A Priori and A Posteriori Error Identities for Vectorial Problems via Convex Duality

Abstract

Convex duality has been leveraged in recent years to derive a posteriori error estimates and identities for a wide range of non-linear and non-smooth scalar problems. By employing remarkable compatibility properties of the Crouzeix-Raviart and Raviart-Thomas elements, optimal convergence of non-conforming discretisations and flux reconstruction formulas have also been established. This paper aims to extend these results to the vectorial setting, focusing on the archetypical problems of incompressible Stokes and Navier-Lam\'e. Moreover, unlike most previous results, we consider inhomogeneous mixed boundary conditions and loads in the topological dual of the energy space. At the discrete level, we derive error identities and estimates that enable to prove quasi-optimal error estimates for a Crouzeix-Raviart discretisation with minimal regularity assumptions and no data oscillation terms.