We consider a linear symmetric and elliptic PDE and a linear goal functional. 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.
We consider a linear symmetric and elliptic PDE and a linear goal functional. We design and analyze a goal-oriented adaptive finite element method , 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 . Unlike prior work, we do not only consider rates with respect to the number of degrees of freedom but even prove optimal complexity, i.e., optimal convergence rates with respect to the total computational cost.
A
and analyze
D
(GOAFEM)
R
(PCG)
or geometric multigrid
A
. Unlike prior work, we do not only consider rates
R
as well as the
but even prove optimal complexity, i.e., optimal convergence rates with respect to the total