The finite element approximation of the nonlinear Poisson-Boltzmann equation

被引:129
作者
Chen, Long [1 ]
Holst, Michael J. [1 ]
Xu, Jinchao [2 ]
机构
[1] Univ Calif San Diego, Dept Math, La Jolla, CA 92093 USA
[2] Peking Univ, Sch Math Sci, Beijing 100871, Peoples R China
关键词
nonlinear Poisson-Boltzmann equation; finite element methods; a priori and a posteriori error estimate; convergence of adaptive methods;
D O I
10.1137/060675514
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
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.
引用
收藏
页码:2298 / 2320
页数:23
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