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

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