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Least-squares collocation for higher-index linear differential-algebraic equations: Estimating the instability threshold


Authors: Michael Hanke, Roswitha März and Caren Tischendorf
Journal: Math. Comp. 88 (2019), 1647-1683
MSC (2010): Primary 65L80; Secondary 65L08, 65L10
DOI: https://doi.org/10.1090/mcom/3393
Published electronically: November 5, 2018
MathSciNet review: 3925480
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Abstract: Differential-algebraic equations with higher-index give rise to essentially ill-posed problems. The overdetermined least-squares collocation for differential-algebraic equations which has been proposed recently is not much more computationally expensive than standard collocation methods for ordinary differential equations. This approach has displayed impressive convergence properties in numerical experiments, however, theoretically, till now convergence could be established merely for regular linear differential-algebraic equations with constant coefficients. We present now an estimate of the instability threshold which serves as the basic key for proving convergence for general regular linear differential-algebraic equations.


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Additional Information

Michael Hanke
Affiliation: Department of Mathematics, School of Engineering Sciences, KTH Royal Institute of Technology, S-100 44 Stockholm, Sweden
Email: hanke@nada.kth.se

Roswitha März
Affiliation: Institute of Mathematics, Humboldt University of Berlin, D-10099 Berlin, Germany
Email: maerz@math.hu-berlin.de

Caren Tischendorf
Affiliation: Institute of Mathematics, Humboldt University of Berlin, D-10099 Berlin, Germany
Email: caren@math.hu-berlin.de

DOI: https://doi.org/10.1090/mcom/3393
Keywords: Differential-algebraic equation, higher-index, essentially ill-posed problem, collocation, boundary value problem, initial value problem
Received by editor(s): October 9, 2017
Received by editor(s) in revised form: March 16, 2018, and May 18, 2018
Published electronically: November 5, 2018
Additional Notes: The work of the third author was supported by the German Research Foundation (DFG CRC/Transregio 154, Subproject C02).
Article copyright: © Copyright 2018 American Mathematical Society