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

Abstract:

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 \[ \eta _M:=\min \{\|p_h-q_h\|_{L^2(\Omega )}: q_h\in \mathcal {Q}_h\} \] in the sense that $\eta _M$ is a lower and upper bound of the flux error $\|p-p_h\|_{L^2(\Omega )}$ 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 $\eta _M\leq \eta _A:=\|p_h-Ap_h\|_{L^2(\Omega )}$. 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.