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

TL;DR

This paper extends convex duality-based error identities to vectorial problems like Stokes and Navier-Lamé, incorporating inhomogeneous boundary conditions and loads, and establishes quasi-optimal error estimates for specific discretizations.

Contribution

It introduces new a posteriori error identities for vectorial PDEs, considering inhomogeneous conditions and minimal regularity, advancing error analysis in finite element methods.

Findings

Derived error identities for vectorial problems

Proved quasi-optimal error estimates for Crouzeix-Raviart discretization

Extended results to inhomogeneous boundary conditions and loads

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.