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 Rᵈ. Given a piecewise constant discrete flux pₕ∈ Pₕ (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 \[ η _M:=min \{\|p_h-q_h\|L^2(Ω ): q_h∈ Q_h\} \] in the sense that η M is a lower and upper bound of the flux error \|p-pₕ\|L²(Ω ) up to multiplicative constants and higher-order terms. The averaging space Qₕ 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: Pₕ→ Qₕ applied within a simple post-processing to pₕ. The result qₕ:=Apₕ∈ Qₕ provides a reliable error bound with η M≤ η A:=\|pₕ-Apₕ\|L²(Ω ). This paper establishes η A≤ Ceff 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 Ceff≤ 3.88 established for tetrahedral P₁ finite elements appears striking in that the shape of the elements does