remove extension ideas from step-12b

diff --git a/examples/step-12b/doc/results.dox b/examples/step-12b/doc/results.dox
index 22e824300fa6b96060b25b1a1b842d7ab0201f83..ee39c3ffe515f05a988508344a5460ad870af3b4 100644 (file)
--- a/examples/step-12b/doc/results.dox
+++ b/examples/step-12b/doc/results.dox
@@ -26,113 +26,7 @@ And finally we show a plot of a 3d computation.
 <img src="https://www.dealii.org/images/steps/developer/step-12.sol-5-3d.png" alt="">
 
 
-<a name="dg-vs-cg"></a>
-<h3>Why use discontinuous elements</h3>
-
-In this program we have used discontinuous elements. It is a legitimate
-question to ask why not simply use the normal, continuous ones. Of course, to
-everyone with a background in numerical methods, the answer is obvious: the
-continuous Galerkin (cG) method is not stable for the transport equation,
-unless one specifically adds stabilization terms. The DG method, however,
-<i>is</i> stable. Illustrating this with the current program is not very
-difficult; in fact, only the following minor modifications are necessary:
-- Change the element to FE_Q instead of FE_DGQ.
-- Add handling of hanging node constraints in exactly the same way as step-6.
-- We need a different solver; the direct solver in step-29 is a convenient
-  choice.
-An experienced deal.II user will be able to do this in less than 10 minutes.
-
-While the 2d solution has been shown above, containing a number of small
-spikes at the interface that are, however, stable in height under mesh
-refinement, results look much different when using a continuous element:
-
-<table align="center">
-  <tr>
-    <td valign="top">
-      0 &nbsp;
-    </td>
-    <td valign="top">
-      <img src="https://www.dealii.org/images/steps/developer/step-12.cg.sol-0.png" alt="">
-    </td>
-    <td valign="top">
-      1 &nbsp;
-    </td>
-    <td valign="top">
-      <img src="https://www.dealii.org/images/steps/developer/step-12.cg.sol-1.png" alt="">
-    </td>
-  </tr>
-  <tr>
-    <td valign="top">
-      2 &nbsp;
-    </td>
-    <td valign="top">
-      <img src="https://www.dealii.org/images/steps/developer/step-12.cg.sol-2.png" alt="">
-    </td>
-    <td valign="top">
-      3 &nbsp;
-    </td>
-    <td valign="top">
-      <img src="https://www.dealii.org/images/steps/developer/step-12.cg.sol-3.png" alt="">
-    </td>
-  </tr>
-  <tr>
-    <td valign="top">
-      4 &nbsp;
-    </td>
-    <td valign="top">
-      <img src="https://www.dealii.org/images/steps/developer/step-12.cg.sol-4.png" alt="">
-    </td>
-    <td valign="top">
-      5 &nbsp;
-    </td>
-    <td valign="top">
-      <img src="https://www.dealii.org/images/steps/developer/step-12.cg.sol-5.png" alt="">
-    </td>
-  </tr>
-</table>
-
-In refinement iteration 5, the image can't be plotted in a reasonable way any
-more as a 3d plot. We thus show a color plot with a range of $[-1,2]$ (the
-solution values of the exact solution lie in $[0,1]$, of course). In any case,
-it is clear that the continuous Galerkin solution exhibits oscillatory
-behavior that gets worse and worse as the mesh is refined more and more.
-
-There are a number of strategies to stabilize the cG method, if one wants to
-use continuous elements for some reason. Discussing these methods is beyond
-the scope of this tutorial program; an interested reader could, for example,
-take a look at step-31.
-
-
-
 <a name="extensions"></a>
 <h3>Possibilities for extensions</h3>
 
-Given that the exact solution is known in this case, one interesting
-avenue for further extensions would be to confirm the order of
-convergence for this program. In the current case, the solution is
-non-smooth, and so we can not expect to get a particularly high order
-of convergence, even if we used higher order elements. But even if the
-solution <i>is</i> smooth, the equation is not elliptic and so it is not
-immediately clear that we should obtain a convergence order that
-equals that of the optimal interpolation estimates (i.e. for example
-that we would get $h^3$ convergence in the $L^2$ norm by using
-quadratic elements).
-
-In fact, for hyperbolic equations, theoretical predictions often
-indicate that the best one can hope for is an order one half below the
-interpolation estimate. For example, for the streamline diffusion
-method (an alternative method to the DG method used here to stabilize
-the solution of the transport equation), one can prove that for
-elements of degree $p$, the order of convergence is $p+\frac 12$ on
-arbitrary meshes. While the observed order is frequently $p+1$ on
-uniformly refined meshes, one can construct so-called Peterson meshes
-on which the worse theoretical bound is actually attained. This should
-be relatively simple to verify, for example using the
-VectorTools::integrate_difference function.
-
-A different direction is to observe that the solution of transport problems
-often has discontinuities and that therefore a mesh in which we <i>bisect</i>
-every cell in every coordinate direction may not be optimal. Rather, a better
-strategy would be to only cut cells in the direction parallel to the
-discontinuity. This is called <i>anisotropic mesh refinement</i> and is the
-subject of step-30.
+For ideas for further extensions, please see see step-12.