Hybrid mixed discontinuous Galerkin finite element analysis of time-dependent convection-diffusion equations and its application in chemotaxis model

A novel class of hybrid mixed discontinuous Galerkin finite element methods is proposed for solving time-dependent convection-diffusion problems. By introducing Lagrange multipliers at the element interfaces, we establish new hybrid mixed finite element procedures specifically for reaction-diffusion equations, parabolic equations, and time-dependent convection-diffusion equations. Theoretical analysis focuses on the stability and local mass conservation of these methods, with corresponding error estimates derived. Numerical examples are provided to validate the theoretical findings. Additionally, the application of these methods to chemotaxis models is explored. Specifically, a hybrid mixed discontinuous Galerkin method is developed for parabolic-parabolic type chemotaxis models, and numerical experiments are conducted to observe the blow-up phenomenon within a finite time frame.