the first two cases, our estimates show excellent quantitative agreement with values of the discrete Hardy constant obtained computationally.

Authors with CRIS profile

Additional Organisation(s)

Involved external institutions

Università degli Studi di Napoli Federico II Italy (IT) University of Bucharest / Universitatea din București Romania (RO)

How to cite

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. Journal of Convex Analysis, 31(2), 497-523.

MLA:

Della Pietra, Francesco, et al. "Finite Element Approximation of the Hardy Constant." Journal of Convex Analysis 31.2 (2024): 497-523.

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