Journal of Convex Analysis 31 (2024), No. 2, 497--523
Copyright Heldermann Verlag 2024
Finite Element Approximation of the Hardy Constant
Francesco Della Pietra
Dip. di Matematica e Applicazioni "R. Caccioppoli", Università degli Studi di Napoli "Federico II", Napoli, Italy
f.dellapietra@unina.it
Giovanni Fantuzzi
Dept. of Mathematics, Friedrich-Alexander-Universität, Erlangen-Nürnberg, Germany
giovanni.fantuzzi@fau.de
Liviu I. Ignat
(1) Institute of Mathematics "Simion Stoilow", Romanian Academy, Bucharest, Romania
(2) Research Institute of the University of Bucharest ICUB, Bucharest, Romania
liviu.ignat@gmail.com
Alba Lia Masiello
Dip. di Matematica e Applicazioni "R. Caccioppoli", Università degli Studi di Napoli "Federico II", Napoli, Italy
albalia.masiello@unina.it
Gloria Paoli
Dip. di Matematica e Applicazioni "R. Caccioppoli", Università degli Studi di Napoli "Federico II", Napoli, Italy
gloria.paoli@unina.it
Enrique Zuazua
(1) Dept. of Mathematics, Friedrich-Alexander-Universität, Erlangen-Nürnberg, Germany
(2) Chair of Computational Mathematics, Fundación Deusto, Bilbao, Spain
(3) Departamento de Matemáticas, Universidad Autónoma de Madrid, Madrid, Spain
enrique.zuazua@fau.de
[Abstract-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 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.
Keywords: Hardy inequality, Hardy constant, finite element method.
MSC: 46E35, 65N30.
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