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.

翻译：泊松-能斯特-普朗克（PNP）方程组是描述离子溶液系统中静电相互作用和扩散过程的最有效模型之一，已广泛应用于生物离子通道、半导体器件和纳米孔系统的数值模拟。由于强耦合、对流主导、非线性和多尺度等特性，经典Gummel迭代求解PNP方程非线性离散系统时收敛缓慢甚至发散。本文聚焦于一般PNP方程非线性离散系统的快速算法，相比Gummel迭代具有更优的自适应性、友好性和高效性。首先，提出几何全近似存储（FAS）算法以改善Gummel迭代的收敛速度问题。其次，设计代数FAS算法，该算法无需多层次几何信息，相比几何FAS算法更适用于实际计算。最后，针对代数FAS算法中粗网格迭代次数过多及对计算域尺寸适应性不足的问题，提出基于加速技术和自适应方法的改进算法。通过数值实验分别展示几何FAS、代数FAS及其改进算法的计算效率。