PNP方程及其改进模型的数值模拟和计算上改进的一些研究

Poisson-Nernst-Planck（PNP）方程是模拟离子输运、电化学系统以及生物细胞膜间的离子转化和静电扩散反应过程最为常用的模型之一。由于方程组具有强耦合非线性性、多点源奇异、对流占优等属性，需要不断改进计算方法来克服分析和计算上的困难。另外，为了更加准确地模拟计算反映生物物理现象, 需要不断改进和发展新的数学模型。VDPNP和PNP-NS模型是根据不同的实际需要而发展的新模型, 在数值计算上存在许多需要深入探究的问题和困难。数值模拟方法由于其可帮助探究实验现象背后的机理、为实验结果提供有力的理论支撑与解释以及为某些实验无法得到的数据提供参考等，是研究离子输运性质、扩散反应过程的重要工具。本项目旨在针对目前数值计算中存在的一些困难，如对流占优、保正性、流守恒、非线性迭代策略等发展和改进一些新的有限元计算方法，为实现有效的大规模并行模拟提供技术支撑。

Poisson-Nernst-Planck（PNP）equations are one of the most commonly used models in simulating the ionic transport, electrochemical system, ionic transformation between biological cell membranes and the electro-diffusion reaction process. Due to the strong coupling nonlinearity, singularity of multi-point sources and convection-dominated of the equations, it is necessary to continuously improve the computing methods to overcome the difficulties in analysis and computation. In addition, in order to simulate and reflect biophysical phenomena more accurately, new mathematical models need to be improved and developed. VDPNP and PNP-NS models are two new models developed according to different practical needs. There are many problems and difficulties to be further studied in numerical computation for them. Numerical simulation is an important tool to study ionic transport properties and diffusion reaction process because it can help to explore the mechanism behind experimental phenomena, provide strong theoretical support and interpretation for experimental results, and provide reference for some data that cannot be obtained in experiments. This program aims at developing and improving some new finite element methods to resolve the difficulties in current numerical computing, such as convection-dominated, positivity-preserving, flux conservation and nonlinear iteration strategies, etc., so as to provide technical support for the realization of the effective large-scale parallel simulation.