Monotone combined edge finite volume–finite element scheme for Anisotropic Keller–Segel model

Abstract

In this article, a new numerical scheme for a degenerate Keller–Segel model with heterogeneous anisotropic tensors is treated. It is well‐known that standard finite volume scheme not permit to handle anisotropic diffusion without any restrictions on meshes. Therefore, a combined finite volume‐nonconforming finite element scheme is introduced, developed, and studied. The unknowns of this scheme are the values at the center of cell edges. Convergence of the approximate solution to the continuous solution is proved only supposing the shape regularity condition for the primal mesh. 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, as a correction of the diffusive flux, to provide a monotone scheme for general tensors. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1030–1065, 2014

Positive nonlinear CVFE scheme for degenerate anisotropic Keller-Segel system

“…However, there is no sufficient conditions for nonnegativity of transmissibility coefficients and therefore the schemes do not permit to tackle general anisotropic diffusion problems. Nevertheless, in [12] the authors propose a combined nonconforming finite elements finite volumes scheme for which they add a monotone regularization permitting positiveness of discrete solution; the convergence of the scheme, introduced in [11], is ensured under a numerical condition depending on the mesh size and on the discrete solutions.…”

Preserving monotony of combined edge finite volume–finite element scheme for a bone healing model on general mesh

“…It is clear that the assumption (19) is verified by this particular example. Concerning condition (20), it can be verified numerically as detailed in [7] and [8].…”

“…Since m → V (m) is a nonnegative decreasing function for m ∈ [0, 1], using assumption (6) and following the same computations as in [8,Lemma 4.2], we get…”

“…This method was then extended in [14] to inhomogeneous and anisotropic diffusion-dispersion tensors and to very general meshes only satisfying the shape regularity condition (6). Such discretizations were then applied to different physical models, such as the anisotropic Keller-Segel chemotaxis system [8] or the two compressible phase flow in porous media [17]. However, it is well-known that the discrete maximum principle is no more guaranteed if there exist negative transmissibilities.…”

A corrected decoupled scheme for chemotaxis models

“…We require also that the scheme converges without needing to fulfill any CFL condition, which is not the case of the fully explicit schemes. In the literature, a number of decoupled methods for the Keller-Segel model and its variants have been proposed (see, e.g., [13,12,14,4,17,16,2,3]). In all these works, the time discretization is based on the classical backward Euler scheme with an explicit approximation of some terms to avoid coupling of the system.…”