A new block preconditioner and improved finite element solver of Poisson-Nernst-Planck equation

Abstract

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, the commonly-used solution decomposition schemes in classic continuum models for isolating the singularities were presented and then one was chosen for solving PNP when zero initial guess was used for the Newton method. 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.