\section{The maths}
The Heidelberg group of Professor Rolf Rannacher, to which the three main
-authors of the deal.II library belonged for the PhD time and partly also
+authors of the deal.II library belonged during their PhD time and partly also
afterwards, has been involved with adaptivity and error estimation for finite
element discretizations since the mid-90ies. The main achievement is the
development of error estimates for arbitrary functionals of the solution, and
\begin{align*}
J(e)
&=
- \sum_K (f+u_h), z-\varphi_h)_K
+ \sum_K (f+\Delta u_h, z-\varphi_h)_K
- (\partial_n u_h, z-\varphi_h)_{\partial K\backslash \partial\Omega}.
\end{align*}
In a final step, note that when taking the normal derivative of $u_h$, we mean
\begin{align*}
J(e)
&=
- \sum_K (f+u_h), z-\varphi_h)_K
+ \sum_K (f+\Delta u_h, z-\varphi_h)_K
- \frac 12 (\partial_n u_h|_K + \partial_{n'} u_h|_{K'},
z-\varphi_h)_{\partial K\backslash \partial\Omega}.
\end{align*}
\begin{align*}
J(e)
&=
- \sum_K (f+u_h), z-I_h z)_K
+ \sum_K (f+\Delta u_h, z-I_h z)_K
- \frac 12 ([\partial_n u_h],
z-I_h z)_{\partial K\backslash \partial\Omega}.
\end{align*}