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 ( h 4 ) 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 ( h 4 ) 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 ( h 3‐J ) global estimate for the j th derivative of the error is derived. Further, O ( h 4‐J ) error bounds for the j th derivative are obtained for certain superconvergence points. It should be observed that standard collocation at midpoints gives O ( h 2‐J ) bounds. Results from numerical experiments are reported that verify the theoretical behaviour of the method.