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 the primal formulation and the dual formulation of the scalar Signorini problem. 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 quasi-optimal error decay rates without imposing additional assumptions on the contact set and in arbitrary space dimensions.

翻译：本文基于连续层面的（Fenchel）对偶理论，导出了标量Signorini问题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，我们推导了适用于Crouzeix-Raviart元逼近原始形式及Raviart-Thomas元逼近对偶形式的先验误差恒等式，该恒等式可在任意空间维度下，无需对接触集附加任何假设条件，即获得拟最优误差衰减率。