A positivity-preserving and free energy dissipative hybrid scheme for the Poisson-Nernst-Planck equations on polygonal and polyhedral meshes

This paper develops a positivity-preserving and free energy dissipative hybrid scheme for the multi-dimensional Poisson-Nernst-Planck equations with multiple ions on unstructured polygonal and polyhedral meshes. It is built on the linear virtual element method for the Poisson-type equation for the electrostatic potential and a cell-centered finite volume scheme for the Nernst-Planck equations for the ion concentrations, and preserves the positivity of the ion concentrations, the mass conservation, the free energy dissipation and steady-state properties. The existence of the numerical solutions is also discussed. Numerical results demonstrate the effectiveness of the proposed two- and three-dimensional schemes.