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


Book Volume: 31

Pages Range: 497-523

Journal Issue: 2


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.

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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.


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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