Goal-oriented r-adaptivity based on variational arguments in the physical and material spaces

The error in a local quantity of interest depends on the error in the corresponding dual solution or generalized Green’s function. This fact is used in dual-weighted based goal-oriented error estimation techniques and h- or p-adaptive algorithms. This contribution is concerned with a novel approach for goal-oriented r-adaptivity based on energy minimization principles for the primal and the dual problems. The minimization of the energy of the primal problem as well as the minimization of the energy of the dual problem with respect to a design function lead to the primal and dual material residuals, respectively. These residuals are error indicators for the discretization error on the current mesh. We propose a goal-oriented mesh optimization algorithm which ends in an optimized mesh with respect to a chosen quantity of interest. By means of selected computational examples, we demonstrate the capability of the proposed framework.