Quadratic‐spline collocation methods for two‐point boundary value problems

A new collocation method based on quadratic splines is presented for second order two-point boundary value problems. First, O(h4) approximations to the first and second derivative of a function are derived using a quadratic-spline interpolant of u. Then these approximations are used to define an O(h4) perturbation of the given boundary value problem. Second, the perturbed problem is used to define a collocation approximation at interval midpoints for which an optimal O(h3-J) global estimate for the jth derivative of the error is derived. Further, O(h4-J) error bounds for the jth derivative are obtained for certain superconvergence points. It should be observed that standard collocation at midpoints gives O(h2-J) bounds. Results from numerical experiments are reported that verify the theoretical behaviour of the method.