A virtual element methods for a class of time-dependent Poisson-Nernst-Planck equations

For a class of time-dependent Poisson-Nernst-Planck(PNP) equations, in order to avoid the mesh adaptability problem in finite element method when solving practical problems, a virtual element algorithm combining L² projection operator and Gummel iteration was proposed. This method allows new formats to be designed and analyzed in a simpler manner and flexible handling of various meshes. Virtual element method can handle polygon or polyhedron elements well, even the mesh division consisting of non-convex elements, which makes virtual element method can adapt to arbitrary polygon mesh, greatly reducing the difficulty of mesh generation. Numerical examples of virtual element algorithm under triangular mesh, quadrilateral mesh, non-convex mesh are given. Numerical experiments show that on three different grids, the convergence order of the L² and H¹ norms are second order and first order respectively, and them both reach the optimal order.