Smooth spline collocation methods for solving Boundary Value Problems (BVPs) have some advantages, in that they combine the simplicity of the collocation methodology, with the high efficiency of solving the smallest possible linear system for the chosen degree of piecewise polynomials. However, the standard formulation of smooth spline collocation methods leads to suboptimal convergence approximations. Some optimal smooth spline collocation methods have been developed for certain classes of BVPs. All these methods, though, have been developed for uniform partitions. In this paper, we extend the optimal Quadratic and Cubic Spline Collocation (QSC and CSC) methods for linear second-order two-point BVPs 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. Optimal global and local orders of convergence of the QSC and CSC approximations and derivatives are derived, similar to those of the respective methods for uniform partitions.

We then explain how to get rid of the mapping function in the CSC case and turn the optimal CSC method into a mapping-free method. The mapping-free optimal CSC methods are integrated with adaptive grid techniques, and grid size and error estimators. These techniques are quite similar to those of COLSYS, with a few differences, which we elaborate. We also discuss the differences between the optimal non-uniform CSC methods and the respective QSC methods, as far as the iterative procedures that construct the adaptive grid are concerned. We compare the performance of the new CSC methods with that of Hermite cubic piecewise polynomial collocation methods as implemented in COLSYS, on a variety of problems, including problems with boundary or interior layers, and singular perturbation problems. For most problems, the CSC methods require less computational effort for the same error tolerance, and have equally (if not more) reliable error estimators.