  This chapter provides an overview of state-of-the-art adaptive finite element
methods (AFEMs) for the numerical solution of second-order elliptic partial
differential equations (PDEs), where the primary focus is on the optimal
interplay of local mesh refinement and iterative solution of the arising
discrete systems. Particular emphasis is placed on the thorough description of
the essential ingredients necessary to design adaptive algorithms of optimal
complexity, i.e., algorithms that mathematically guarantee the optimal rate of
convergence with respect to the overall computational cost and, hence, time.
Crucially, adaptivity induces reliability of the computed numerical
approximations by means of a-posteriori error control. This ensures that the
error committed by the numerical scheme is bounded from above by computable
quantities. The analysis of the adaptive algorithms is based on the study of
appropriate quasi-error quantities that include and balance different
components of the overall error. Importantly, the quasi-errors stemming from an
adaptive algorithm with contractive iterative solver satisfy a centerpiece
concept, namely, full R-linear convergence. This guarantees that the adaptive
algorithm is essentially contracting this quasi-error at each step and it turns
out to be the cornerstone for the optimal complexity of AFEM. The unified
analysis of the adaptive algorithms is presented in the context of symmetric
linear PDEs. Extensions to goal-oriented, non-symmetric, as well as non-linear
PDEs are presented with suitable nested iterative solvers fitting into the
general analytical framework of the linear symmetric case. Numerical
experiments highlight the theoretical results and emphasize the practical
relevance and gain of adaptivity with iterative solvers for numerical
simulations with optimal complexity.
