P4

Long Chen, Michael Holst, and Jinchao Xu. The Finite Element Approximation of the Nonlinear Poisson-Boltzmann Equation.

Chen.L;Holst.M;Xu.J2006a

ABSTRACT: 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 function sources. {\it A priori} error estimates for the finite element approximation is 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 {\it 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-function 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.