Error Analysis of Virtual Element Methods for the Time-dependent Poisson-Nernst-Planck Equations

Abstract: We discuss and analyze the virtual element method on general polygonal meshes for the time-dependent Poisson-Nernst-Planck equations, which are a nonlinear coupled system widely used in semiconductors and ion channels. The spatial discretization is based on the elliptic projection and the $L^2$ projection operator, and for the temporal discretization, the backward Euler scheme is employed. After presenting the semi and fully discrete schemes, we derive the a priori error estimates in the $L^2$ and $H^1$ norms. Finally, a numerical experiment verifies the theoretical convergence results.