Finite element approximation of the Hardy constant

TL;DR

This work analyzes finite element approximations to the Hardy constant for $p=2$ in domains with $n=1$ or $n\ge 3$. It proves that the discrete Hardy constant $S_h$ converges to the optimal constant $S$ at a square-log rate, with explicit asymptotics in 1D and analogous results in the unit ball for $n\ge 3$, including the 3D spherical and non-spherical triangulations. The proofs combine a calibration-based lower bound and carefully constructed near-minimizing sequences to obtain sharp $O(1/|\log h|^2)$ error terms, complemented by interpolation error estimates for FE spaces. Numerical experiments in 1D and 3D with radial symmetry confirm the predicted rates and demonstrate good quantitative agreement with the theoretical bounds. The results illuminate the distinctive challenges of approximating optimal constants for inequalities that are not attained, and they open avenues for extending the approach to other exponents and dimensions.

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.

Figures (6)

• Figure 1: Left: Ratio of $\delta_h$ to the leading-order term $\pi/(6+{\left\vert \log h \right\vert})$ in its asymptotic expansion. Right: The error between this ratio and the value $1$.
• Figure 2: The gap $E_h = S_{h} - 1/4$ between the discrete and exact Hardy constant in dimension $n=1$, scaled by the factors $\pi^2/(6+{\left\vert \log h \right\vert})^2$ (left) and $\pi^2/ ( {\left\vert \log h \right\vert} - 3 \log {\left\vert \log h \right\vert} )^2$ (right) predicted by the lower and upper bounds in \ref{['e:lb-1d-statement', 'e:ub-1d-statement']}, respectively.
• Figure 3: Left: Minimizer $u \in V_h$ for \ref{['e:discrete-hardy-min']} and $h=10^{-3}$, normalized so that $u(1)=1$. Right: Error $\|u-u_h\|_{L^\infty(0,1)}$ between the minimizer $u$ of \ref{['e:discrete-hardy-min']} and the function $u_h$ in \ref{['e:near-opt-u-1d']}. Both functions are normalized so that $u(1)=u_h(1)=1$.
• Figure 4: The gap $E_h^3 = S_{h}^3 - 1/4$ between the discrete Hardy constant for $n=3$ dimensions, computed with \ref{['Snh']}, and the exact value $S^3 = 1/4$. Results are plotted after scaling by the functions $9\pi^2/(16 + 3h + 3{\left\vert \log h \right\vert})^2$ (right) and $4\pi^2/ ( {\left\vert \log h \right\vert} - 3 \log {\left\vert \log h \right\vert} )^2$ (right) predicted by the lower and upper bounds in \ref{['e:lb-3d-statement', 'e:ub-3d-statement']}, respectively.
• Figure 5: Left: Minimizer $v \in V_h$ for \ref{['Snh']} with $n=3$ and $h=10^{-2}$, normalized so that $v(0)=1$. Right: Error $\|v-v_h\|_{L^\infty(0,1)}$ between the minimizer $v$ of \ref{['Snh']} and the function $v_h$ in \ref{['e:optimal-function-3d']} for $n=3$. Both functions are normalized so that $v(0)=v_h(0)=1$.

Theorems & Definitions (33)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Definition 2.2
• Theorem 2.1
• Lemma 2.2
• Theorem 2.3
• Theorem 3.1
• Lemma 3.2: Lower bound on $\mu_h$