Goal-oriented adaptive method for Fredholm partial integro-differential equations

In this article, a goal-oriented adaptive scheme is proposed for Fredholm partial integro-differential equations (FPIDEs). The aim of this work is to obtain higher accuracy approximations for the unknown function and its derivative at a given fixed point. In the proposed goal-oriented scheme, as opposed to the typical energy norm, the finite element approximated error is computed based on the quantities of interest. These designated quantities are identified by linear goal functions. The formula of the goal error and the a priori estimate of its upper bound are derived. An a posteriori error estimate is formulated and then approximated by using the recovered gradient. Some numerical tests are presented to show the efficiency of the proposed scheme in hastening the achievement of regional aspects of the solution.