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

Frank, Florian; Knabner, Peter

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 5, pp. 1883-1902.

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 $L^{\infty }$ 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.

Classification : 65M12, 65M15, 65M60, 65L06, 76Rxx, 76Wxx
Mots clés : Stokes / Darcy–Nernst–Planck–Poisson system, mixed finite elements, backward difference formula, error analysis, porous media

R.A. Adams, Sobolev Spaces. Number Bd. 65 in Pure Appl. Math. Academic Press (1975). | Zbl

G. Allaire, R. Brizzi, J.F. Dufrêche, A. Mikelić and A. Piatnitski, Ion transport in porous media: derivation of the macroscopic equations using upscaling and properties of the effective coefficients. Comput. Geosci. 17 (2013) 479–495. | DOI | Zbl

T. Arbogast, M.F. Wheeler and N.Y. Zhang, A nonlinear mixed finite element method for a degenerate parabolic equation arising in flow in porous media. SIAM J. Numer. Anal. 33 (1996) 1669–1687. | DOI | Zbl

S. Barbeiro and M.F. Wheeler, A priori error estimates for the numerical solution of a coupled geomechanics and reservoir flow model with stress-dependent permeability. Comput. Geosci. 14 (2010) 755–768. | DOI | Zbl

P. Berg and J. Findlay, Analytical solution of the poisson–nernst–planck–stokes equations in a cylindrical channel. Proc. R. Soc. A: Math., Phys. Eng. Sci. 467 (2011) 3157–3169. | DOI | Zbl

F. Brezzi and M. Fortin,Mixed and hybrid finite elements methods. Springer Series in Computational Mathematics. Springer, New York (1991). | Zbl

M. Bukal, E. Emmrich and A. Jüngel, Entropy-stable and entropy-dissipative approximations of a fourth-order quantum diffusion equation. Numer. Math. 127 (2013) 365–396. | DOI | Zbl

Z. Chen, Finite Element Methods and Their Applications. Springer (2005). | Zbl

P.G. Ciarlet, Basic error estimates for elliptic problems. Finite Element Methods (Part 1), Vol. 2. In Handbook of Numerical Analysis. Edited by P.G. Ciarlet and J. L. Lions. North-Holland, Amsterdam (1991). | Zbl

D. Cioranescu and P. Donato, An Introduction to Homogenization, in vol. 17 of Oxford lecture series in mathematics and its applications. Edited by J. Ball and D. Welsh. Oxford University Press (1999). | Zbl

Z. Ding, A proof of the trace theorem of Sobolev spaces on Lipschitz domains. Vol. 124 of Proc. of the American Mathematical Society (1996) 591–600. | Zbl

R.G. Durán, Mixed finite element methods. In Mixed Finite Elements, Compatibility Conditions, and Applications: Lectures given at the C.I.M.E. Springer (2008). | Zbl

A. Ern and J. L. Guermond, Theory and Practice of Finite Elements. Applied Mathematical Sciences. Springer (2004). | Zbl

L.C. Evans,Partial Differential Equations. Graduate Studies in Mathematics. American Mathematical Society (2010). | Zbl

F. Frank, Numerical studies of models for electrokinetic flow and charged solute transport in periodic porous media. Ph.D. thesis, University of Erlangen-Nürnberg (2013).

F. Frank, N. Ray and P. Knabner, Numerical investigation of homogenized stokes–nernst–planck–poisson systems. Comput. Visualiz. Sci. 14 (2013) 385–400. | DOI | Zbl

V. Girault and P.A. Raviart, Finite Element Approximation of the Navier–Stokes Equations. Lecture Notes in Mathematics Springer (1979). | Zbl

M. Herz, N. Ray and P. Knabner, Existence and uniqueness of a global weak solution of a Darcy–Nernst–Planck–Poisson system. GAMM-Mitteilungen 35 (2012) 191–208. | DOI | Zbl

B. Kirby, Micro- And Nanoscale Fluid Mechanics: Transport in Microfluidic Devices. Cambridge University Press (2010). | Zbl

P. Knabner and W. Barth, Lineare Algebra: Grundlagen und Anwendungen. Springer (2012). | Zbl

J.R. Looker and S.L. Carnie, Homogenization of the ionic transport equations in periodic porous media. Transp. Porous Media 65 (2006) 107–131. | DOI

J.H. Masliyah and S. Bhattacharjee, Electrokinetic and Colloid Transport Phenomena. Wiley-Interscience (2006).

C. Moyne and M.A. Murad, Electro-chemo-mechanical couplings in swelling clays derived from a micro / macro-homogenization procedure. Inter. J. Solids Structures 39 (2002) 6159–6190. | DOI | Zbl

C. Moyne and M.A. Murad, A two-scale model for coupled electro-chemo-mechanical phenomena and Onsager’s reciprocity relations in expansive clays: I Homogenization analysis. Transp. Porous Media 62 (2006) 333–380. | DOI

J.C. Nédélec, Mixed finite elements in $R^3$. Numer. Math. 35 (1980) 315–341. | DOI | Zbl

A. Prohl and M. Schmuck, Convergent discretizations for the Nernst–Planck–Poisson system. Numer. Math. 111 (2009) 591–630. | DOI | Zbl

A. Quarteroni and A. Valli. Numerical Approximation of Partial Differential Equations. Springer Series in Computational Mathematics. Springer (1994). | Zbl

F. Radu, I.S. Pop and P. Knabner, Order of convergence estimates for an Euler implicit, mixed finite element discretization of Richards’ equation. SIAM J. Numer. Anal. 42 (2004) 1452–1478. | DOI | Zbl

F.A. Radu, Mixed finite element discretization of Richard’s equation: error analysis and application to realistic infiltration problems. Ph.D. thesis, Chair of Applied Mathematics 1, University of Erlangen–Nuremberg (2004).

F.A. Radu, I.S. Pop and S. Attinger, Analysis of an Euler implicit-mixed finite element scheme for reactive solute transport in porous media. Numer. Methods for Partial Differ. Equ. 26 (2010) 320–344. | Zbl

P.A. Raviart and J.M. Thomas, A mixed finite element method for second-order elliptic problems. Mathematical Aspects of the Finite Element Method. In vol. 606 of Lectures Notes in Mathematics (1977) 292–315. | Zbl

N. Ray, A. Muntean and P. Knabner, Rigorous homogenization of a Stokes–Nernst–Planck–Poisson system. J. Math. Anal. Appl. 390 (2012) 374–393. | DOI | Zbl

T. Roubcíˇek, Incompressible ionized fluid mixtures. Contin. Mech. Thermodyn. 17 (2006) 493–509. | Zbl

M. Schmuck, Modeling and deriving porous media Stokes–Poisson–Nernst–Planck equations by a multiple-scale approach. Commun. Math. Sci. 9 (2011) 685–710. | DOI | Zbl

S. Sun, B. Rivière, and M.F. Wheeler, A combined mixed finite element and discontinuous galerkin method for miscible displacement problem in porous media. In Recent Progress in Computational and Applied PDES, edited by Tony F. Chan, Yunqing Huang, Tao Tang, Jinchao Xu, and Long-An Ying (2002) 323–351. Springer. | Zbl

J.M. Thomas, Sur l’analyse numérique des méthodes d’éléments finis hybrides et mixtes. Thèse d’Etat, Université Pierre et Marie Curie, Paris (1977).

V. Thomée, Galerkin Finite Element Methods for Parabolic Problems. Springer Series in Computational Mathematics. Springer (2006). | Zbl

MATLAB, The MathWorks, Inc., Natick, Massachusetts, United States. Available at: http://www.mathworks.com/products/matlab (2017).

Z. Wu, J. Yin and C. Wang, Elliptic and Parabolic Equations. World Scientific (2006). | Zbl