Linearized Conservative Finite Element Methods for the Nernst-Planck-Poisson Equations

The aim of this paper is to present and study new linearized conservative schemes with finite element approximations for the Nernst-Planck-Poisson equations. For the linearized backward Euler FEM, an optimal error L-2 estimate is provided almost unconditionally (i.e., when the mesh size h and time step tau are less than a small constant). Global mass conservation and electric energy decay of the schemes are also proved. Extension to second-order time discretizations is given. Numerical results in both two- and three-dimensional spaces are provided to confirm our theoretical analysis and show the optimal convergence, unconditional stability, global mass conservation and electric energy decay properties of the proposed schemes.