Discrete Least Squares Approximations for Ordinary Differential Equations

The application of the least squares method, using Cq piecewise polynomials of order k + m,k m,q m, for obtaining approximations to an isolated solution of a nonlinear mth order ordinary differential equation, involves integrals which in practice need to be discretized. Using for this latter purpose the k-point Gaussian quadrature rule in each subinterval, the discrete least squares schemes obtained are close to collocation, on the same points, by piecewise polynomials from Cm - 1. We prove here that under smoothness assumptions similar to those made by de Boor and Swartz for the collocation procedure, i.e. that the solution be in Cm + 2k, an optimal global rate of convergence O(|Δ |k + m ) is obtained in the uniform norm for the discrete least squares schemes, provided that the partitions Δ are quasiuniform. In addition, a superconvergence rate of O(|Δ |²ᵏ ) is obtained at the knots for those derivatives l which satisfy 0 l 2(m - 1) - q.