Discrete Least Squares Approximations for Ordinary Differential Equations

Abstract

The application of the least squares method, using $C^q $ piecewise polynomials of order $k + m,k \geqq m,q \geqq m$, 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 $C^{m - 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 $C^{m + 2k} $, an optimal global rate of convergence $O(|\Delta |^{k + m} )$ is obtained in the uniform norm for the discrete least squares schemes, provided that the partitions $\Delta $ are quasiuniform. In addition, a superconvergence rate of $O(|\Delta |^{2k} )$ is obtained at the knots for those derivatives l which satisfy $0 \leqq l \leqq 2(m - 1) - q$.