Convergent finite element discretizations of the Navier-Stokes-Nernst-Planck-Poisson system
Prohl, Andreas ; Schmuck, Markus
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 44 (2010), p. 531-571 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

We propose and analyse two convergent fully discrete schemes to solve the incompressible Navier-Stokes-Nernst-Planck-Poisson system. The first scheme converges to weak solutions satisfying an energy and an entropy dissipation law. The second scheme uses Chorin's projection method to obtain an efficient approximation that converges to strong solutions at optimal rates.

Publié le : 2010-01-01
DOI : https://doi.org/10.1051/m2an/2010013
Classification:  65N30,  35L60,  35L65 
@article{M2AN_2010__44_3_531_0,
     author = {Prohl, Andreas and Schmuck, Markus},
     title = {Convergent finite element discretizations of the Navier-Stokes-Nernst-Planck-Poisson system},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     volume = {44},
     year = {2010},
     pages = {531-571},
     doi = {10.1051/m2an/2010013},
     mrnumber = {2666654},
     zbl = {pre05711746},
     language = {en},
     url = {http://dml.mathdoc.fr/item/M2AN_2010__44_3_531_0}
}

Prohl, Andreas; Schmuck, Markus. Convergent finite element discretizations of the Navier-Stokes-Nernst-Planck-Poisson system. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 44 (2010) pp. 531-571. doi : 10.1051/m2an/2010013. http://gdmltest.u-ga.fr/item/M2AN_2010__44_3_531_0/

[1] R.A. Adams and J.J.F. Fournier, Sobolev Spaces. Elsevier (2003). | Zbl 1098.46001

[2] D.N. Arnold, F. Brezzi and M. Fortin, A stable finite element for the Stokes equations. Calcolo 23 (1984) 337-344. | Zbl 0593.76039

[3] M.Z. Bazant, K. Thornton and A. Ajdari, Diffuse-charge dynamics in electrochemical systems. Phys. Rev. E 70 (2004) 021506.

[4] S.C. Brenner and L. Scott, The Mathematical Theory of Finite Element Methods. Second edition, Springer (2002). | Zbl 0804.65101

[5] A. Chorin, On the convergence of discrete approximations of the Navier-Stokes Equations. Math. Com. 23 (1969) 341-353. | Zbl 0184.20103

[6] P.G. Ciarlet and P.A. Raviart, Maximum principle and uniform convergence for the finite element method. Comput. Methods Appl. Mech. Eng. 2 (1973) 17-31. | Zbl 0251.65069

[7] J.F. Ciavaldini, Analyse numérique d'un problème de Stefan à deux phases par une méthode d'éléments finis. SIAM J. Numer. Anal. 12 (1975) 464-487. | Zbl 0272.65101

[8] P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman Advanced Publishing Program, Boston, USA (1985). | Zbl 0695.35060

[9] J.L. Guermond, J. Minev and J. Shen, An overview of projection methods for incompressible flows. Comput. Meth. Appl. Mech. Engrg. 195 (2006) 6011-6045. | Zbl 1122.76072

[10] J.G. Heywood and R. Rannacher, Finite element approximation of the non-stationary Navier-Stokes problem I: Regularity of solutions and second-order error estimates for spatial discretization. SIAM J. Numer. Anal. 19 (1982) 275-311. | Zbl 0487.76035

[11] R.J. Hunter, Foundations of Colloidal Science. Oxford University Press, UK (2000).

[12] M.S. Kilic, M.Z. Bazant and A. Ajdari, Steric effects in the dynamics of electrolytes at large applied voltages. II. Modified Poisson-Nernst-Planck equations. Phys. Rev. E 75 (2007) 021503.

[13] J.L. Lions, On some questions in boundary value problems of mathematical physics, in Contemporary Developments in Continuum Mechanics and Partial Differential equations, Math. Stud. 30, Amsterdam, North-Holland (1978) 283-346. | Zbl 0404.35002

[14] J.L. Lions and E. Magenes, Nonhomogeneous boundary value problems and applications, Grundlehren der Mathematischen Wissenschaften 181. Springer-Verlag, Berlin-New York (1972). | Zbl 0223.35039

[15] P.L. Lions, Mathematical Topics in Fluid Mechanics, Volume 1: Incompressible Models. Oxford University Press, UK (1996). | Zbl 0908.76004

[16] R.H. Nochetto and C. Verdi, Convergence past singularities for a fully discrete approximation of curvature-driven interfaces. SIAM J. Numer. Anal. 34 (1997) 490-512. | Zbl 0876.35053

[17] R.F. Probstein, Physiochemical Hydrodynamics, An introduction. John Wiley and Sons, Inc. (1994).

[18] A. Prohl, Projection and Quasi-Compressibility Methods for Solving the Incompressible Navier-Stokes Equations. Teubner (1997). | Zbl 0874.76002

[19] A. Prohl, On pressure approximation via projection methods for the nonstationary incompressible Navier-Stokes equations. SIAM J. Numer. Anal. 47 (2008) 158-180. | Zbl pre05686544

[20] A. Prohl and M. Schmuck, Convergent discretizations for the Nernst-Planck-Poisson system. Numer. Math. 111 (2009) 591-630. | Zbl 1178.65106

[21] M. Schmuck, Modeling, Analysis and Numerics in Electrohydrodynamics. Ph.D. Thesis, University of Tübingen, Germany (2008).

[22] M. Schmuck, Analysis of the Navier-Stokes-Nernst-Planck-Poisson system. M3AS 19 (2009) 1-23. | Zbl pre05580895

[23] J. Simon, Sobolev, Besov and Nikolskii fractional spaces: Imbeddings and comparisons for vector valued spaces on an interval. Ann. Mat. Pura Appl. 157 (1990) 117-148. | Zbl 0727.46018

[24] R. Temam, Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires ii. Arch. Ration. Mech. Anal. 33 (1969) 377-385. | Zbl 0207.16904

[25] R. Temam, Navier-Stokes equations - theory and numerical analysis. AMS Chelsea Publishing, Providence, USA (2001). | Zbl 0981.35001