A posteriori error estimators, gradient recovery by averaging, and superconvergence

For the linear finite element solution to a linear elliptic model problem, we derive an error estimator based upon appropriate gradient recovery by local averaging. In contrast to popular variants like the ZZ estimator, our estimator contains some additional terms that ensure reliability also on coarse meshes. Moreover, the enhanced estimator is proved to be (locally) efficient and asymptotically exact whenever the recovered gradient is superconvergent. We formulate an adaptive algorithm that is directed by this estimator and illustrate its aforementioned properties, as well as their importance, in numerical tests.