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Convergence of collocation schemes for boundary value problems in nonlinear index 1 DAEs with a singular point


Authors: Alexander Dick, Othmar Koch, Roswitha März and Ewa Weinmüller
Journal: Math. Comp. 82 (2013), 893-918
MSC (2010): Primary 65L80; Secondary 65L70
DOI: https://doi.org/10.1090/S0025-5718-2012-02637-8
Published electronically: September 10, 2012
MathSciNet review: 3008842
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Abstract: We analyze the convergence behavior of collocation schemes applied to approximate solutions of BVPs in nonlinear index 1 DAEs, which exhibit a critical point at the left boundary. Such a critical point of the DAE causes a singularity in the inherent nonlinear ODE system. In particular, we focus on the case when the inherent ODE system is singular with a singularity of the first kind and apply polynomial collocation to the original DAE system. We show that for a certain class of well-posed boundary value problems in DAEs having a sufficiently smooth solution, the global error of the collocation scheme converges uniformly with the so-called stage order. Due to the singularity, superconvergence at the mesh points does not hold in general. The theoretical results are supported by numerical experiments.


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Additional Information

Alexander Dick
Affiliation: Vienna University of Technology, Institute for Analysis and Scientific Computing, Wiedner Hauptstraße 8–10, A-1040 Wien
Email: alex.dick@aon.at

Othmar Koch
Affiliation: Vienna University of Technology, Institute for Analysis and Scientific Computing, Wiedner Hauptstraße 8–10, A-1040 Wien
Email: othmar@othmar-koch.org

Roswitha März
Affiliation: Humboldt-Universität of Berlin, Institute for Mathematics, Unter den Linden 6, D-10099 Berlin, Germany
Email: maerz@mathematik.hu-berlin.de

Ewa Weinmüller
Affiliation: Vienna University of Technology, Institute for Analysis and Scientific Computing, Wiedner Hauptstraße 8–10, A-1040 Wien
Email: e.weinmueller@tuwien.ac.at

DOI: https://doi.org/10.1090/S0025-5718-2012-02637-8
Received by editor(s): April 10, 2011
Received by editor(s) in revised form: September 27, 2011
Published electronically: September 10, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society