the numerical solution of nonlinear elliptic equations is important in many applications; however, it is a challenging task to develop efficient software to solve general problems. this talk will describe the basic finite element method and develop an adaptive framework based on the solve-estimate-mark-refine iteration. a theory of convergence for this iteration, which allows the solver to be inexact, will be given as well as a new error estimator. numerical results will be shown for a model problem arising in computational biochemistry.