This is again a joint effort with Michael Hanke (KTH Stockholm) and ties in with the results we both presented at NUMDIFF-15.

We are looking for an approximate solution \(x_{\pi}\in X_{\pi}\) of the initial- or boundary-value problem \[\begin{aligned} f((Dx)'(t), x(t), t)=0,\;t\in [a,b],\quad g(x(a), x(b))=0.\end{aligned}\] The DAE in it can be of arbitrarily high index. The ansatz-space \(X_{\pi}\) consists of componentwise and piecewise polynomial functions \(x_{\pi}\) on the grid \(\pi: a=t_{0}<t_{1}<\cdots<t_{n}=b\), with continuously connected part \(Dx_{\pi}\). We use polynomials of degree \(N>1\) for the component \(Dx_{\pi}\) but for the nondifferentiated part degree \(N-1\). Introducing \(M\geq N+1\) so-called collocation nodes \(0\leq \tau_{1}<\dots<\tau_{M}\leq 1\) and in turn \(t_{ji}=t_{j-1}+\tau_{i}h_{j}\), we form the overdetermined collocation system \[\begin{aligned} f((Dx_{\pi})'(t_{ji}), x_{\pi}(t_{ji}), t_{ji})=0,\;i=1,\dots M,\; j=1,\dots n,\quad g(x_{\pi}(a), x_{\pi}(b))=0,\end{aligned}\] which is then solved into a special least-squares sense for \(x_{\pi}\). The procedure is inherently simple, the numerical tests are surprisingly good, but the underlying theory is quite demanding. Considering the fact that we are dealing here with an essentially ill-posed problem, it is important to implement it very carefully. Many questions are still open. We describe achievements, difficulties and surprises.