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Summary: In goal‐oriented adaptive algorithms for partial differential equations, we adapt the finite element mesh to reduce the error of the solution in some quantity of interest. In time‐dependent problems, this adaptive algorithm involves solving a dual problem that runs backward in time. This process is, in general, computationally expensive in terms of memory storage. In this work, we define a pseudo‐dual problem that runs forward in time. We also describe a forward‐in‐time adaptive algorithm that works for some specific problems. Although it is not possible to define a general dual problem running forwards in time that provides information about future states, we provide numerical evidence via one‐dimensional problems in space to illustrate the efficiency of our algorithm as well as its limitations. Finally, we propose a hybrid algorithm that employs the classical backward‐in‐time dual problem once and then performs the adaptive process forwards in time. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]