  We consider adaptive finite element methods for second-order elliptic PDEs,
where the arising discrete systems are not solved exactly. For contractive
iterative solvers, we formulate an adaptive algorithm which monitors and steers
the adaptive mesh-refinement as well as the inexact solution of the arising
discrete systems. We prove that the proposed strategy leads to linear
convergence with optimal algebraic rates. Unlike prior works, however, we focus
on convergence rates with respect to the overall computational costs. In
explicit terms, the proposed adaptive strategy thus guarantees quasi-optimal
computational time. In particular, our analysis covers linear problems, where
the linear systems are solved by an optimally preconditioned CG method as well
as nonlinear problems with strongly monotone nonlinearity which are linearized
by the so-called Zarantonello iteration.
