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arXiv · 2308.01580

Finite element approximation of the Hardy constant

Francesco Della Pietra·Giovanni Fantuzzi·Liviu I. Ignat·Alba Lia Masiello·Gloria Paoli·Enrique Zuazua
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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 $n \geq 3$. For finite element spaces of piecewise linear and continuous functions on a mesh of size $h$, we prove that the approximate Hardy constant 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 $n \geq 3$ 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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Francesco Della Pietra, Giovanni Fantuzzi, Liviu I. Ignat, Alba Lia Masiello, Gloria Paoli, Enrique Zuazua. 2023-08-03. Finite element approximation of the Hardy constant. https://arxiv.org/abs/2308.01580

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