An a posteriori estimate for the global error of collocation
methods for two-point boundary value problems in ordinary
differential equations is discussed.
The estimate was first introduced
and analyzed for regular problems in [1], and is
based on the defect correction principle. Our analysis is
concerned with problems with a singularity of the first kind,
In [2], this error estimate was discussed for a restricted class
of singular boundary value problems, where the coefficient matrix
M(0) has only eigenvalues with nonpositive real parts. In that
case, a shooting argument can be used to derive bounds for the
error of the collocation solution (see also [3])
which enable an analysis of the
error estimate. Here, we extend the results to the most general
case, where eigenvalues of M(0) with positive real parts are
permitted. To analyze the collocation solution and error estimate
in this case, we derive a new representation of the global error.
To accurately describe the structure of the terms in this representation,
we have to derive new results for analytical, piecewise defined
singular boundary value problems. Moreover, we have to extend the stability
analysis for collocation methods (see [2])
to the case where eigenvalues of M(0)
with positive real parts are present. To this end, we adapt some
results from [4]
for our purpose. Using the results for the collocation solution,
we can analyze the error estimate similarly as in the regular case
([1]).
We conclude that the error estimate is asymptotically correct when applied
to our problem class. Consequently, collocation methods together with
our a posteriori error estimation procedure can successfully be used
in our Matlab code sbvp, which was designed especially
for the efficient solution of singular boundary value problems, see
[5].