Goal-oriented adaptivity using unconventional error representations

Abstract

In Goal-Oriented Adaptivity (GOA), the error in the Quantity of Interest (QoI) is represented using the error functions of the direct and adjoint problems. This error representation is subsequently bounded above by element-wise error indicators that are used to drive optimal refinements. In this work, we propose to replace, in the error representation, the adjoint problem by an alternative operator. The main advantage of the proposed approach is that, when judiciously selecting such alternative operator, the corresponding upper bound of the error representation becomes sharper, leading to a more efficient GOA.While the method can be applied to a variety of problems, we focused on one-, two- and three dimensional Helmholtz and one- and two-dimensional convection-dominated diffusion problems. Weshow via extensive numerical experimentation that the upper bounds provided by the alternative error representations are sharper than the classical ones and lead to a more robust p-adaptive process. Weprovide guidelines for finding operators delivering sharp error representation upper bounds. We further extend the results to problems with discontinuous material coefficients. Finally, we consider a sonic Logging-While-Drilling (LWD) problem to illustrate the applicability of the proposed method.