All first-order averaging techniques for a posteriori finite element error control on unstructured grids are efficient and reliable

All first-order averaging or gradient-recovery operators for lowest-order finite element methods are shown to allow for an efficient a posteriori error estimation in an isotropic, elliptic model problem in a bounded Lipschitz domain Ω in R d . Given a piecewise constant discrete flux p h E P h (that is the gradient of a discrete displacement) as an approximation to the unknown exact flux p (that is the gradient of the exact displacement), recent results verify efficiency and reliability of ηM:= min{∥p h - q h ∥ L 2 (Ω) : q h ∈ Qh} in the sense that η M is a lower and upper bound of the flux error ∥p - p h ∥ L 2 (Ω) up to multiplicative constants and higher-order terms. The averaging space Q h consists of piecewise polynomial and globally continuous finite element functions in d components with carefully designed boundary conditions. The minimal value η M is frequently replaced by some averaging operator A: P h → Q h applied within a simple post-processing to p h . The result q h := Ap h E Q h provides a reliable error bound with η M < η A := ∥p h - Ap h ∥ L 2 (Ω) . This paper establishes η A < C eff η M and so equivalence of η M and η A . This implies efficiency of η A for a large class of patchwise averaging techniques which includes the ZZ-gradient-recovery technique. The bound C eff < 3.88 established for tetrahedral P 1 finite elements appears striking in that the shape of the elements does not enter: The equivalence η A η M is robust with respect to anisotropic meshes. The main arguments in the proof are Ascoli's lemma, a strengthened Cauchy inequality, and elementary calculations with mass matrices.