  In this article we study adaptive finite element methods (AFEM) with inexact
solvers for a class of semilinear elliptic interface problems. We are
particularly interested in nonlinear problems with discontinuous diffusion
coefficients, such as the nonlinear Poisson-Boltzmann equation and its
regularizations. The algorithm we study consists of the standard
SOLVE-ESTIMATE-MARK-REFINE procedure common to many adaptive finite element
algorithms, but where the SOLVE step involves only a full solve on the coarsest
level, and the remaining levels involve only single Newton updates to the
previous approximate solution. We summarize a recently developed AFEM
convergence theory for inexact solvers, and present a sequence of numerical
experiments that give evidence that the theory does in fact predict the
contraction properties of AFEM with inexact solvers. The various routines used
are all designed to maintain a linear-time computational complexity.
