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

• Published in: Computational & applied mathematics Vol. 39; no. 3
• Main Authors: Jiang, Maosheng, Bevilacqua, Luiz, Zhu, Jiang, Yu, Xijun
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.09.2020; Springer Nature B.V
• Subjects: Acceleration; Applications of Mathematics; Applied physics; Boundary conditions; Computational mathematics; Computational Mathematics and Numerical Analysis; Computer simulation; Convergence; Dirichlet problem; Domains; Finite element method; Flux; Galerkin method; Hermite polynomials; Mathematical Applications in Computer Science; Mathematical Applications in the Physical Sciences; Mathematical models; Mathematics; Mathematics and Statistics; Model accuracy; Hermite element; Reaction diffusion; Nonlinear Galerkin method; 65M60; Bi-flux diffusion; Numerical experiments
• ISSN: 2238-3603; 1807-0302
• DOI: 10.1007/s40314-020-01168-w

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.