A Direct Discontinuous Galerkin (ddg) Method for Solving the Nonlinear Poisson Boltzmann Equation

Abstract

In this work, we propose a novel approach to solve the nonlinear Poisson Boltzmann (PB) equation. We identify a function space in which the solution of the nonlinear PB equation is iteratively approximated through solving a series of linear PB equations, while an appropriate initial guess and a suitable iterative parameter are matched 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 (DDG) method to those linear PB equations. More precisely, we provide one initial guess when the Debye parameter λ = O(1), and for λ 1 a special initial guess is adopted to ensure convergence. The choice of iterative parameter is carefully made to guarantee the existence, uniqueness, and convergence of the iterative approximation. In particular, we show that the variable iterative parameter can reduce iteration steps for convergence. Both one and two-dimensional numerical results are carried out to demonstrate accuracy and capacity of the iteration approximation method when combined with the DDG method for both cases of λ = O(1) and λ 1. The (m + 1)th order of accuracy for L and mth order of accuracy for H for P elements are numerically obtained.

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Cite this paper

@inproceedings{YIN2013ADD, title={A Direct Discontinuous Galerkin (ddg) Method for Solving the Nonlinear Poisson Boltzmann Equation}, author={PEIMENG YIN and Yunqing Huang and Hailiang Liu}, year={2013} }