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 $$\begin{array}{r}f((Dx{)}^{\prime }(t),x(t),t)=0,t\in [a,b],g(x(a),x(b))=0.\end{array}$$ 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 $D{x}_{\pi }$. We use polynomials of degree $N>1$ for the component $D{x}_{\pi }$ but for the nondifferentiated part degree $N-1$. Introducing $M\ge N+1$ so-called collocation nodes $0\le {\tau }_1<\cdots <{\tau }_M\le 1$ and in turn $t_{ji}=t_{j-1}+{\tau }_i{h}_j$, we form the overdetermined collocation system $$\begin{array}{r}f((D{x}_{\pi }{)}^{\prime }(t_{ji}),x_{\pi }(t_{ji}),t_{ji})=0,i=1,\dots M,j=1,\dots n,g(x_{\pi }(a),x_{\pi }(b))=0,\end{array}$$ 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.