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 λ=O(1), and a special initial guess for λ≪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 λ=O(1) and λ≪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.