<a name="Results"></a>
<h1>Results</h1>
+
+<p>
+The output of this program consist of the console output, the eps
+files including the grids, and the solutions given in gnuplot format.
+<code>
+<pre>
+Cycle 0:
+ Number of active cells: 64
+ Number of degrees of freedom: 256
+Time of assemble_system1: 0.26
+Time of assemble_system2: 0.22
+solution1 and solution2 do not differ.
+Writing grid to <grid-0.eps>...
+Writing solution to <sol-0.gnuplot>...
+Cycle 1:
+ Number of active cells: 112
+ Number of degrees of freedom: 448
+Time of assemble_system1: 0.47
+Time of assemble_system2: 0.39
+solution1 and solution2 do not differ.
+Writing grid to <grid-1.eps>...
+Writing solution to <sol-1.gnuplot>...
+Cycle 2:
+ Number of active cells: 214
+ Number of degrees of freedom: 856
+Time of assemble_system1: 0.89
+Time of assemble_system2: 0.73
+solution1 and solution2 do not differ.
+Writing grid to <grid-2.eps>...
+Writing solution to <sol-2.gnuplot>...
+Cycle 3:
+ Number of active cells: 415
+ Number of degrees of freedom: 1660
+Time of assemble_system1: 1.78
+Time of assemble_system2: 1.43
+solution1 and solution2 do not differ.
+Writing grid to <grid-3.eps>...
+Writing solution to <sol-3.gnuplot>...
+Cycle 4:
+ Number of active cells: 796
+ Number of degrees of freedom: 3184
+Time of assemble_system1: 3.29
+Time of assemble_system2: 2.76
+solution1 and solution2 do not differ.
+Writing grid to <grid-4.eps>...
+Writing solution to <sol-4.gnuplot>...
+Cycle 5:
+ Number of active cells: 1561
+ Number of degrees of freedom: 6244
+Time of assemble_system1: 6.61
+Time of assemble_system2: 5.4
+solution1 and solution2 do not differ.
+Writing grid to <grid-5.eps>...
+Writing solution to <sol-5.gnuplot>...
+
+</pre>
+</code>
+We see that, as expected, on each refinement step the two solutions
+coincide. The difference measured in time of treating each face only
+once (second version of the DG method) in comparison with treating
+each face twice within a nested loop over all cells and all faces of
+each cell (first version), is much less than one might have
+expected. The gain is less than 20% on all refinement steps.
+</p>
+
+<p> First we show the solutions on the initial mesh, the mesh after two
+and after five adaptive refinement steps.
+<p align="center">
+ <img src="step-12.data/sol-0.gif" alt="sol-0.gnuplot" width="75%">
+ <img src="step-12.data/sol-2.gif" alt="sol-2.gnuplot" width="75%">
+ <img src="step-12.data/sol-5.gif" alt="sol-5.gnuplot" width="75%">
+</p>
+
+<p> And finally we show the final grid (after 5 refinement steps). The
+grid is largely concentrated in the vicinity of the jump of the
+solution.
+<p align="center">
+ <img src="step-12.data/grid-5.gif" alt="grid-5.eps" width="50%">
+</p>