  We propose a new two-grid approach based on Bellman-Kalaba quasilinearization
and Axelsson-Xu finite element two-grid method for the solution of singularly
perturbed reaction-diffusion equations. The algorithms involve solving one
inexpensive problem on coarse grid and solving on fine grid one linear problem
obtained by quasilinearization of the differential equation about an
interpolant of the computed solution on the coarse grid. Different meshes (of
Bakhvalov, Shishkin and Vulanovi\'c types) are examined. All the schemes are
uniformly convergent with respect to the small parameter. We show theoretically
and numerically that the global error of the two-grid method is the same as of
the nonlinear problem solved directly on the fine layer-adapted mesh.
