Nonlinear Galerkin finite element methods for fourth-order Bi-flux diffusion model with nonlinear reaction term

A fourth-order diffusion model is presented with a nonlinear reaction term to simulate some special chemical and biological phenomenon. To obtain the solutions to those problems, the nonlinear Galerkin finite element method under the framework of the Hermite polynomial function for the spatial domain is utilized. The Euler backward difference method is used to solve the equation in the temporal domain. Subject to the Dirichlet and Navier boundary conditions, the numerical experiments for Bi-flux Fisher–Kolmogorov model present excellent convergence, accuracy and acceleration behavior. Also, the numerical solutions to the Bi-flux Gray–Scott model, subject to no flux boundary conditions, show excellent convergence, accuracy and symmetry.