The equilibrated residual method and the method of hypercircle are popular methods for a posteriori error estimation for linear elliptic problems. Both these methods are intended to produce
guaranteed upper bounds of the energy norm of the error, but the equilibrated residual method is guaranteed only theoretically. The disadvantage of the hypercircle method is its globality, hence slowness. The combination of these two methods leads to local, hence fast, and guaranteed a posteriori error estimator.