O N H - P Collocation

We derive a simple accuracy matching strategy for inexact Gauss Newton methods and apply it to the numerical solution of boundary value problems of ordinary differential equations by collocation. The matching strategy is based on an affine contravariant convergence theorem, i.e., the characteristic constants are invariant under affine transformations of the domain. The inexact Gauss Newton method is applied to an integral formulation of the BVP. As discretization for the arising linear subproblems we employ adaptive collocation at Gaussian nodes with varying local orders and stepsizes. The grids are chosen via adaptive efinement and order selection. Introduction. The standard approach to the solution of nonlinear boundary value problems is to first discretize them, e.g. using collocation on a suitable grid, and then to solve the arising finite dimensional nonlinear equation by Newton' method up to the discetization error. This procedur may be iterated using moe and more suitable discretizations until the requied tolerance is achieved. Due to the mesh independence principle, only very few Newton iterations ae necessary on the fine grids, which makes these methods very efficient. This is clearly demonstrated e.g. by the code COL-SYs/CoLNEW by Ascher, Christiansen, Russell [CR79] and Bader [Bad88] as the most prominent example.