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 η 𝒜 := ‖ p h ‐ Ap h ‖ as an approximation of the error ‖ p ‐ p h ‖ in (an energy norm) ‖⋅‖. 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.