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M. Hanke, R. März, C. Tischendorf, E. Weinmüller, S. Wurm:
"Least-Squares Collocation for Higher Index Differential-Algebraic Equations";
in: "ASC Report 10/2016", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2016, ISBN: 978-3-902627-09-4, 1 - 36.

Differential-algebraic equations with higher index give rise to essentially ill-posed problems. Therefore, their numerical approximation requires special care. In the present paper, we state the notion of ill-posedness for differential-algebraic equations more precisely. Based on this property, we construct a regularization procedure us-ing a least-squares collocation approach by discretizing the pre-image space. Numer-ical experiments show that the resulting method has excellent convergence properties and is not much more computationally expensive than standard collocation methods used in the numerical solution of ordinary differential equations or index-1 differential-algebraic equations. Convergence is shown for a limited class of higher index differential-algebraic equations.

differential-algebraic equation, higher index, essentially ill-posed problem, collocation, boundary value problem

http://www.asc.tuwien.ac.at/preprint/2016/asc10x2016.pdf