Collocation at Gaussian Points

• Published in: SIAM journal on numerical analysis 1973-09, Vol.10 (4), p.582-606
• Main Authors: de Boor, Carl, Swartz, Blair
• Format: Article
• Language: English
• Subjects: Approximation; Degrees of polynomials; Differential equations; Error rates; Interpolation; Linear transformations; Mathematical functions; Newton approximation methods; Polynomials; Projectors; Spline functions; Textual collocation
• ISSN: 0036-1429; 1095-7170
• DOI: 10.1137/0710052

Approximations to an isolated solution of an mth order nonlinear ordinary differential equation with m linear side conditions are determined. These approximations are piecewise polynomial functions of order m + k (degree less than m + k) possessing m - 1 continuous derivatives. They are determined by collocation, i.e., by the requirement that they satisfy the differential equation at k points in each subinterval, together with the m side conditions. If the solution of the sufficiently smooth differential equation problem has m + 2k continuous derivatives and if the collocation points are the zeroes of the kth Legendre polynomial relative to each subinterval, then the global error in these approximations is O(|Δ|m + k) with |Δ| the maximum subinterval length. Moreover, at the ends of each subinterval, the approximation and its first m - 1 derivatives are O(|Δ|2k) accurate. The solution of the nonlinear collocation equations may itself be approximated by solving the sequence of linear collocation problems associated with a Newton iteration; convergence of this process is locally quadratic.