Monotone Combined Finite Volume-Finite Element Scheme for Anisotropic Keller-Segel Model

Abstract

The directed movement of cells and organisms in response to chemical gradients, Chemotaxis, has attracted signicant interest due to its critical role in a wide range of biological phenomena. The Keller-Segel model has provided a cornerstone for mathematical modeling of chemotaxis. We are interested by the numerical analysis of the В degenerate Keller-Sege lmodel. A scheme recently developed in the nite volume framework В treats the discretization of the model in homogeneous domains where the diffusion tensors are considered to be the identity matrix. However, standard nite volume scheme not permit to handle anisotropic diffusion on general,possibly nonconforming meshes. In the other hand, it is well-known that nite element discretization allows a very simple discretization of full diffusion tensors and does not impose any restrictions on the meshes but many numerical instabilities may arise in the convection-dominated case. A quite intuitive idea is hence to combine a nite element discretization of the diffusion term with a nite volume discretization of the other terms. Hence, we construct and study the convergence analysis of a combined scheme discretizing the system. This scheme ensures the validity of the discrete maximum principle under the classical condition that all transmissibilities coefficients are positive. Therefore, a nonlinear technique is presented, in the spirit of well-known methods as a correction to provide a monotone scheme for generaltensors.