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

Dick, Alexander; Koch, Othmar; März, Roswitha; Weinmüller, Ewa

doi:10.1090/S0025-5718-2012-02637-8

Convergence of collocation schemes for boundary value problems in nonlinear index 1 DAEs with a singular point
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by Alexander Dick, Othmar Koch, Roswitha März and Ewa Weinmüller PDF

Math. Comp. 82 (2013), 893-918 Request permission

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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