In this paper, on the basis of a (Fenchel) duality theory on the continuous level, we derive an $\textit{a posteriori}$ error identity for arbitrary conforming approximations of a primal formulation and a dual formulation of variational problems involving gradient constraints. In addition, on the basis of a (Fenchel) duality theory on the discrete level, we derive an $\textit{a priori}$ error identity that applies to the approximation of the primal formulation using the Crouzeix-Raviart element and to the approximation of the dual formulation using the Raviart-Thomas element, and leads to error decay rates that are optimal with respect to the regularity of a dual solution.

翻译：本文基于连续层面的（Fenchel）对偶理论，导出了涉及梯度约束的变分问题之原始形式与对偶形式任意保形逼近的后验误差恒等式。此外，基于离散层面的（Fenchel）对偶理论，我们导出了适用于采用Crouzeix-Raviart元逼近原始形式及采用Raviart-Thomas元逼近对偶形式的先验误差恒等式，该恒等式导出的误差衰减率相对于对偶解的正则性而言是最优的。