Classical residual type error estimators approximate the error flux around the elements and yield upper bounds of the exact (or reference) error. Lower bounds of the error are also needed in goal oriented adaptivity and for bounds on functional outputs. This work introduces a simple and cheap strategy to recover a lower bound estimate from standard upper bound estimates. This lower bound may also be used to assess the effectivity of the former estimate and to improve it.
Díez, P.; Pares, N.; Huerta, A. Recovering lower bounds of the error by postprocessing implicit residual a posteriori error estimates. "International journal for numerical methods in engineering (Recurs electrònic)", Març 2003, vol. 56, núm. 10, The definitive version is available at http://www3.interscience.wiley.com/journal/102522978/abstract, p. 1465-1488.