  A widely used electrostatics model in the biomolecular modeling community,
the nonlinear Poisson-Boltzmann equation, along with its finite element
approximation, are analyzed in this paper. A regularized Poisson-Boltzmann
equation is introduced as an auxiliary problem, making it possible to study the
original nonlinear equation with delta distribution sources. A priori error
estimates for the finite element approximation are obtained for the regularized
Poisson-Boltzmann equation based on certain quasi-uniform grids in two and
three dimensions. Adaptive finite element approximation through local
refinement driven by an a posteriori error estimate is shown to converge. The
Poisson-Boltzmann equation does not appear to have been previously studied in
detail theoretically, and it is hoped that this paper will help provide
molecular modelers with a better foundation for their analytical and
computational work with the Poisson-Boltzmann equation. Note that this article
apparently gives the first rigorous convergence result for a numerical
discretization technique for the nonlinear Poisson-Boltzmann equation with
delta distribution sources, and it also introduces the first provably
convergent adaptive method for the equation. This last result is currently one
of only a handful of existing convergence results of this type for nonlinear
problems.
