A Linearized Local Conservative Mixed Finite Element Method for Poisson–Nernst–Planck Equations

Abstract

In this paper, a linearized local conservative mixed finite element method is proposed and analyzed for Poisson–Nernst–Planck (PNP) equations, where the mass fluxes and the potential flux are introduced as new vector-valued variables to equations of ionic concentrations (Nernst–Planck equations) and equation of the electrostatic potential (Poisson equation), respectively. These flux variables are crucial to PNP equations on determining the Debye layer and computing the electric current in an accurate fashion. The Raviart–Thomas mixed finite element is employed for the spatial discretization, while the backward Euler scheme with linearization is adopted for the temporal discretization and decoupling nonlinear terms, thus three linear equations are separately solved at each time step. The proposed method is more efficient in practice, and locally preserves the mass conservation. By deriving the boundedness of numerical solutions in certain strong norms, an unconditionally optimal error analysis is obtained for all six unknowns: the concentrations p and n, the mass fluxes $${{\varvec{J}}}_p=\nabla p + p {\varvec{\sigma }}$$ and $${{\varvec{J}}}_n=\nabla n - n {\varvec{\sigma }}$$ , the potential $$\psi $$ and the potential flux $${\varvec{\sigma }}= \nabla \psi $$ in $$L^{\infty }(L^2)$$ norm. Numerical experiments are carried out to demonstrate the efficiency and to validate the convergence theorem of the proposed method.