O. Koch, R. März, D. Praetorius, E. Weinmüller:

"Collocation methods for index 1 DAEs with a singularity of the first kind";

in: "ASC Report 10/2008", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2008, ISBN: 978-3-902627-01-8.

We study the convergence behavior of collocation schemes applied to

approximate solutions of BVPs in linear index 1 DAEs which exhibit a

critical point at the left boundary. Such a critical point of the DAE

causes a singularity within the inherent ODE system. We focus our

attention on the case when the inherent ODE system is singular with

singularity of the first kind, apply polynomial collocation to the

original DAE system and consider different choices of the collocation

points such as equidistant, Gaussian or Radau points. We show that

for a well-posed boundary value problem for DAEs having a sufficiently

smooth solution the global error of the collocation scheme converges

with the so-called stage order, or equivalently, it is O(h^m), where

m is the number of collocation points. Superconvergence cannot be

expected in general due to the singularity, not even for the

differential components of the solution. The theoretical results are

illustrated by numerical experiments.

http://www.asc.tuwien.ac.at/preprint/2008/asc10x2008.pdf

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