Convergence analysis of a BDF2 / mixed finite element discretization of a Darcy–Nernst–Planck–Poisson system

¹ Rice University, CAAM Department, 6100 Main Street, Houston, TX 77005, USA
florian.frank@rice.edu
² Friedrich–Alexander University of Erlangen–Nürnberg, Department of Mathematics, Cauerstr. 11, 91058 Erlangen, Germany

Received: 25 January 2016
Revised: 24 August 2016
Accepted: 17 January 2017

Abstract

This paper presents an a priori error analysis of a fully discrete scheme for the numerical solution of the transient, nonlinear Darcy–Nernst–Planck–Poisson system. The scheme uses the second order backward difference formula (BDF2) in time and the mixed finite element method with Raviart–Thomas elements in space. In the first step, we show that the solution of the underlying weak continuous problem is also a solution of a third problem for which an existence result is already established. Thereby a stability estimate arises, which provides an L^∞ bound of the concentrations / masses of the system. This bound is used as a level for a cut-off operator that enables a proper formulation of the fully discrete scheme. The error analysis works without semi-discrete intermediate formulations and reveals convergence rates of optimal orders in time and space. Numerical simulations validate the theoretical results for lowest order finite element spaces in two dimensions.