The directed movement of cells and organisms in response to chemical gradients ''Chemotaxis'' has attracted significant interest due to its critical role in a wide range of biological phenomena. The Keller-Segel model has provided a cornerstone for theoretical and mathematical modelling of chemotaxis. For instance, this model can be used to model the displacement of some bacteria, the bone regeneration mechanism and the breast cancer evolve.
Our aim is to analyze the convergence of a combined finite volume finite element scheme for a degenerate Keller-Segel model with general tensors over general meshes. We focus on the construction of positives schemes to ensure the discrete maximum principle and therefore the confinement of the density of cells and the positivity of the chemical concentration.
For that, firstly, a nonlinear monotone correction is introduced for a combined scheme which defined as a compromise between the nonconforming finite elements, and between the finite volumes enabling to avoid spurious oscillations in the convection dominated regime.
Next, we consider a nonlinear Control Volume Finite Element (CVFE) scheme, whose construction is based on the Godunov scheme to approximate the degenerate diffusion fluxes and on a nonclassical upwind finite
volume scheme. This scheme ensures the discrete maximum principle whatever the anisotropy of the problem.
Finally, we carry out the convergence analysis of an efficient monotone DDFV method for approximating solutions of degenerate parabolic equations and some numerical tests are then presented.