Multiscale Basis Functions for Singular Perturbation on Adaptively Graded Meshes

We apply the multiscale basis functions for the singularly perturbedreaction-diffusion problem on adaptively graded meshes, which can provide a goodbalance between the numerical accuracy and computational cost. The multiscale spaceis built through standard finite element basis functions enriched with multiscale basisfunctions. The multiscale basis functions have abilities to capture originally perturbedinformation in the local problem, as a result, our method is capable of reducing theboundary layer errors remarkably on graded meshes, where the layer-adapted meshesare generated by a given parameter. Through numerical experiments we demonstratethat the multiscale method can acquire second order convergence in the L² norm andfirst order convergence in the energy norm on graded meshes, which is independentof ε. In contrast with the conventional methods, our method is much more accurateand effective.