  The Poisson-Nernst-Planck (PNP) equations are one of the most effective model
for describing electrostatic interactions and diffusion processes in ion
solution systems, and have been widely used in the numerical simulations of
biological ion channels, semiconductor devices, and nanopore systems. Due to
the characteristics of strong coupling, convection dominance, nonlinearity and
multiscale, the classic Gummel iteration for the nonlinear discrete system of
PNP equations converges slowly or even diverges. We focus on fast algorithms of
nonlinear discrete system for the general PNP equations, which have better
adaptability, friendliness and efficiency than the Gummel iteration. First, a
geometric full approximation storage (FAS) algorithm is proposed to improve the
slow convergence speed of the Gummel iteration. Second, an algebraic FAS
algorithm is designed, which does not require multi-level geometric information
and is more suitable for practical computation compared with the geometric one.
Finally, improved algorithms based on the acceleration technique and adaptive
method are proposed to solve the problems of excessive coarse grid iterations
and insufficient adaptability to the size of computational domain in the
algebraic FAS algorithm. The numerical experiments are shown for the geometric,
algebraic FAS and improved algorithms respectively to illustrate the effiency
of the algorithms.
