We examine a nonlinear PDE model of electrostatics phenomena arising
in biophysics. Through use of a two-scale expansion we establish
well-posedness and a priori max-norm estimates for the continuous
and discrete problems. We derive a priori and a posteriori
estimates for Galerkin approximations, and describe a nonlinear
approximation algorithm based on error indicator-driven adaptive
refinement. We then prove that the adaptive algorithm converges,
establishing one of only a handful of results of this type for
nonlinear elliptic equations. We finish by illustrating the adaptive
algorithm with examples using the Finite Element ToolKit (FETK).