We discuss an error estimation procedure for the
global error of collocation schemes applied to solve
singular boundary value problems with a singularity of
the first kind. This a posteriori estimate of the global error was
proposed by Stetter in 1978 and is based on the idea
of Defect Correction, originally due to Zadunaisky.
Here, we present a new, carefully designed modification of this error estimate
which not only results in less computational work but also appears
to perform satisfactorily for singular problems.
We give a full analytical justification for the
asymptotical correctness of the error estimate when it is applied
to a general nonlinear regular problem. For the singular case,
we are presently only able to provide
computational evidence for the full convergence order,
the related analysis is still work in progress.
This global estimate is the basis for a
grid selection routine in which the grid is modified
with the aim to equidistribute the global error. This
procedure yields meshes suitable for an efficient
numerical solution. Most importantly, we observe
that the grid is refined in a way reflecting only the
behavior of the solution and remains unaffected by the
unsmooth direction field close to the singular point.