The finite element approximation of the nonlinear Poisson-Boltzmann equation
The finite element approximation of the nonlinear Poisson-Boltzmann equation
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DOI：
10.1137/060675514
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发表时间：
2007-01-01
影响因子：
2.9
通讯作者：
Xu, Jinchao
中科院分区：
文献类型：
--
作者：
Chen, Long;Holst, Michael J.;Xu, Jinchao
本文分析了生物分子建模界广泛使用的静电模型、非线性泊松-玻尔兹曼方程及其有限元近似。引入正则化泊松-玻尔兹曼方程作为辅助问题，使得研究具有δ分布源的原始非线性方程成为可能。基于某些二维和三维准均匀网格，获得了正则化泊松-玻尔兹曼方程的有限元近似的先验误差估计。通过后验误差估计驱动的局部细化的自适应有限元近似被证明是收敛的。泊松-玻尔兹曼方程以前似乎没有在理论上进行过详细研究，希望本文能够帮助分子建模者利用泊松-玻尔兹曼方程进行分析和计算工作提供更好的基础。请注意，本文显然给出了具有 delta 分布源的非线性 Poisson-Boltzmann 方程的数值离散化技术的第一个严格收敛结果，并且还介绍了该方程的第一个可证明收敛的自适应方法。最后一个结果是目前为数不多的此类非线性问题收敛结果之一。
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.