Advertisement

Entropy-Diminishing CVFE Scheme for Solving Anisotropic Degenerate Diffusion Equations

  • Clément Cancès
  • Cindy Guichard
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 77)

Abstract

We consider a Control Volume Finite Elements (CVFE) scheme for solving possibly degenerated parabolic equations. This scheme does not require the introduction of the so-called Kirchhoff transform in its definition. The discrete solution obtained via the scheme remains in the physical range whatever the anisotropy of the problem, while the natural entropy of the problem decreases with time. Moreover, the discrete solution converges towards the unique weak solution of the continuous problem. Numerical results are provided and discussed.

Keywords

Dirichlet Boundary Condition Continuous Problem Discrete Solution Permeability Tensor Degenerate Parabolic Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

This work was supported by the French National Research Agency ANR (project GeoPor, grant ANR-13-JS01-0007-01).

References

  1. 1.
    Alt, H.W., Luckhaus, S.: Quasilinear elliptic-parabolic differential equations. Math. Z. 183(3), 311–341 (1983)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Baliga, B.R., Patankar, S.V.: A control volume finite-element method for two-dimensional fluid flow and heat transfer. Numer. Heat Transfer 6(3), 245–261 (1983)MATHGoogle Scholar
  3. 3.
    Bear, J.: Dynamic of Fluids in Porous Media. American Elsevier, New York (1972)Google Scholar
  4. 4.
    Cancès, C., Guichard, C.: Convergence of a nonlinear entropy diminishing control volume finite element scheme for solving anisotropic degenerate parabolic equations (2014). HAL: hal-00955091Google Scholar
  5. 5.
    Chainais-Hillairet, C.: Entropy method and asymptotic behaviours of finite volume schemes. In: FVCA7 conference proceedings (2014).Google Scholar
  6. 6.
    Chainais-Hillairet, C., Jüngel, A.S.S.: Entropy-dissipative discretization of nonlinear diffusion equations and discrete Beckner inequalities (2014). HAL: hal-00924282Google Scholar
  7. 7.
    Eymard, R., Gallouët, T., Ghilani, M., Herbin, R.: Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes. IMA J. Numer. Anal. 18(4), 563–594 (1998)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. In: Ciarlet, P.G. et al. (ed.) Handbook of numerical analysis, pp. 713–1020. North-Holland, Amsterdam (2000)Google Scholar
  9. 9.
    Herbin, R., Hubert, F.: Benchmark on discretization schemes for anisotropic diffusion problems on general grids. In: Eymard, R, Herard, J.M. (eds.) Finite volumes for complex applications V, pp. 659–692. Wiley (2008)Google Scholar
  10. 10.
    Otto, F.: \({L}^1\)-contraction and uniqueness for quasilinear elliptic-parabolic equations. J. Diff. Equat. 131, 20–38 (1996)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Otto, F.: The geometry of dissipative evolution equations: the porous medium equation. Commun. Partial Differ. Equ. 26(1–2), 101–174 (2001)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Laboratoire Jacques-Louis LionsSorbonne Universités, UPMC University Paris 06, UMR 7598ParisFrance
  2. 2.Laboratoire Jacques-Louis LionsCNRS, UMR 7598ParisFrance

Personalised recommendations