We present the enriched multiscale finite element method on Bakhvalov grid for solving the singularly perturbed problem effectively, and this model is characterized with the scale oscillations and boundary layers. To capture the large scale components of solution without resolving all the small scale features, the method is accomplished to obtain the high-quality basis functions from the differential operator. These multiscale basis functions are used to capture the multiscale oscillations, and the standard basis functions on Bakhvalov grid are used to resolve the boundary layers, as a consequence, our enriched multiscale method may acquire a nice simulation. Through the numerical experiment, we achieve the accurate and stable results, and remove the resonance effect and boundary layers by the multiscale on Bakhvalov grid, which provides an excellent balance between the numerical accuracy and computational cost in the material applications.

multiscale finite element, enriched multiscale space, Bakhvalov grid, scale oscillation, boundary layer.