Commun. Comput. Phys., 16 (2014), pp. 491-515. |
An Iterative Discontinuous Galerkin Method for Solving the Nonlinear Poisson Boltzmann Equation Peimeng Yin ^{1}, Yunqing Huang ^{1}, Hailiang Liu ^{2*} 1 Hunan Key Laboratory for Computation and Simulation in Science and Engineering; School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, P.R. China.2 Iowa State University, Mathematics Department, Ames, IA 50011, USA. Received 27 July 2013; Accepted (in revised version) 28 February 2014 Available online 12 June 2014 doi:10.4208/cicp.270713.280214a 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 $\lambda=O(1)$, and a special initial guess for $\lambda \ll 1$ 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 $\lambda=O(1)$ and $\lambda \ll 1$. The $(m+1)$th order of accuracy for $L^2$ and $m$th order of accuracy for $H^1$ for $P^m$ elements are numerically obtained. AMS subject classifications: 65D15, 65N30, 35J05, 35J25Notice: Undefined variable: pac in /var/www/html/readabs.php on line 165 Key words: Poisson-Boltzmann equation, nonlinear, existence, uniqueness, DDG methods, numerical flux. *Corresponding author. Email: pemyin@gmail.com (P. Yin), huangyq@xtu.edu.cn (Y. Huang), hliu@iastate.edu (H. Liu) |