  We propose a new heuristic goal-oriented a posteriori error estimator that
connects the dual weighted residual method with equilibrated a posteriori error
estimation. Our numerical experiments demonstrate the practical reliability of
the error estimator, confirming theoretical predictions, as well as optimally
convergent adaptivity even over singular domains and coarse meshes. The central
algorithm is a localized flux reconstruction, which has been implemented in the
finite element library deal.II. For a solid preparation we assess the
performance of the equilibrated a posteriori error estimator of the energy norm
in numerical experiments. Moreover, we give what seems to be first rigorous
discussion in the numerical literature of localized flux reconstruction over
quadrilateral meshes with hanging nodes.
