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CONTINUOUS AND DISCRETE LEAST SQUARES METHODS FOR BOUNDARY VALUE PROBLEMS

A priori error estimates in both the L and L norms are derived for continuous and discrete least squares approximates to solutions of two-point boundary value problems from approximating spaces of Hermite splines. Superconvergence at knots is established, and the estimates are shown to be optimal in the sense that no better rate of convergence is possible in the spaces employed. 1. Introduction. In this paper we consider the numerical solution of a general class of two-point boundary value problems by both the method of continuous least squares and the method of discrete least squares using an approximating space of Hermite splines. In particular, we study the problem of approximating the solution of the equation (1.1) Lu =f where L is an nth order linear differential operator in L2(a, b) and f is a given function belonging to L2(a, hi. Let H'(a, b) denote the space of all functions u C-(a, b) such that u '-1 is absolutely continuous on (a,b) and ut'L2(a,b). Given a formal differential operator