Analysis of a New Error Estimate for Collocation Methods Applied to Singular Boundary Value Problems

We discuss an a-posteriori error estimate for the numerical solution of boundary value problems for nonlinear systems of ordinary differential equations with a singularity of the first kind. The estimate for the global error of an approximation obtained by collocation with piecewise polynomial functions is based on the defect correction principle. We prove that for collocation methods which are not superconvergent, the error estimate is asymptotically correct. As an essential prerequisite we derive convergence results for collocation methods applied to nonlinear singular problems.