An Iterative Discontinuous Galerkin Method for Solving the Nonlinear Poisson Boltzmann Equation (Communications in Computational Physics, 2014)
An iterative discontinuous Galerkin (DG) method is proposed to solve thenonlinear Poisson Boltzmann (PB) equation. We first identify a function space in whichthe solution of the nonlinear PB equation is iteratively approximated through a seriesof linear PB equations, while an appropriate initial guess and a suitable iterative parameterare selected so that the solutions of linear PB equations are monotone withinthe identified solution space. For the spatial discretization we apply the direct discontinuousGalerkin method to those linear PB equations. More precisely, we use oneinitial guess when the Debye parameter λ=O(1), and a special initial guess for λ≪1to ensure convergence. The iterative parameter is carefully chosen to guarantee the existence,uniqueness, and convergence of the iteration. In particular, iteration steps canbe reduced for a variable iterative parameter. Both one and two-dimensional numericalresults are carried out to demonstrate both accuracy and capacity of the iterativeDG method for both cases of λ=O(1) and λ≪1. The (m+1)th order of accuracy forL2 and mth order of accuracy for H1 for Pm elements are numerically obtained.