Skip to main content
SpringerLink
Log in

On the Discretization of the Coupled Heat and Electrical Diffusion Problems

  • Conference paper
Numerical Methods and Applications (NMA 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4310))

Included in the following conference series:

Abstract

We consider a nonlinear system of elliptic equations, which arises when modelling the heat diffusion problem coupled with the electrical diffusion problem. The ohmic losses which appear as a source term in the heat diffusion equation yield a nonlinear term which lies in L 1. A finite volume scheme is proposed for the discretization of the system; we show that the approximate solution obtained with the scheme converges, up to a subsequence, to a solution of the coupled elliptic system.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Eymard, R., Gallouët, T., Herbin, R.: A hybrid finite volume schemes for the discretization of anistropic diffusion operator on general meshes (submitted)

    Google Scholar 

  2. Bern, M., Eppstein, D.: Mesh generation and optimal triangulation. In: Computing in Euclidean geometry. Lecture Notes Ser. Comput., vol. 1, pp. 23–90. World Sci. Publ., River Edge (1992)

    Google Scholar 

  3. Boccardo, L., Gallouët, T., Vazquez, J.-L.: Nonlinear Elliptic Equations in R n without Growth Restrictions on the Data. Journal of Differential Equations 105(2), 334–363 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bénilan, P., Boccardo, L., Gallouët, T., Gariepy, R., Pierre, M., Vazquez, J.-L.: An L 1 theory of existence and uniqueness of solutions of nonlinear elliptic equations. Ann. Scuola. Norm. Sup. Pisa 22, 241–273 (1995)

    MATH  Google Scholar 

  5. Bradji, A., Herbin, R.: Discretization of ohmic losses by the finite element and the finite volume method (in preparation)

    Google Scholar 

  6. Casado-Diaz, J., Chacon Rebollo, T., Girault, V., Gomez, M., Murat, F.: Finite elements approximation of second order linear elliptic equations in divergence form with right-hand side in L 1 (submitted)

    Google Scholar 

  7. Ciarlet, P.A.: Basic error estimates for elliptic problems. In: Handbook of Numerical Analysis, vol. II, pp. 17–352. North-Holland, Amsterdam (1991)

    Google Scholar 

  8. Clain, S.: Finite element approximations for Laplace operator with right hand side measure. Math. Models Meth. App. Sci. 6 n° 5 (1996) 713–719

    Article  MathSciNet  MATH  Google Scholar 

  9. Coudière, Y., Gallouët, T., Herbin, R.: Discrete Sobolev inequalities and L p error estimates for approximate finite volume solutions of convection diffusion equations. M2AN 35(4), 767–778 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dall’aglio, A.: Approximated solutions of equations with L 1 solutions wiht L 1 data. Application to the H–convergence of quasi–linear parabolic equations. Ann. Mat. Pura Appl. 170, 207–240 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  11. Droniou, J., Gallouët, T., Herbin, R.: A finite volumes schemes for a noncoercive elliptic equation with measure data. SIAM J. Numer. Anal. 41 n° 6 (2003) 1997-2031

    Article  MathSciNet  MATH  Google Scholar 

  12. Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, vol. II, pp. 723–1020 (2000)

    Google Scholar 

  13. Eymard, R., Gallouët, T., Herbin, R.: Convergence of finite volume schemes for semilinear convection diffusion equations. Numer. Math. 82, 91–116 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  14. Eymard, R., Gallouët, T.: H-convergence and numerical schemes for elliptic problems. SIAM J. Numer. Anal. 41, 539–562 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Eymard, R., Gallouët, T., Herbin, R.: A cell-centered finite-volume approximation for anisotropic diffusion operators on unstructured meshes in any space dimension. IMA J. of Numer. Anal. 26, 326–353 (2006)

    Article  MATH  Google Scholar 

  16. Ferguson, J.R., Fiard, J.M., Herbin, R.: A mathematical model of solid oxide fuel cells. Journal of Power Sources 58, 109–122 (1996)

    Article  Google Scholar 

  17. Fiard, J.M., Herbin, R.: Comparison between finite volume and finite element methods for an elliptic system arising in electrochemical engineerings. Comput. Methods App. Mech. Engrg. 115, 315–338 (1994)

    Article  Google Scholar 

  18. Gallouët, T., Herbin, R.: Convergence of linear finite element for diffusion equations with measure data. CRAS 338, 81–84 (2004)

    MATH  Google Scholar 

  19. Gallouët, T., Herbin, R.: Existence of a solution to a coupled elliptic system. Appl. Math. Lett. 7, 49–55 (1994)

    Article  MATH  Google Scholar 

  20. Gallouët, T., Herbin, R.: Finite volume approximation of elliptic problems with irregular data. In: Benkhaldoun, F., Hänel, M., Vilsmeier, R. (eds.) Finite Volumes for Complex Applications, II, pp. 155–162. Hermes, Paris (1999)

    Google Scholar 

  21. Gallouët, T.: Measure data and numerical schemes for elliptic problems. Progress in Nonlinear Differential Equations and their Applications 63, 1–13 (2005)

    Article  Google Scholar 

  22. Murat, F.: Equations elliptiques non linéaires avec second membre L 1 ou mesure. In: Actes du 26-ème Congrès d’Analyse Numérique, Les Karellis, Université de Lyon (1994)

    Google Scholar 

  23. Stampacchia, G.: Le problème de Dirichlet pour les équations elliptiques du second orders à coefficients discontinus. Ann. Inst. Fourier 15, 189–258 (1965)

    MathSciNet  MATH  Google Scholar 

  24. Scott, R.: Finite element convergence for singular data. Numer. Math. 21, 317–327 (1973)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39 10117 Berlin, Germany

    Abdallah Bradji

  2. Laboratoire d’Analyse, Topologie et Probabilités, Université Aix–Marseille 1, 39 rue Joliot Curie 13453 Marseille, France

    Raphaèle Herbin

Authors
  1. Abdallah Bradji
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Raphaèle Herbin
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Todor Boyanov Stefka Dimova Krassimir Georgiev Geno Nikolov

Rights and permissions

Reprints and permissions

Copyright information

© 2007 Springer Berlin Heidelberg

About this paper

Cite this paper

Bradji, A., Herbin, R. (2007). On the Discretization of the Coupled Heat and Electrical Diffusion Problems. In: Boyanov, T., Dimova, S., Georgiev, K., Nikolov, G. (eds) Numerical Methods and Applications. NMA 2006. Lecture Notes in Computer Science, vol 4310. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70942-8_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-70942-8_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-70940-4

  • Online ISBN: 978-3-540-70942-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation