Numer. Math. (2009) 111:591–630 DOI 10.1007/s00211-008-0194-2 Numerische Mathematik Convergent discretizations for the Nernst–Planck–Poisson system Andreas Prohl · Markus Schmuck Received: 2 November 2007 / Revised: 7 August 2008 / Published online: 26 November 2008 © Springer-Verlag 2008 Abstract We propose and compare two classes of convergent finite element based approximations of the nonstationary Nernst–Planck–Poisson equations, whose constructions are motivated from energy versus entropy decay properties for the limi- ting system. Solutions of both schemes converge to weak solutions of the limiting problem for discretization parameters tending to zero. Our main focus is to study qualitative properties for the different approaches at finite discretization scales, like conservation of mass, non-negativity, discrete maximum principle, decay of discrete energies, and entropies to study long-time asymptotics. Mathematics Subject Classification (2000) 65N30 · 35L60 · 35L65 1 Introduction Let  ⊂ R d , for d = 2, 3 be a bounded Lipschitz domain. The classical drift- diffusion system describes evolution of positively, and negatively charged particles p, n : (0, T ]×  → R + 0 , and the electric potential ψ : (0, T ]×  → R, p t = div (∇ p + p∇ψ) in  T := (0, T ]× , (1.1) n t = div (∇n - n∇ψ) in  T , (1.2) -ψ = p - n in  T . (1.3) A. Prohl (B ) · M. Schmuck Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany e-mail: prohl@na.uni-tuebingen.de M. Schmuck e-mail: schmuck@na.uni-tuebingen.de 123