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 $\Omega$ in $\mathbb{R}^d$. Given a piecewise constant discrete flux $p_h\in P_h$(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

in the sense that $\eta_M$ is a lower and upper bound of the flux error $\Vert p-p_h\Vert _{L^2(\Omega)}$ up to multiplicative constants and higher-order terms. The averaging space $\mathcal{Q}_h$ consists of piecewise polynomial and globally continuous finite element functions in $d$ components with carefully designed boundary conditions. The minimal value $\eta_M$ is frequently replaced by some averaging operator $A: P_h\rightarrow\mathcal{Q}_h$applied within a simple post-processing to $p_h$. The result $q_h:=Ap_h\in\mathcal{Q}_h$ provides a reliable error bound with $\eta_M\leq\eta_A:=\Vert p_h-Ap_h\Vert _{L^2(\Omega)}$.

This paper establishes $\eta_A\leq C_{\mbox{\tiny eff}}\,\eta_M$ and so equivalence of $\eta_M$ and $\eta_A$. This implies efficiency of $\eta_A$ for a large class of patchwise averaging techniques which includes the ZZ-gradient-recovery technique. The bound $C_{\mbox{\tiny eff}}\le 3.88$established for tetrahedral $P_1$ finite elements appears striking in that the shape of the elements does not enter: The equivalence $\eta_A\approx\eta_M$ 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.