The Prager-Synge theorem in reconstruction based a posteriori error estimation

In this paper we review the hypercircle method of Prager and Synge. This theory inspired several studies and induced an active research in the area of a posteriori error analysis. In particular, we review the Braess--Sch\"oberl error estimator in the context of the Poisson problem. We discuss adaptive finite element schemes based on two variants of the estimator and we prove the convergence and optimality of the resulting algorithms

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

Interior Penalty Discontinuous Galerkin (IPDG) methods for second order elliptic boundary value problems have been derived from a mixed hybrid 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 a convergence analysis of the resulting adaptive scheme. 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 an extension operator for BDM elements, and this can be done in the same way for all the DG methods presented in a unified framework. This construction enables to establish the efficiency of the equilibrated estimator, whereas the reliability can be shown by standard arguments. In contrast to the residual-type estimators, the new estimators do not contain unknown generic constants. Numerical results are given that illustrate the performance of the suggested approach