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An Iterative Discontinuous Galerkin Method for Solving the Nonlinear Poisson Boltzmann Equation

Published online by Cambridge University Press:  03 June 2015

Peimeng Yin ,
Yunqing Huang  and
Hailiang Liu
Peimeng Yin*
Affiliation:
Hunan Key Laboratory for Computation and Simulation in Science and Engineering; School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, P.R. China
Yunqing Huang*
Affiliation:
Hunan Key Laboratory for Computation and Simulation in Science and Engineering; School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, P.R. China
Hailiang Liu*
Affiliation:
Iowa State University, Mathematics Department, Ames, IA 50011, USA
*
Email:pemyin@gmail.com
Email:huangyq@xtu.edu.cn
Corresponding author.Email:hliu@iastate.edu
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Abstract

An iterative discontinuous Galerkin (DG) method is proposed to solve the nonlinear Poisson Boltzmann (PB) equation. We first identify a function space in which the solution of the nonlinear PB equation is iteratively approximated through a series of linear PB equations, while an appropriate initial guess and a suitable iterative parameter are selected so that the solutions of linear PB equations are monotone within the identified solution space. For the spatial discretization we apply the direct discontinuous Galerkin method to those linear PB equations. More precisely, we use one initial guess when the Debye parameter λ = (1), and a special initial guess for λ ≫1 to ensure convergence. The iterative parameter is carefully chosen to guarantee the existence, uniqueness, and convergence of the iteration. In particular, iteration steps can be reduced for a variable iterative parameter. Both one and two-dimensional numerical results are carried out to demonstrate both accuracy and capacity of the iterative DG method for both cases of λ = (1) and λ ≪ 1. The (m + 1)th order of accuracy for L2 and mth order of accuracy for H1 for Pm elements are numerically obtained.

Keywords

65D1565N3035J0535J25Poisson-Boltzmann equationnonlinearexistenceuniquenessDDG methodsnumerical flux.
Type
Research Article
Information
Communications in Computational Physics , Volume 16 , Issue 2 , August 2014 , pp. 491 - 515
Copyright
Copyright © Global Science Press Limited 2014

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