Some Remarks on the History and Future of Averaging Techniques in A~Posteriori Finite Element Error Analysis

Given a flux or stress approximation $p_h$ from a low-order finite
element simulation of an elliptic boundary value problem, averaging or
(gradient-) recovery techniques aim the computation of an improved
approximation $Ap_h$ by a (simple) post processing of
$p_h$. For instance, frequently named after Zienkiewicz and Zhu,
$Ap_h$ is the elementwise interpolation of the nodal values
$(Ap_h)(z)$ obtained as the integral mean of $p_h$ on a
neighbourhood of $z$. This paper gives an
overview over old and new arguments in the proof of reliability and
efficiency of the error estimator $ eta_{ cal A}:= |p_h-{A}p_h |$
as an approximation of the error $ |p-p_h |$ in (an energy norm)
$ | cdot |$. High-lighted are the general class of meshes, averaging
operators, or finite elements [conforming, nonconforming, or mixed].
Emphasis is on old and new aspects of superconvergence and arguments
to circumvent superconvergence at all within proofs of a~posteriori
finite element error estimates.