equation, while the second gives many examples of applications of these
concepts to a large number of other, more complicated equations. For
applications to individual types of equations, see also the publications by
-Becker @ref step_14_Bec95 "[Bec95]", @ref step_14_Bec98 "[Bec98]",
+Becker @ref step_14_Bec95 "[Bec95]", @ref step_14_Bec98 "[Bec98]",
Kanschat @ref step_14_Kan96 "[Kan96]", @ref step_14_FK97 "[FK97]",
Suttmeier @ref step_14_Sut96 "[Sut96]", @ref step_14_RS97 "[RS97]", @ref step_14_RS98c "[RS98c]",
-@ref step_14_RS99 "[RS99]",
-Bangerth @ref step_14_BR99b "[BR99b]", @ref step_14_Ban00w "[Ban00w]",
-@ref step_14_BR01a "[BR01a]", @ref step_14_Ban02 "[Ban02]", and
-Hartmann @ref step_14_Har02 "[Har02]", @ref step_14_HH01 "[HH01]",
+@ref step_14_RS99 "[RS99]",
+Bangerth @ref step_14_BR99b "[BR99b]", @ref step_14_Ban00w "[Ban00w]",
+@ref step_14_BR01a "[BR01a]", @ref step_14_Ban02 "[Ban02]", and
+Hartmann @ref step_14_Har02 "[Har02]", @ref step_14_HH01 "[HH01]",
@ref step_14_HH01b "[HH01b]".
All of these works, from the original introduction by Becker and Rannacher to
individual contributions to particular equations, have later been summarized
-in a book by Bangerth and Rannacher that covers all of these topics, see
+in a book by Bangerth and Rannacher that covers all of these topics, see
@ref step_14_BR03 "[BR03]".
zero, since $z$ has zero boundary values, and $\varphi_h$ can be chosen to
have the same.
-Thus, we have
+Thus, we have
@f{eqnarray*}
J(e)
&=&
&=&
\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}.
+ z-\varphi_h)_{\partial K\backslash \partial\Omega}.
@f}
Using that for the normal vectors $n'=-n$ holds, we define the jump of the
normal derivative by
&=&
\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}.
+ z-I_h z)_{\partial K\backslash \partial\Omega}.
@f}
With this, we have obtained an exact representation of the error of the finite
<dd>Roland Becker and Rolf Rannacher.
<br> Weighted a posteriori error control in FE methods.
<br> In H. G. Bock et al., ed.s, <em>ENUMATH 95</em>, pages 621–637,
- Paris, September 1998. World Scientific Publ., Singapure.
+ Paris, September 1998. World Scientific Publ., Singapore.
<br> in @ref step_14_enumath97 "[enumath97]".