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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
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)
Cances, C., Ibrahim, M., Saad, M.: Positive nonlinear CVFE scheme for degenerate anisotropic Keller-Segel system. https://hal.archives-ouvertes.fr/hal-01119210 (2015)
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)
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)
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)
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)
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)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Foucher, F., Ibrahim, M., Saad, M. (2017). Convergence of a Nonlinear Control Volume Finite Element Scheme for Simulating Degenerate Breast Cancer Equations. In: Quintela, P., et al. Progress in Industrial Mathematics at ECMI 2016. ECMI 2016. Mathematics in Industry(), vol 26. Springer, Cham. https://doi.org/10.1007/978-3-319-63082-3_33
Download citation
DOI: https://doi.org/10.1007/978-3-319-63082-3_33
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-63081-6
Online ISBN: 978-3-319-63082-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)