Skip to main content Accesibility Help
×
×
Home

Convergent finite element discretizations of the Navier-Stokes-Nernst-Planck-Poisson system

  • Andreas Prohl (a1) and Markus Schmuck (a2)
Abstract

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.

Copyright
References
Hide All
[1] R.A. Adams and J.J.F. Fournier, Sobolev Spaces. Elsevier (2003).
[2] Arnold, D.N., Brezzi, F. and Fortin, M., A stable finite element for the Stokes equations. Calcolo 23 (1984) 337344.
[3] Bazant, M.Z., Thornton, K. and Ajdari, A., 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).
[5] Chorin, A., On the convergence of discrete approximations of the Navier-Stokes Equations. Math. Com. 23 (1969) 341353.
[6] Ciarlet, P.G. and Raviart, P.A., Maximum principle and uniform convergence for the finite element method. Comput. Methods Appl. Mech. Eng. 2 (1973) 1731.
[7] Ciavaldini, J.F., Analyse numérique d'un problème de Stefan à deux phases par une méthode d'éléments finis. SIAM J. Numer. Anal. 12 (1975) 464487.
[8] P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman Advanced Publishing Program, Boston, USA (1985).
[9] Guermond, J.L., Minev, J. and Shen, J., An overview of projection methods for incompressible flows. Comput. Meth. Appl. Mech. Engrg. 195 (2006) 60116045.
[10] Heywood, J.G. and Rannacher, R., 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) 275311.
[11] R.J. Hunter, Foundations of Colloidal Science. Oxford University Press, UK (2000).
[12] Kilic, M.S., Bazant, M.Z. and Ajdari, A., 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.
[14] J.L. Lions and E. Magenes, Nonhomogeneous boundary value problems and applications, Grundlehren der Mathematischen Wissenschaften 181. Springer-Verlag, Berlin-New York (1972).
[15] P.L. Lions, Mathematical Topics in Fluid Mechanics, Volume 1: Incompressible Models. Oxford University Press, UK (1996).
[16] Nochetto, R.H. and Verdi, C., Convergence past singularities for a fully discrete approximation of curvature-driven interfaces. SIAM J. Numer. Anal. 34 (1997) 490512.
[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).
[19] Prohl, A., On pressure approximation via projection methods for the nonstationary incompressible Navier-Stokes equations. SIAM J. Numer. Anal. 47 (2008) 158180.
[20] Prohl, A. and Schmuck, M., Convergent discretizations for the Nernst-Planck-Poisson system. Numer. Math. 111 (2009) 591630.
[21] M. Schmuck, Modeling, Analysis and Numerics in Electrohydrodynamics. Ph.D. Thesis, University of Tübingen, Germany (2008).
[22] Schmuck, M., Analysis of the Navier-Stokes-Nernst-Planck-Poisson system. M3AS 19 (2009) 123.
[23] Simon, J., Sobolev, Besov and Nikolskii fractional spaces: Imbeddings and comparisons for vector valued spaces on an interval. Ann. Mat. Pura Appl. 157 (1990) 117148.
[24] Temam, R., Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires ii. Arch. Ration. Mech. Anal. 33 (1969) 377385.
[25] R. Temam, Navier-Stokes equations – theory and numerical analysis. AMS Chelsea Publishing, Providence, USA (2001).
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

ESAIM: Mathematical Modelling and Numerical Analysis
  • ISSN: 0764-583X
  • EISSN: 1290-3841
  • URL: /core/journals/esaim-mathematical-modelling-and-numerical-analysis
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed