The Evolution of Finite Element Approaches in Reaction-Diffusion Modeling

This paper presents a comprehensive review of finite element methods (FEMs) applied to a diverse range of reaction-diffusion equations (RDEs). Beginning with a historical overview of FEMs, we then provide a summary of various FEMs, including standard Galerkin (both conforming and non-conforming), mixed Galerkin, discontinuous Galerkin, and weak Galerkin. Additionally, a priori and a posteriori error have been discussed for standard Galerkin. In further discussion related to RDEs, we provide insights into the evolution of these equations and their significance in various fields. We then systematically review these FEMs for solving different types of RDEs, including more recent advances pertaining to RDEs with nonlinear reaction terms, and advection reaction-diffusion equations. Finally, we briefly highlight the applications of machine learning and deep neural networks to FEM.