Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T20:58:55.648Z Has data issue: false hasContentIssue false
  • English
  • Français

Convergent finite element discretizationsof the Navier-Stokes-Nernst-Planck-Poisson system

Published online by Cambridge University Press:  23 February 2010

Andreas Prohl  and
Markus Schmuck
Andreas Prohl
Affiliation:
Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany. prohl@na.uni-tuebingen.de
Markus Schmuck
Affiliation:
Department of Chemical Engineering, MIT Boston, 25 Ames Street, Cambridge, MA 02139, USA. schmuck@mit.edu
Get access

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.

Keywords

Electrohydrodynamicsspace-time discretizationfinite elementsconvergence
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 44 , Issue 3 , May 2010 , pp. 531 - 571
Copyright
© EDP Sciences, SMAI, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

R.A. Adams and J.J.F. Fournier, Sobolev Spaces. Elsevier (2003).
Arnold, D.N., Brezzi, F. and Fortin, M., A stable finite element for the Stokes equations. Calcolo 23 (1984) 337–344. CrossRef
Bazant, M.Z., Thornton, K. and Ajdari, A., Diffuse-charge dynamics in electrochemical systems. Phys. Rev. E 70 (2004) 021506. CrossRef
S.C. Brenner and L. Scott, The Mathematical Theory of Finite Element Methods. Second edition, Springer (2002).
Chorin, A., On the convergence of discrete approximations of the Navier-Stokes Equations. Math. Com. 23 (1969) 341–353.
Ciarlet, P.G. and Raviart, P.A., Maximum principle and uniform convergence for the finite element method. Comput. Methods Appl. Mech. Eng. 2 (1973) 17–31.
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) 464–487. CrossRef
P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman Advanced Publishing Program, Boston, USA (1985).
Guermond, J.L., Minev, J. and Shen, J., An overview of projection methods for incompressible flows. Comput. Meth. Appl. Mech. Engrg. 195 (2006) 6011–6045. CrossRef
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) 275–311. CrossRef
R.J. Hunter, Foundations of Colloidal Science. Oxford University Press, UK (2000).
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. CrossRef
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.
J.L. Lions and E. Magenes, Nonhomogeneous boundary value problems and applications, Grundlehren der Mathematischen Wissenschaften 181. Springer-Verlag, Berlin-New York (1972).
P.L. Lions, Mathematical Topics in Fluid Mechanics, Volume 1: Incompressible Models. Oxford University Press, UK (1996).
Nochetto, R.H. and Verdi, C., Convergence past singularities for a fully discrete approximation of curvature-driven interfaces. SIAM J. Numer. Anal. 34 (1997) 490–512. CrossRef
R.F. Probstein, Physiochemical Hydrodynamics, An introduction. John Wiley and Sons, Inc. (1994).
A. Prohl, Projection and Quasi-Compressibility Methods for Solving the Incompressible Navier-Stokes Equations. Teubner (1997).
Prohl, A., On pressure approximation via projection methods for the nonstationary incompressible Navier-Stokes equations. SIAM J. Numer. Anal. 47 (2008) 158–180. CrossRef
Prohl, A. and Schmuck, M., Convergent discretizations for the Nernst-Planck-Poisson system. Numer. Math. 111 (2009) 591–630. CrossRef
M. Schmuck, Modeling, Analysis and Numerics in Electrohydrodynamics. Ph.D. Thesis, University of Tübingen, Germany (2008).
Schmuck, M., Analysis of the Navier-Stokes-Nernst-Planck-Poisson system. M3AS 19 (2009) 1–23.
Simon, J., Sobolev, Besov and Nikolskii fractional spaces: Imbeddings and comparisons for vector valued spaces on an interval. Ann. Mat. Pura Appl. 157 (1990) 117–148. CrossRef
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) 377–385. CrossRef
R. Temam, Navier-Stokes equations – theory and numerical analysis. AMS Chelsea Publishing, Providence, USA (2001).