  A wide variety of (fixed-point) iterative methods for the solution of
nonlinear equations (in Hilbert spaces) exists. In many cases, such schemes can
be interpreted as iterative local linearization methods, which, as will be
shown, can be obtained by applying a suitable preconditioning operator to the
original (nonlinear) equation. Based on this observation, we will derive a
unified abstract framework which recovers some prominent iterative schemes. In
particular, for Lipschitz continuous and strongly monotone operators, we derive
a general convergence analysis. Furthermore, in the context of numerical
solution schemes for nonlinear partial differential equations, we propose a
combination of the iterative linearization approach and the classical Galerkin
discretization method, thereby giving rise to the so-called iterative
linearization Galerkin (ILG) methodology. Moreover, still on an abstract level,
based on two different elliptic reconstruction techniques, we derive a
posteriori error estimates which separately take into account the
discretization and linearization errors. Furthermore, we propose an adaptive
algorithm, which provides an efficient interplay between these two effects. In
addition, the ILG approach will be applied to the specific context of finite
element discretizations of quasilinear elliptic equations, and some numerical
experiments will be performed.
