Notice of Retraction Enriched multiscale space in the boundary layers for problems with singular solution: Convergent on the coarse uniform grid

We present an enriched multiscale finite element method (MFEM) for solving the singularly perturbed reaction-diffusion problem, and our multiscale space is constructed by two parts, which are the multiscale basis in the optimal-width boundary layers domain, plus the standard linear basis in the smooth domain. Since the multiscale basis can express the local perturbed information automatically, our method is capable of reducing the boundary layer errors remarkably. And through numerical experiments the method not only acquires good accuracy, but also obtains second order convergence rate for the L 2 norm and first order convergence rate for the energy norm on the coarse uniform grid.