We propose the multiscale finite element method for the singularly perturbed reaction-diffusion problem, and it can recover the boundary layer errors on the graded meshes, which are generated layer-adapted meshes defined by parameters. The multiscale space is constructed by the standard basis functions enriched with the multiscale basis ones, and the latter have abilities to express the local perturbed information naturally. It is illustrated in the numerical experiment that our multiscale method on the graded meshes acquires the second order convergence of  norm error, and it is more accurate numerical method than the adaptive finite element method.

multiscale finite element method, singular perturbation, reaction-diffusion model, boundary layers, graded meshes.