Goal oriented a posteriori error estimators for problems with modified discrete formulations based on the dual weighted residual method

The article at hand focuses on finite element discretizations, where the continuous and the discrete formulations differ. We introduce a general approach based on the dual weighted residual method for estimating on the one hand the discretization error in a user specified quantity of interest and on the other hand the discrete model error induced by using different discrete techniques. Here, the usual error identities are obtained plus some additional terms. Furthermore, the numerical approximation of the error identities is discussed. As a simple example, we consider selective reduced integration for stabilizing the finite element discretization of linear elastic problems with nearly incompressible material behavior. This example fits well in the general setting. However, one has to be very careful in the numerical approximation of the error identities, where different reconstruction techniques have to be used for the additional terms due to the deviating discrete bi-linear form. Numerical examples substantiate the accuracy of the a posteriori error estimators and the efficiency of the adaptive methods based on them