High-order space–time finite element methods for the Poisson–Nernst–Planck equations: Positivity and unconditional energy stability

Abstract

We present a novel class of high-order space–time finite element schemes for the Poisson–Nernst–Planck (PNP) equations. We prove that our schemes are mass conservative, positivity preserving, and unconditionally energy stable for any order of approximation. To the best of our knowledge, this is the first class of (arbitrarily) high-order accurate schemes for the PNP equations that simultaneously achieve all these three properties.This is accomplished via (1) using finite elements to directly approximate the so-called entropy variableui≔U′(ci)=log(ci) instead of the density variable ci, where U(ci)=(log(ci)−1)ci is the corresponding entropy, and (2) using a discontinuous Galerkin (DG) discretization in time. The entropy variable formulation, which was originally developed by Metti et al. (2016) under the name of a log-density formulation, guarantees both positivity of densities ci=exp(ui)>0 and a continuous-in-time energy stability result. The DG in time discretization further ensures an unconditional energy stability in the fully discrete level for any approximation order, where the lowest order case is exactly the backward Euler discretization and in this case we recover the method of Metti et al. (2016).