MiS Preprint Repository

The use of dual/adjoint problems for approximating functionals of solutions of PDEs with great accuracy or to merely drive a goal-oriented adaptive refinement scheme has become well-accepted, and it continues to be an active area of research. The traditional approach involves dual residual weighting (DRW). In this work we present two new functional error estimators and give conditions under which we can expect them to be asymptotically exact. The first is of DRW type and is derived for meshes in which most triangles satisfy an $\text{cO}(h^2)$-approximate parallelogram property. The second functional estimator involves dual error estimate weighting (DEW) using any superconvergent gradient recovery technique for the primal and dual solutions. Several experiments are done which demonstrate the asymptotic exactness of a DEW estimator which uses a gradient recovery scheme proposed by Bank and Xu, and the effectiveness of refinement done with respect to the corresponding local error indicators.