On the efficacy of a control volume finite element method for the capture of patterns for a volume-filling chemotaxis model

In this paper, a control volume finite element scheme   for the capture of spatial patterns for a volume-filling chemotaxis model is proposed and analyzed. The diffusion term, which generally involves an anisotropic and heterogeneous diffusion tensor, is discretized by piecewise linear conforming triangular finite elements (P1-FEM). The other terms are discretized by means of an upstream finite volume scheme on a dual mesh, where the dual volumes are constructed around the vertices of each element of the original mesh. The scheme ensures the validity of the discrete maximum principle under the assumption that the transmissibility coefficients are nonnegative. The convergence analysis is based on the establishment of a priori estimates on the cell density, these estimates lead to some compactness arguments in L2L2 based on the use of the Kolmogorov compactness theorem. Finally, we show some numerical results to illustrate the effectiveness of the scheme to capture the pattern formation for the mathematical model.