The finite element approximation of the nonlinear Poisson-Boltzmann equation

Long Chen, Michael J. Holst, Jinchao Xu

Research output: Contribution to journalArticle

87 Citations (Scopus)

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.

Original languageEnglish (US)
Pages (from-to)2298-2320
Number of pages23
JournalSIAM Journal on Numerical Analysis
Volume45
Issue number6
DOIs
StatePublished - Dec 1 2007

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Poisson-Boltzmann Equation
Boltzmann equation
Finite Element Approximation
Nonlinear Equations
Convergence Results
Local Refinement
Adaptive Finite Elements
A Priori Error Estimates
A Posteriori Error Estimates
Adaptive Method
Nonlinear equations
Electrostatics
Three-dimension
Nonlinear Problem
Two Dimensions
Discretization
Grid
Converge
Modeling

All Science Journal Classification (ASJC) codes

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

Cite this

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The finite element approximation of the nonlinear Poisson-Boltzmann equation. / Chen, Long; Holst, Michael J.; Xu, Jinchao.

In: SIAM Journal on Numerical Analysis, Vol. 45, No. 6, 01.12.2007, p. 2298-2320.

Research output: Contribution to journalArticle

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