We develop an efficient and reliable adaptive finite element method (AFEM)
for the nonlinear Poisson-Boltzmann equation (PBE). We first examine the
regularization technique of Chen, Holst, and Xu; this technique made possible
the first a priori pointwise estimates and the first complete solution and
approximation theory for the Poisson-Boltzmann equation. It also made possible
the first provably convergent discretization of the PBE, and allowed for the
development of a provably convergent AFEM for the PBE. However, in practice the
regularization turns out to be numerically ill-conditioned. In this article, we
examine a second regularization, and establish a number of basic results to
ensure that the new approach produces the same mathematical advantages of the
original regularization, without the ill-conditioning property. We then design
an AFEM scheme based on the new regularized problem, and show that the
resulting AFEM scheme is accurate and reliable, by proving a contraction result
for the error. This result, which is one of the first results of this type for
nonlinear elliptic problems, is based on using continuous and discrete a priori
pointwise estimates to establish quasi-orthogonality. To provide a high-quality
geometric model as input to the AFEM algorithm, we also describe a class of
feature-preserving adaptive mesh generation algorithms designed specifically
for constructing meshes of biomolecular structures, based on the intrinsic
local structure tensor of the molecular surface. The stability advantages of
the new regularization are demonstrated using an FETK-based implementation,
through comparisons with the original regularization approach for a model
problem. The convergence and accuracy of the overall AFEM algorithm is also
illustrated by numerical approximation of electrostatic solvation energy for an
insulin protein.