Abstract: We proposed a  two-level algorithm based on the virtual element discretization for both  steady-state and time-dependent PNP (Poisson-Nernst-Planck) equations. This algorithm first decoupled and linearized the PNP equations using the linear virtual element solution, and then solved them in the quadratic virtual element space. Compared with the commonly used Gummel algorithm for solving the PNP equations, this algorithm could accelerate the solving speed. Numerical experimental results, including both 
 steady-state and time-dependent PNP equations, show that compared with the Gummel algorithm with the linear virtual element, the two-level algorithm has  higher accuracy, and consumes less CPU time and is more efficient at  comparable accuracy.

Key words: Poisson-Nernst-Planck equations, virtual element method, two-level algorithm, Gummel algorithm