The Poisson-Nernst-Planck (PNP) equation is one important continuum model for studying charge transport in ion channels, which is a common phenomenon and plays a key role in molecular biosciences. In this paper, to improve the current PNP solvers, according to the equation structure, a new block preconditioner was proposed and proved that the preconditioned linear system has bounded eigenvalues independent of mesh sizes under some conditions, thus guaranteeing that the convergence rate of the preconditioned linear solver is independent of mesh sizes. Meanwhile, comparisons of the commonly-used solution decomposition schemes in classic continuum models for isolating the singularities were carried out as the first time to demonstrate the disadvantage/advantage in solving PNP. Furthermore, an efficient and improved finite element PNP solver was proposed by further combining the two-grid method as an acceleration technique. Then the new program package was fulfilled based on the state-of-the-art finite element library FEniCS and the efficient scientific library PETSc. Finally, numerical simulations on a test model with analytical solution as well as some tests on protein cases were carried out to validate the new program package and our theoretical results.

Poisson-Nernst-Planck（PNP）方程是一种重要的连续模型，用于研究离子通道中的电荷传输，这是一种常见现象，在分子生物学中起着关键作用。为了改进当前的PNP求解器，根据方程结构，提出了一种新的块预处理器，并证明了预处理线性系统在某些条件下具有与网格大小无关的有界特征值，从而保证了预处理器的收敛速度。线性求解器与网格尺寸无关。同时，比较了经典连续谱模型中用于分离奇异性的常用解法的比较，这是第一次证明解决PNP的优缺点。此外，通过进一步结合两网格法作为一种加速技术，提出了一种有效的改进的有限元PNP求解器。然后，基于最新的有限元库完成新的程序包FEniCS和高效的科学图书馆PETSc。最后，对具有解析解的测试模型进行了数值模拟，并对蛋白质盒进行了一些测试，以验证新程序包和我们的理论结果。