QUADRATIC-SPLINE COLLOCATION METHODS FOR 2-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** minus **j) global estimate for the jth derivative of the error is derived. Further, O(h**4** minus **j) error bounds for the jth derivative are obtained for certain superconvergence points. It should be observed that standard collocation at midpoints gives O(h**2** minus **j) bounds. Results from numerical experiements are reported that verify the theoretical behaviour of the method.