where B0, B1 and β are suitably chosen to yield a
Collocation methods of variable order p are used to obtain
the basic solution and different estimators of the global error
are investigated. An estimator due to Zadunaisky works
satisfactorily as in
the classical case - for collocation schemes where no superconvergence
effects are present - if only mesh points are used and collocation points
neglected. However, this means that most of the computed information
is discarded and that the step-sizes for the implicit Euler
method which is involved in the error estimation are undesirably
large. However, the same estimation procedure is not asymptotically
correct if the collocation points are included in the process.
Therefore, we propose a modification of the classical Zadunaisky
method which makes it possible to obtain an asymptotically
correct estimate in the collocation and mesh points.
The information about the global error obtained thus is
subsequently used for mesh selection. Our method turns out
to yield meshes that are well suited to equidistribute the global
error on the interval [0,1] and is not negatively affected
by the singularity. This is an improvement upon classical
mesh selection strategies which usually yield unnecessarily
fine meshes near the singularity. The theory is supported
by numerous numerical examples.