An Iterative Discontinuous Galerkin Method for Solving the Nonlinear Poisson Boltzmann Equation
An Iterative Discontinuous Galerkin Method for Solving the Nonlinear Poisson Boltzmann Equation
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DOI：
10.4208/cicp.270713.280214a
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发表时间：
2014-08
影响因子：
3.7
通讯作者：
Peimeng Yin;Yunqing Huang;Hailiang Liu
中科院分区：
文献类型：
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作者：
Peimeng Yin;Yunqing Huang;Hailiang Liu
提出了一种求解非线性Poisson Boltzmann方程的迭代间断Galerkin（DG）方法.我们首先确定一个函数空间，其中非线性PB方程的解通过一系列线性PB方程迭代逼近，同时选择适当的初始猜测和适当的迭代参数，使得线性PB方程的解在所确定的解空间内是单调的。对于空间离散，我们采用直接间断Galerkin方法对线性PB方程进行离散.更准确地说，当德拜参数l=O（1）时，我们使用一个初始猜测，并且对于l <$1使用一个特殊的初始猜测以确保收敛。对迭代参数进行了仔细的选择，以保证迭代的存在性、唯一性和收敛性。特别地，对于可变迭代参数，可以减少迭代步骤。对l=O（1）和l ≠ 1两种情况，用一维和二维数值结果证明了迭代DG方法的精度和计算能力.数值计算得到Pm元L2的m+1阶精度和H1的m阶精度.
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 pa- rameter 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 dis- continuous Galerkin method to those linear PB equations. More precisely, we use one initial guess when the Debye parameter l=O(1), and a special initial guess for l≪1 to ensure convergence. The iterative parameter is carefully chosen to guarantee the ex- istence, uniqueness, and convergence of the iteration. In particular, iteration steps can be reduced for a variable iterative parameter. Both one and two-dimensional numer- ical results are carried out to demonstrate both accuracy and capacity of the iterative DG method for both cases of l=O(1) and l≪1. The (m+1)th order of accuracy for L 2 and mth order of accuracy for H 1 for P m elements are numerically obtained.