Linear/linear rational spline collocation for linear boundary value problems

We investigate the collocation method with linear/linear rational spline S of smoothness class C1 for the numerical solution of two-point boundary value problems if the solution y of the boundary value problem is a strictly monotone function. We show that for the linear/linear rational splines on a uniform mesh it holds ‖S−y‖∞=O(h2). Established bound of error for the collocation method gives a dependence on the solution of the boundary value problem and its coefficients. We prove also convergence rates ‖S′−y′‖∞=O(h2), ‖S″−y″‖∞=O(h) and the superconvergence of order h2 for the second derivative of S in certain points. Numerical examples support the obtained theoretical results.