Fourth order accuracy collocation method for singularly perturbed boundary value problems

It is well known that, for nonrigid boundary value problems, the collocation method basing on the use of cubic splines gives schemata up to fourth order precision with respect to grid size $h$. We show that collocation schemata of fourth order, converging uniformly with respect to a small parameter, can be constructed for singularly perturbed problems as well. For their construction, we use N. S. Bakhvalov optimal grids and cubic splines of mixed defect.