Approximation of singularly perturbed reaction‐diffusion problems by quadratic C 1‐splines

Collocation with quadratic C ¹‐splines for a singularly perturbed reaction‐diffusion problem in one dimension is studied. A modified Shishkin mesh is used to resolve the layers. The resulting method is shown to be almost second order accurate in the maximum norm, uniformly in the perturbation parameter. Furthermore, a posteriori error bounds are derived for the collocation method on arbitrary meshes. These bounds are used to drive an adaptive mesh moving algorithm. Numerical results are presented.