Convergence of a positive nonlinear Control Volume Finite Element scheme for solving an anisotropic degenerate breast cancer development model

Abstract

In this paper, a nonlinear control volume finite element (CVFE) scheme for solving an anisotropic degenerate breast cancer development model is introduced and analyzed. This model includes both ordinary differential equations and convection–diffusion–reaction equations modeling the stepwise mutations from a normal breast stem cell to a tumor cell. The diffusion term, which generally involves an anisotropic and heterogeneous diffusion tensor, is discretized on a dual mesh by means of the piecewise linear conforming finite element method and using the Godunov scheme to approximate the diffusion fluxes provided by the conforming finite element reconstruction. The other terms are discretized using a nonclassical upwind finite volume scheme on the dual mesh, where the dual volumes are constructed around the vertices of the original mesh. This technique ensures the positivity and boundedness of discrete solutions without any restriction on the diffusion tensor nor the transmissibility coefficients. The convergence of the scheme is proved, only supposing the shape regularity condition for the original mesh and using a priori estimates as well as the Kolmogorov relative compactness theorem. The proposed scheme is robust, locally conservative, efficient, and stable, which is confirmed by numerical experiments over a general mesh.