In this paper, we discuss the asymptotic properties and efficiency
of several a posteriori estimates for the global error of
collocation methods. Proofs of the asymptotic correctness are given for
regular problems and for problems with a singularity of the first kind.
Our main focus, however, is on the applicability and
performance of the estimates when applied to boundary value
problems in ordinary differential equations with an essential
singularity. Particularly, we compare estimates based on the
defect correction principle with a strategy based on mesh halving.