Uniformly Superconvergent Approximations for Linear Two-Point Boundary Value Problems

A local collocation scheme for solutions of first-order systems of linear two-point boundary value problems is developed that preserves uniformly the superconvergent accuracy at meshpoints achieved by collocation at Gauss points over piecewise polynomials. The method uses the superconvergent meshpoint values from the first collocation along with a second collocation that can be done independently on each piece. A complete mathematical analysis is provided for the special cases when the points for the second collocation are equally spaced, either open or closed. It is shown that the choice of Gauss points for this second collocation is always undesirable.