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All rst-order averaging or gradient-recovery operators for lowest- order nite element methods are shown to allow for an ecient a posteriori error estimation in an isotropic, elliptic model problem in a bounded Lipschitz domain in R d . Given a piecewise constant discrete ux ph 2 Ph (that is the gradient of a discrete displacement) as an approximation to the unknown exact ux p (that is the gradient of the exact displacement), recent results verify eciency and reliability of M := minfkph qhkL2() : qh2Qhg in the sense that M is a lower and upper bound of the ux errorkp phkL2() up to multiplicative constants and higher-order terms. The averaging space Qh consists of piecewise polynomial and globally continuous nite element functions in d components with carefully designed boundary conditions. The minimal value M is frequently replaced by some averaging operator A : Ph! Qh applied within a simple post-processing to ph .T he resultqh :=Aph2Qh provides a reliable error bound with M A :=kph Aphk L2() . This paper establishesA CeM and so equivalence ofM andA .T his implies eciency of A for a large class of patchwise averaging techniques which includes the ZZ-gradient-recovery technique. The bound Ce 3:88 established for tetrahedral P1 nite elements appears striking in that the shape of the elements does not enter: The equivalenceA 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 matri- ces.

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