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.

翻译：我们考虑有界域中指数$p=2$的Hardy不等式最优常数的有限元逼近，其中维数$n=1$或$n \geq 3$。针对网格尺寸为$h$的分片线性连续函数有限元空间，我们证明了近似Hardy常数以$1/| \log h |^2$的速率收敛到最优Hardy常数。该结果在一维情形、当区域为单位球且有限元离散利用问题旋转对称性的任意维数$n \geq 3$情形，以及单位球一般有限元离散的三维情形中均成立。在前两种情形中，我们的估计与通过计算获得的离散Hardy常数值显示出极好的一致性。