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 , 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.