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

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-Lamé. 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.

💡 Research Summary

This paper extends the powerful framework of convex duality–based a‑posteriori error identities, originally developed for scalar nonlinear and nonsmooth problems, to vectorial problems, specifically the incompressible Stokes and Navier‑Lamé equations. The authors treat inhomogeneous mixed Dirichlet–Neumann boundary conditions and loads that belong to the topological dual of the natural energy space (i.e., H⁻¹(Ω) for both problems), thereby covering a much broader class of data than previous works that required L²‑type loads.

The core idea is to rewrite the primal variational formulation and its associated dual problem in a convex‑conjugate setting. For the Stokes problem, the primal functional is the viscous dissipation plus a divergence constraint, while the dual functional involves the deviatoric part of the stress tensor. By exploiting the strong duality relation I(u)=D(T) (where u is the velocity and T the Cauchy stress), the authors derive an exact error identity \