Variational problems with gradient constraints: A priori and a posteriori error identities

In this paper, on the basis of a (Fenchel) duality theory on the continuous level, we derive an <italic>a posteriori</italic> 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 <italic>a priori</italic> 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.