A Decoupling Two-grid Method for the Time-dependent Poisson-Nernst-Planck Equations

TL;DR

This paper introduces a decoupling two-grid method for efficiently solving the time-dependent Poisson-Nernst-Planck equations, maintaining accuracy while reducing computational complexity through coarse space solutions.

Contribution

The paper develops a novel two-grid decoupling strategy for the Poisson-Nernst-Planck equations, providing optimal error estimates and demonstrating efficiency and accuracy improvements.

Findings

The method achieves optimal $L^2$ and $H^1$ error estimates.

Numerical experiments confirm the efficiency and accuracy of the approach.

The decoupling strategy speeds up computations compared to traditional methods.

Abstract

We study a two-grid strategy for decoupling the time-dependent Poisson-Nernst-Planck equations describing the mass concentration of ions and the electrostatic potential. The computational system is decoupled to smaller systems by using coarse space solutions at each time level, which can speed up the solution process compared with the finite element method combined with the Gummel iteration. We derive the optimal error estimates in $L^2$ norm for both semi- and fully discrete finite element approximations. Based on the a priori error estimates, the error estimates in $H^1$ norm are presented for the two-grid algorithm. The theoretical results indicate this decoupling method can retain the same accuracy as the finite element method. Numerical experiments including the Poisson-Nernst-Planck equations for an ion channel show the efficiency and effectiveness of the decoupling two-grid method.