Variational problems with gradient constraints: $\text{A priori}$ and $\text{a posteriori}$ error identities

Abstract

In this paper, on the basis of a (Fenchel) duality theory on the continuous level, we derive an $\text{a posteriori}$ 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 $\text{a priori}$ 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.

26 pages, 3 figures. Several typographical errors and minor inaccuracies have been corrected compared with the published version; the main results remain unchanged