Fourth Order Accuracy for a Cubic Spline Collocation Method

This note is inspired by the paper of Bialecki, Fairweather, and Bennett [1] which studies a method for the numerical solution of u00 = f on an interval, and u = f on a rectangle, with zero boundary data; their scheme calculates that C piecewise cubic U for which U 00 (or U) agrees with f at the two (or four) Gauss quadrature points of each subdivision of the original domain. They observe that not only does their U approximate u to order h at mesh nodes { h is the linear dimension of the subdivisions { but also U 0 agrees with u0 to order h in one dimension, and Ux, Uy, and Uxy agree to order h with ux, uy, and uxy in two dimensions. Agreement of U with u to fourth order is perhaps not so surprising { Gaussian quadrature of this order is, after all, fourth order accurate and one could well expect this order of accuracy to be re ected in a Gauss-inspired di erential equation solver. But fourth order agreement also for the derivatives is a surprise, and is due to the particular nature of the collocation scheme. Indeed, U agrees with u to fourth order uniformly in , but away from the nodal points their derivatives need not be so close. It is in this sense that the derivatives of U exhibit superconvergence at the nodes.