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 ,
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.
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.