A Synthesis of A Posteriori Error Estimation Techniques for Conforming, Non-Conforming and Discontinuous Galerkin Finite Element Methods

A posteriori error estimation for conforming, non-conforming and discontinuous finite element schemes are discussed within a single framework. By dealing with three ostensibly different schemes under the same umbrella, the same common underlying principles at work in each case are highlighted leading to a clearer understanding of the issues involved. The ideas are presented in the context of piecewise affine finite element approximation of a second-order elliptic problem. It is found that the framework leads to three different known a posteriori error estimators: the equilibrated residual method in the case of conforming Galerkin FEM; the estimator of Ainsworth [3] in the case of the Crouzeix-Raviart scheme, and a new estimator [1] recently derived in case of discontinuous Galerkin approximation. In all cases one has computable upper bounds on the error measured in the energy norm and corresponding local lower bounds showing the efficiency of the schemes.