Research On More Efficient Numerical Algorithm For Solving The Nonlinear Poisson Boltzmann Equation

In this work, we propose a novel approach to solve the nonlinear Poisson Boltz-mann (PB) equation. Firstly, we find out the space that the solution of the nonlinear PB equation. In order to overcome the challenge of the nonlinear PB equation, we utilize the solution of a series of linear PB equations with appropriate guess u0and appropriate iterative coefficient kn to approximate the solution of nonlinear PB e-quation, which assure that the solutions of those linear PB equations still lie in the former space, and we use the Direct Discontinuous method to discretize those linear PB equations. In this paper, we provide two kinds of way to find u0which work well when λ=O(1)or λ<<1; also we show two kinds of way to choose kn, which both can guarantee the existence, uniqueness, and convergence of the iteration ap-proximation but differing in the steps for convergence. We propose relationship of the parameters of the flux formulae which can assure the stability as well as the ex-istence and uniqueness of the DDG scheme when utilize it to discretize those linear PB equations whose Dirichlet problems with boundary value not completely equal to zero. Both one and two-dimensional numerical results are carried out to demon-strate the splendid virtue of the iteration approximation method when combined with the DDG method for both case λ=O(1) and λ<<1.(m+1)th order for L2and mth order for H1of accuracy for Pm elements are obtained.