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, $S_h^n$, converges to the optimal Hardy constant $S^n$ no slower than $O(1/\vert \log h \vert)$. We also show that the convergence is no faster than $O(1/\vert \log h \vert^2)$ if $n=1$ or if $n\geq 3$, 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.

翻译：我们考虑在维数$n=1$或$n\geq 3$的有界区域上，指数$p=2$的Hardy不等式最优常数的有限元逼近。对于网格尺寸为$h$的分片线性连续函数有限元空间，我们证明近似Hardy常数$S_h^n$收敛到最优Hardy常数$S^n$的速度不低于$O(1/\vert \log h \vert)$。同时，当$n=1$或$n\geq 3$且区域为单位球、有限元离散利用问题旋转对称性时，我们证明收敛速度不超过$O(1/\vert \log h \vert^2)$。我们的估计与通过计算获得的$S_h^n$精确值进行了比较。