Geodesic
Finite Element Approximation of the Hardy Constant
Journal of convex analysis, Tome 31 (2024) no. 2, pp. 497-523
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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.
Classification : 46E35, 65N30
Mots-clés : Hardy inequality, Hardy constant, finite element method 
@article{JCA_2024_31_2_JCA_2024_31_2_a7,
     author = {F. Della Pietra and G. Fantuzzi and L. I. Ignat and A. L. Masiello and G. Paoli and E. Zuazua},
     title = {Finite {Element} {Approximation} of the {Hardy} {Constant}},
     journal = {Journal of convex analysis},
     pages = {497--523},
     year = {2024},
     volume = {31},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/JCA_2024_31_2_JCA_2024_31_2_a7/}
}
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F. Della Pietra; G. Fantuzzi; L. I. Ignat; A. L. Masiello; G. Paoli; E. Zuazua. Finite Element Approximation of the Hardy Constant. Journal of convex analysis, Tome 31 (2024) no. 2, pp. 497-523. http://geodesic.mathdoc.fr/item/JCA_2024_31_2_JCA_2024_31_2_a7/