A 4TH-ORDER CUBIC SPLINE METHOD FOR LINEAR 2ND-ORDER 2-POINT BOUNDARY-VALUE-PROBLEMS

A cubic spline method for linear second-order two-point boundary-value problems is analysed. The method is a Petrov-Galerkin method using a cubic spline trial space, a piecewise-linear test space, and a simple quadrature rule for the integration, and may be considered a discrete version of the H-1-Galerkin method. The method is fully discrete, allows an arbitrary mesh, yields a linear system with bandwidth five, and under suitable conditions is shown to have an o(h4-i) rate of convergence in the W(p)i norm for i = 0, 1, 2, 1 less-than-or-equal-to p less-than-or-equal-to infinity. The H-1-Galerkin method and orthogonal spline collocation with Hermite cubics are also discussed.