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An efficient asymptotically correct error estimator for collocation solutions to singular index-1 DAEs

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Abstract

A computationally efficient a posteriori error estimator is introduced and analyzed for collocation solutions to linear index-1 DAEs (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 stiffly accurate 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 DAEs 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 the DAE case.

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Authors and Affiliations

  1. Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstrasse 8–10/E101, 1040, Wien, Austria

    Winfried Auzinger, Herbert Lehner & Ewa Weinmüller

Authors
  1. Winfried Auzinger
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  2. Herbert Lehner
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  3. Ewa Weinmüller
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Correspondence to Winfried Auzinger.

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Communicated by Christian Lubich.

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Auzinger, W., Lehner, H. & Weinmüller, E. An efficient asymptotically correct error estimator for collocation solutions to singular index-1 DAEs. Bit Numer Math 51, 43–65 (2011). https://doi.org/10.1007/s10543-011-0321-9

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