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W. Auzinger, H. Lehner, E. Weinmüller:
"An efficient asymptotically correct error estimator for collocation solutions to singular index-1 DAEs";
in: "ASC Report 18/2010", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2010, ISBN: 978-3-902627-03-2.

A computationally efficient a posteriori error estimator is
introduced and analyzed for collocation solutions to linear
index-1 differential-algebraic equations with properly stated
leading term exhibiting a singularity of the first kind. The
procedure is based on a modified defect correction principle,
extending an established technique from the context of ordinary differential
equations to the differential-algebraic case.
Using recent convergence results for collocation methods, we prove
that the resulting error estimate is asymptotically correct.
Numerical examples demonstrate the performance of this approach.
To keep the presentation reasonably self-contained, some
arguments from the literature on differential-algebraic equations
concerning the decoupling of the problem and its discretization,
which is essential for our analysis, are also briefly reviewed.
The appendix contains a remark about the interrelation between
collocation and implicit Runge-Kutta methods for
differential-algebraic equations.

Differential algebraic equations ¢ singularity of the first kind ¢ collocation ¢ a posteriori error estimation ¢ defect correction

http://www.asc.tuwien.ac.at/preprint/2010/asc18x2010.pdf