The Finite Element Approximation of the Nonlinear Poisson-Boltzmann Equation

TL;DR

This paper provides the first rigorous convergence analysis and adaptive finite element method for the nonlinear Poisson-Boltzmann equation, a key model in biomolecular electrostatics, including error estimates and handling of delta sources.

Contribution

It introduces a regularized formulation, proves convergence of finite element approximations, and develops the first adaptive method with provable convergence for this nonlinear equation.

Findings

First rigorous convergence result for discretization of nonlinear Poisson-Boltzmann equation with delta sources.

Development of an adaptive finite element method with proven convergence.

Error estimates for finite element approximation in 2D and 3D.

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 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.