Error Analysis of Virtual Element Methods for the Time-Dependent Poisson-Nernst-Planck Equations (Journal of Computational Mathematics, 2024)
We discuss and analyze the virtual element method on general polygonal meshes forthe time-dependent Poisson-Nernst-Planck (PNP) equations, which are a nonlinear coupledsystem widely used in semiconductors and ion channels. After presenting the semi-discretescheme, the optimal $H^1$ norm error estimates are presented for the time-dependent PNPequations, which are based on some error estimates of a virtual element energy projection.The Gummel iteration is used to decouple and linearize the PNP equations and the erroranalysis is also given for the iteration of fully discrete virtual element approximation. Thenumerical experiment on different polygonal meshes verifies the theoretical convergenceresults and shows the efficiency of the virtual element method.