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Interior penalty discontinuous Galerkin (IPDG) methods for second order elliptic boundary value problems have been derived from a mixed variational formulation of the problem. Numerical flux functions across interelement boundaries play an important role in that theory. Residual type a posteriori error estimators for IPDG methods have been derived and analyzed by many authors including the convergence analysis of the resulting adaptive schemes. Typically, the effectivity indices deteriorate with increasing polynomial order of the IPDG methods. The situation is more favorable for a posteriori error estimators derived by means of the so-called hypercircle method. Equilibrated fluxes are obtained by using a mixed method and an extension operator for Brezzi--Douglas--Marini elements, and this can be done in the same way for all the discontinuous Galerkin methods that fit into a known unified framework. The hypercircle method immediately provides the reliability of the estimator, whereas its efficiency can be ...

An Equilibrated A Posteriori Error Estimator for the Interior Penalty Discontinuous Galerkin Method