This paper is concerned with the numerical solution of singularly perturbed elliptic two point boundary value problems. The authors propose a Galerkin method with Hermite splines where the knots have been adapted to the boundary layer behaviour of the solution. A sufficient condition on the mesh that ensures that the approximate solution has optimal order of convergence in the energy norm with respect to the perturbation parameter is given.
These optimal meshes are constructed with the aim to have an equal distribution of the errors in the subintervals and improve the orders of convergence of N. S. Bakhvalov [Zh. Vychisl. Mat. Mat. Fiz. 9, 841–859 (1969; Zbl 0208.19103)] and G. I. Shishkin [Sov. J. Numer. Anal. Math. Model. 3, 393–407 (1988; Zbl 0825.65062)].
Further an approach for the effective construction of these optimal meshes is given. Finally, the paper includes the results of some numerical experiments to test the orders of the theoretical estimates.