Finite element approximation of the Hardy constant

08/03/2023

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by   Francesco Della Pietra, et al.

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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≥ 3. For finite element spaces of piecewise linear and continuous functions on a mesh of size h, we prove that the approximate Hardy constant, S_h^n, converges to the optimal Hardy constant S^n no slower than O(1/|log h |). We also show that the convergence is no faster than O(1/|log h |^2) if n=1 or if n≥ 3, the domain is the unit ball, and the finite element discretization exploits the rotational symmetry of the problem. Our estimates are compared to exact values for S_h^n obtained computationally.