A Gradient Discretisation Method for Anisotropic Reaction–Diffusion Models with Applications to the Dynamics of Brain Tumors

Abstract

Abstract A gradient discretisation method (GDM) is an abstract setting that designs the unified convergence analysis of several numerical methods for partial differential equations and their corresponding models. In this paper, we study the GDM for anisotropic reaction–diffusion problems, based on a general reaction term, with Neumann boundary condition. With natural regularity assumptions on the exact solution, the framework enables us to provide proof of the existence of weak solutions for the problem, and to obtain a uniform-in-time convergence for the discrete solution and a strong convergence for its discrete gradient. It also allows us to apply non-conforming numerical schemes to the model on a generic grid (the non-conforming ℙ ⁢ 1 {\mathbb{P}1} finite element scheme and the hybrid mixed mimetic (HMM) methods). Numerical experiments using the HMM method are performed to assess the accuracy of the proposed scheme and to study the growth of glioma tumors in heterogeneous brain environment. The dynamics of their highly diffusive nature is also measured using the fraction anisotropic measure. The validity of the HMM is examined further using four different mesh types. The results indicate that the dynamics of the brain tumor is still captured by the HMM scheme, even in the event of a highly heterogeneous anisotropic case performed on the mesh with extreme distortions.