  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.
