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

TL;DR

This paper develops both a priori and a posteriori error identities for variational problems with gradient constraints, utilizing duality theories at continuous and discrete levels to improve approximation accuracy.

Contribution

It introduces novel error identities based on Fenchel duality for both continuous and discrete variational problems with gradient constraints, enabling optimal error estimates.

Findings

Derives an a posteriori error identity for conforming approximations.

Establishes an a priori error identity with optimal decay rates.

Applies to primal and dual formulations using specific finite elements.

Abstract

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.