  We consider scalar semilinear elliptic PDEs where the nonlinearity is
strongly monotone, but only locally Lipschitz continuous. We formulate an
adaptive iterative linearized finite element method (AILFEM) which steers the
local mesh refinement as well as the iterative linearization of the arising
nonlinear discrete equations. To this end, we employ a damped Zarantonello
iteration so that, in each step of the algorithm, only a linear Poisson-type
equation has to be solved. We prove that the proposed AILFEM strategy
guarantees convergence with optimal rates, where rates are understood with
respect to the overall computational complexity (i.e., the computational time).
Moreover, we formulate and test an adaptive algorithm where also the damping
parameter of the Zarantonello iteration is adaptively adjusted. Numerical
experiments underline the theoretical findings.
