Adaptive Finite Element Modeling Techniques for the Poisson-Boltzmann Equation (Communications in Computational Physics, 2012)
We consider the design of an effective and reliable adaptive finite elementmethod (AFEM) for the nonlinear Poisson-Boltzmann equation (PBE). We first examine the two-term regularization technique for the continuous problem recently proposed by Chen, Holst and Xu based on the removal of the singular electrostatic potential inside biomolecules; this technique made possible the development of the firstcomplete solution and approximation theory for the Poisson-Boltzmann equation, thefirst provably convergent discretization and also allowed for the development of aprovably convergent AFEM. However, in practical implementation, this two-term regularization exhibits numerical instability. Therefore, we examine a variation of thisregularization technique which can be shown to be less susceptible to such instability.We establish a priori estimates and other basic results for the continuous regularizedproblem, as well as for Galerkin finite element approximations. We show that the newapproach produces regularized continuous and discrete problems with the same mathematical advantages of the original regularization. We then design an AFEM schemefor 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 isone of the first results of this type for nonlinear elliptic problems, is based on using continuous and discrete a priori L∞ estimates. To provide a high-quality geometricmodel as input to the AFEM algorithm, we also describe a class of feature-preservingadaptive mesh generation algorithms designed specifically for constructing meshes ofbiomolecular structures, based on the intrinsic local structure tensor of the molecularsurface. All of the algorithms described in the article are implemented in the Finite Element Toolkit (FETK), developed and maintained at UCSD. The stability advantagesof the new regularization scheme are demonstrated with FETK through comparisonswith the original regularization approach for a model problem. The convergence andaccuracy of the overall AFEM algorithm is also illustrated by numerical approximationof electrostatic solvation energy for an insulin protein.