Convergence of Collocation Schemes for Nonlinear Index 1 DAEs with a Singular Point

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 well-posed boundary value problem in DAEs having a sufficiently smooth solution, the global error of the collocation scheme converges with the so-called stage order. Due to the singularity, superconvergence does not hold in general. The theoretical results are supported by numerical experiments.