Goal-oriented adaptivity using unconventional error representations for the multidimensional Helmholtz equation

In goal-oriented adaptivity, the error in the quantity of interest 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 goal-oriented adaptivity. While the method can be applied to a variety of problems, we focus here on two- and three-dimensional (2-D and 3-D) Helmholtz problems. We show 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. We also provide guidelines for finding operators delivering sharp error representation upper bounds. We further extend the results to a convection-dominated diffusion problem as well as to problems with discontinuous material coefficients. Finally, we consider a sonic logging-while-drilling problem to illustrate the applicability of the proposed method.