Convergence and superconvergence analysis for a mass conservative, energy stable and linearized BDF2 scheme of the Poisson–Nernst–Planck equations

Abstract

In this paper, we consider a linearized BDF2 finite element scheme for the Poisson–Nernst–Planck (PNP) equations. By employing a novel approach, we rigorously derive unconditional optimal error estimates of the numerical solutions in the $l^{\infty }\left(L^2\right)$ and $l^{\infty }\left(H^1\right)$ norms, as well as superconvergent results. The key of the convergence and superconvergence analysis lies in deriving the stability of the finite element solutions in some stronger norms. The advantage of this approach is that there is no need to introduce a corresponding time discrete system, so it is more concise than the error split technique in previous literatures. Finally, we carry out two numerical examples to confirm the theoretical findings.