Forward-In-Time Goal-Oriented Adaptivity

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.