An adapted Petrov–Galerkin multi-scale finite element method for singularly perturbed reaction–diffusion problems

Abstract

In this paper, we propose the use of an adapted Petrov–Galerkin (PG) multi-scale finite element method for solving the singularly perturbed problem. The multi-scale basis functions that form the function space are constructed from both homogeneous and nonhomogeneous localized problems, which provide more flexibility. These PG multi-scale basis functions are shown to capture the originally perturbed information for the reaction–diffusion model, and reduce the boundary layer errors on graded (non-uniform) coarse meshes. We present the numerical experiment in order to demonstrate that our method acquires stable and convergent results in the $L^{\mathrm{\infty }}$, $L^2$ and energy norms. Due to the independent construction of the multi-scale bases, and the demonstrated accuracy by removing the resonance effect, the adapted PG multi-scale method is shown to be a suitable method for solving the singular perturbation problem.

Keywords:

• Petrov–Galerkin multi-scale finite element
• singular perturbation
• reaction–diffusion
• graded mesh
• elimination of resonance effect

2010 AMS Subject Classifications:

• 35J25
• 65N12
• 65N30

Acknowledgments

We would like to thank Professor Yalchin Efendiev in Texas A&M University for many useful discussions. And we would appreciate the thanks to the anonymous referees for their insightful comments and helpful suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Research is supported by National Natural Science Foundation of China [grant no. 11301462], University Science Research Project of Jiangsu Province [grant no. 13KJB110030] and Yangzhou University Post-doctoral Program and New Century Talent Project to Shan Jiang.