  We introduce a direct numerical treatment of nonlinear higher-index
differential-algebraic equations by means of overdetermined polynomial
least-squares collocation. The procedure is not much more computationally
expensive than standard collocation methods for regular ordinary differential
equations. The numerical experiments show impressive results. In contrast, the
theoretical basic concept turns out to be considerably challenging. So far,
quite recently convergence proofs for linear problems have been published. In
the present paper we come up to a first convergence result for nonlinear
problems.
