Advertisement

Convergence of a Nonlinear Control Volume Finite Element Scheme for Simulating Degenerate Breast Cancer Equations

  • Françoise Foucher
  • Moustafa Ibrahim
  • Mazen Saad
Conference paper
Part of the Mathematics in Industry book series (MATHINDUSTRY, volume 26)

Abstract

In this paper, a positive nonlinear CVFE scheme is considered for the simulation of an anisotropic degenerate breast cancer model. This scheme includes the use of the finite element method combined with the Godunov scheme to approximate the diffusion terms over a primal mesh, and it uses a nonclassical upwind finite volume scheme to approximate the other terms over a barycentric dual mesh. This scheme ensures the discrete maximum principle and it converges to a weak solution without any restriction on the transmissibility coefficients. Numerical experiment is supplied in order to show the efficiency of the scheme to simulate an anisotropic breast cancer model over a general mesh.

References

  1. 1.
    Cances, C., Guichard, C.: Convergence of a nonlinear entropy diminishing control volume finite element scheme for solving anisotropic degenerate parabolic equations. Math. Comput. 85(298), 549–580 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cances, C., Ibrahim, M., Saad, M.: Positive nonlinear CVFE scheme for degenerate anisotropic Keller-Segel system. https://hal.archives-ouvertes.fr/hal-01119210 (2015)
  3. 3.
    Enderling, H., Anderson, A.R., Chaplain, M.A., Munro, A.J., Vaidya, J.S.: Mathematical modelling of radiotherapy strategies for early breast cancer. J. Theor. Biol. 241(1), 158–171 (2006)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Enderling, H., Chaplain, M.A., Anderson, A.R., Vaidya, J.S.: A mathematical model of breast cancer development, local treatment and recurrence. J. Theor. Biol. 246(2), 245–259 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, vol. 7, pp. 713–1020 (2000)Google Scholar
  6. 6.
    Foucher, F., Ibrahim, M., Saad, M.: Numerical analysis of a finite volume scheme for the simulation of a nonlinear degenerate breast cancer model (submitted for publication, 2016)Google Scholar
  7. 7.
    Gallouët, T., Latché, J.C.: Compactness of discrete approximate solutions to parabolic PDEs – application to a turbulence model. Commun. Pure Appl. Anal. 11(6), 2371–2391 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  • Françoise Foucher
    • 1
  • Moustafa Ibrahim
    • 2
  • Mazen Saad
    • 1
  1. 1.Ecole Centrale de Nantes, UMR 6629 CNRSLaboratoire de Mathématiques Jean LerayNantesFrance
  2. 2.Math and Science DivisionAmerican College of the Middle EastDasmanKuwait

Personalised recommendations