Finite Element Approximation of the Hardy Constant
Francesco Della Pietra, Giovanni Fantuzzi, Liviu I. Ignat, Alba Lia Masiello, Gloria Paoli, Enrique Zuazua · Journal of convex analysis · 2024
We consider finite element approximations to the optimal constant for the Hardy inequality with exponent p=2 p = 2 in bounded domains of dimension n=1 n = 1 or n\geq 3 n ≥ 3 . For finite element spaces of piecewise linear and continuous functions on a mesh of size h h , we prove that the approximate Hardy constant converges to the optimal Hardy constant at a rate proportional to 1/|\log h|^2 1 / ∣ log h ∣ 2 . This result holds in dimension n=1 n = 1 , in any dimension n\geq 3 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 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.