Error analysis for spline collocation methods with application to knot selection

Some collocation schemes used to solve m th order ordinary differential equations are known to display superconvergence at the mesh points. Here we show that some such schemes have additional superconvergence points for the approximate solution and all of its derivatives. Using such points, we argue that a mesh selection scheme introduced by Dodson can be expected to perform well under general circumstances. A numerical example is given to demonstrate the new superconvergence results.