Guaranteed a posteriori error estimator for mixed finite element methods of elliptic problems

In this work we analyze a posteriori error estimator for a general class of mixed finite element methods of elliptic problems which guarantees an upper bound on the vector error, thus extending the recent result obtained for the lowest order triangular Raviart–Thomas mixed finite element method in [M. Ainsworth, A posteriori error estimation for lowest order Raviart–Thomas mixed finite elements, SIAM J. Sci. Comput. 30 (2007/08) 189–204]. The error estimator is constructed through a variant of Stenberg’s postprocessing procedure, and the guaranteed upper bound is readily established by making use of the argument similar to the hypercircle method. However, the proof of the lower bound given in the above reference does not seem to apply to other kinds of mixed elements. So we employ a different technique using the discrete Friedrichs inequality to establish the lower bound of the error estimator.