<a name="Results"></a>
<h1>Results</h1>
-<p>
+
If run, the program generates output like this:
-</p>
-<code><pre>
+
+@code
Realization 0:
Energy: 0.00377302
Energy: 0.00106138
Energy: 0.00723772
Energy: 0.00690564
Energy: 0.00690562
-</pre></code>
+@endcode
+
-<p>
The lowest energy yet seen is in this run (you only get this by increasing the
number of runs):
-</p>
-<code><pre>
+
+@code
Realization 18:
Energy: 0.00200645
Energy: 0.000638519
Energy: 3.18104e-05
Energy: 2.97091e-05
Energy: 3.5011e-05
-</pre></code>
+@endcode
+
-<p>
Apparently something went wrong in the last step (the energy increased, which
it shouldn't - but then this is a strongly nonlinear problem), which is also
why the program aborted after this iteration. Apart from this, the iterations
shown above demonstrate that our program indeed is able to reduce the energy
in the solution in each iteration, as should be.
-</p>
-<p>
+
+
Since the program did not really deliver the goal we had originally intended
for (the computation of the minimal energy of finite element spaces), the
graphical output is also not very exciting. Here are plots of five of the
first 10 solutions (clicking on a picture gives the unscaled version of the
image):
-</p>
-<p align="center">
- <a href="step-15.data/solutions-1.png" target="_top"><img
- src="step-15.data/solutions-1.png" alt="ux" width="60%"></a>
-</p>
-<p>
+@image html step-15.solutions-1.png
+
+
+
And here are the first 100 solutions, where each node in each solution is
represented as a dot. As can be seen, all the solutions cluster somewhat
-around the <i>x<sup>3</sup></i> curve, here shown in turquoise:
-</p>
+around the $x^{1/3}$ curve, here shown in turquoise:
+
+
+@image html step-15.solutions-2.png
+
-<p align="center">
- <a href="step-15.data/solutions-2.png" target="_top"><img
- src="step-15.data/solutions-2.png" alt="ux" width="60%"></a>
-</p>
-<p>
Note that this behavior is mostly independent of the choice of starting data
(which we have chosen to be close to this curve), which a posteriori justfies
our choice. Some of the curves actually show a linear behavior of the solution
close to the origin; this is particularly obvious when the curves are viewed
in a log-log plot (not shown here, but rather left as an exercise to the
reader).
-</p>
-<p>
+
+
Given the almost complete absence of interesting results of this program, we
hope that at least its source code provided some information with respect to
programming with deal.II
-</p>
+