Finite element approximation of the Hardy constant

TL;DR

This paper analyzes the convergence rate of finite element approximations to the Hardy constant in various dimensions, showing a logarithmic rate and confirming the accuracy through computational results.

Contribution

It establishes the convergence rate of finite element approximations to the Hardy constant in specific domains and dimensions, including symmetry considerations.

Findings

Convergence rate proportional to 1/|log h|^2

Excellent agreement with computational values in certain cases

Applicable to domains with symmetry and general discretizations

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.