  Purpose: This paper aims at assessing the effect of (1) the statical
admissibility of the recovered solution; (2) the ability of the recovered
solution to represent the singular solution; on the accuracy, local and global
effectivity of recovery-based error estimators for enriched finite element
methods (e.g. the extended finite element method, XFEM).
Design/methodology/approach: We study the performance of two recovery
techniques. The first is a recently developed superconvergent patch recovery
procedure with equilibration and enrichment (SPR-CX). The second is known as
the extended moving least squares recovery (XMLS), which enriches the recovered
solutions but does not enforce equilibrium constraints. Both are extended
recovery techniques as the polynomial basis used in the recovery process is
enriched with singular terms for a better description of the singular nature of
the solution. Findings: Numerical results comparing the convergence and the
effectivity index of both techniques with those obtained without the enrichment
enhancement clearly show the need for the use of extended recovery techniques
in Zienkiewicz-Zhu type error estimators for this class of problems. The
results also reveal significant improvements in the effectivities yielded by
statically admissible recovered solutions. Originality/value: This work shows
that both extended recovery procedures and statical admissibility are key to an
accurate assessment of the quality of enriched finite element approximations.
