We study the $P_1$ finite element approximation of the best constant in the classical Hardy inequality over bounded domains containing the origin in $\mathbb{R}^N$, for $N \geq 3$. Despite the fact that this constant is not attained in the associated Sobolev space $H^1$, our main result establishes an explicit, sharp, and dimension-independent rate of convergence proportional to $1/|\log h|^2$. The analysis carefully combines an improved Hardy inequality involving a reminder term with logarithmic weights, approximation estimates for Hardy-type singular radial functions constituting minimizing sequences, properties of piecewise linear and continuous finite elements, and weighted Sobolev space techniques. We also consider other closely related spectral problems involving the Laplacian with singular quadratic potentials obtaining sharp convergence rates.

翻译：我们研究了$\mathbb{R}^N$（$N \geq 3$）中包含原点的有界域上经典Hardy不等式中最佳常数的$P_1$有限元逼近。尽管该常数在相关的Sobolev空间$H^1$中不可达，我们的主要结果建立了一个显式的、精确的、与维度无关的收敛速率，其阶为$1/|\log h|^2$。该分析细致地结合了包含对数权重余项的改进Hardy不等式、构成极小化序列的Hardy型奇异径向函数的逼近估计、分段线性连续有限元的性质以及加权Sobolev空间技术。我们还考虑了涉及具有奇异二次势的Laplacian的其他密切相关的谱问题，并获得了精确的收敛速率。