Finite Element Approximation of the Hardy Constant

Pietra FD, Fantuzzi G, Ignat LI, Masiello AL, Paoli G, Zuazua E (2024)

Publication Type: Journal article

Publication year: 2024

Journal

Journal of Convex Analysis Heldermann Verlag

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

Giovanni Fantuzzi Juniorprofessur für Angewandte Mathematik Enrique Zuazua Iriondo Lehrstuhl für Dynamics, Control, Machine Learning and Numerics (Alexander von Humboldt-Professur)

Involved external institutions

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

How to cite

APA:

Pietra, F.D., Fantuzzi, G., Ignat, L.I., Masiello, A.L., Paoli, G., & Zuazua, E. (2024). Finite Element Approximation of the Hardy Constant. Journal of Convex Analysis, 31(2), 497-523.

MLA:

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

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