A Decoupling Two-grid Method for the Time-dependent Poisson-Nernst-Planck Equations
This work provides a faster numerical solver for the Poisson-Nernst-Planck equations, which are important for modeling ion transport in biological and electrochemical systems.
The authors propose a two-grid decoupling method for the time-dependent Poisson-Nernst-Planck equations, achieving computational speedup while maintaining accuracy comparable to the standard finite element method with Gummel iteration. Numerical tests on an ion channel model demonstrate efficiency.
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.