Before revision:  We design  a goal-oriented adaptive finite element method (GOAFEM) , which steers the adaptive mesh-refinement as well as the approximate solution of the arising linear systems by means of a contractive iterative solver like the optimally preconditioned conjugate gradient method (PCG) . We prove linear convergence of the proposed adaptive algorithm with optimal algebraic rates  with respect to the number of degrees of freedom as well as the computational cost.
After revision:  We design a goal-oriented adaptive finite element method (GOAFEM) , which steers the adaptive mesh-refinement as well as the approximate solution of the arising linear systems by means of a contractive iterative solver like the optimally preconditioned conjugate gradient method or geometric multigrid . We prove linear convergence of the proposed adaptive algorithm with optimal algebraic rates with respect to the number of degrees of freedom as well as the computational cost.