Quadratic and Cubic Spline Collocation (QSC and CSC) methods of optimal orders of convergence have been developed for the solution of linear second-order two-point Boundary Value Problems discretized on uniform partitions. In this paper, we extend the optimal QSC and CSC methods to non-uniform partitions. To do this, we use a mapping function between uniform and non-uniform partitions and develop expansions of the error at the non-uniform collocation points of some appropriately defined spline interpolants. We use the Green's function approach to analyze the proposed methods. The existence and uniqueness of the QSC and CSC approximations are shown, under some conditions. Optimal global and local orders of convergence of the spline approximations and derivatives are derived, similar to those of the respective methods for uniform partitions. The jth derivative of the QSC approximation, for j > 0, is O (h3-j) globally and O (h4-j) locally on certain points. The jth derivative of the CSC approximation, is O (h4-j) globally, for j > 0, and O (h5-j) locally on certain points, for j > 0. The non-uniform partition QSC and CSC methods are integrated with adaptive grid techniques, and applied to the solution of a variety of problems, including problems with interior or boundary layers. The numerical results verify the theoretically expected behaviour of the methods.

Joint work with Christina C. Christara, ccc@cs.utoronto.ca