We present an a posteriori error estimate for the global error of
collocation methods for two-point boundary value problems in
ordinary differential equations. The estimate was first introduced
and analyzed for regular problems in [1], and is
based on the defect correction principle, see for example
[2]. Our analysis, cf. [3], is
concerned with problems with a singularity of the first kind,
Part I (by Ewa Weinmüller): The presentation deals with
the linear case, where f(t,z(t))=f(t) in (1).
Here, we recapitulate how the superposition principle is used to
prove basic convergence results for collocation applied to linear
singular problems, cf. [4] and
[5].
With these prerequisites, we derive refined bounds for the
collocation solution and its derivative and use these to analyze
the error estimate. We show that the error estimate is
asymptotically correct on the whole collocation grid if the
underlying collocation method is not superconvergent.
Part II (by Othmar Koch): Here, we extend the above results
to the nonlinear case. In particular, we choose a Banach space
setting and use the stability results derived for collocation for
linear problems in order to prove the convergence of the solution
of the nonlinear collocation equations. This solution is shown to exist in a
suitable neighborhood of an isolated solution of (1),
(2). Moreover, Newton's method converges quadratically
for the computation of this solution. Relying on
these results, asymptotical correctness of our error estimate is
shown in the case where the analytical problem is stable
w.r.t. perturbations in the right-hand side of (1).