Edit the results section of step-61.

diff --git a/examples/step-61/doc/intro.dox b/examples/step-61/doc/intro.dox
index 8173bdbc2352438f2e7409763b6f1b1b6e465129..1709c846ef794bf4047cae2b01b1cf4245daf728 100644 (file)
--- a/examples/step-61/doc/intro.dox
+++ b/examples/step-61/doc/intro.dox
@@ -447,11 +447,11 @@ define the $L_2$-errors of pressure, velocity, and flux
 by the following formulas:
 @f{eqnarray*}
 \|p-p_h^\circ\|^2
-  = \sum_{K \in \mathbb{T}} \|p-p_h^\circ\|_{L_2(K)}^2, \\
+  &= \sum_{K \in \mathbb{T}} \|p-p_h^\circ\|_{L_2(K)}^2, \\
  \|\mathbf{u}-\mathbf{u}_h\|^2
-  = \sum_{K \in \mathbb{T}} \|\mathbf{u}-\mathbf{u}_h\|_{L_2(K)^2}^d,\\
+  &= \sum_{K \in \mathbb{T}} \|\mathbf{u}-\mathbf{u}_h\|_{L_2(K)^2}^d,\\
 \|(\mathbf{u}-\mathbf{u}_h) \cdot \mathbf{n}\|^2
-  = \sum_{K \in \mathbb{T}} \sum_{\gamma \subset \partial K}
+  &= \sum_{K \in \mathbb{T}} \sum_{\gamma \subset \partial K}
     \frac{|K|}{|\gamma|} \|\mathbf{u} \cdot \mathbf{n} - \mathbf{u}_h \cdot \mathbf{n}\|_{L_2(\gamma)}^2,
 @f}
 where $| K |$ is the area of the element, 

diff --git a/examples/step-61/doc/results.dox b/examples/step-61/doc/results.dox
index 41b73af8e57123735c8112a8d7c55301b0be75ec..ed99e7b0ff455f835c9d4bcd2df7916cbfe14193 100644 (file)
--- a/examples/step-61/doc/results.dox
+++ b/examples/step-61/doc/results.dox
@@ -1,20 +1,25 @@
 <h1>Results</h1>
 
-We run the test example $p = \sin(\pi x) \sin(\pi y)$ with homogeneous Dirichelet 
-boundary conditions in the domain $\Omega = (0,1)^2$. And  $\mathbf{K}$ is the 
-identity matrix. We test it on $\mbox{WG}(Q_0,Q_0;RT_{[0]})$, $\mbox{WG}(Q_1,Q_1;RT_{[1]})$ 
-and $\mbox{WG}(Q_2,Q_2;RT_{[2]})$. We will visualize pressure values in interiors 
-and on faces. We want to see the pressure maximum is around 1 and the minimum 
-is around 0. With the mesh refinement, the convergence rates of pressure, 
-velocity and flux should be around 1 on $\mbox{WG}(Q_0,Q_0;RT_{[0]})$ , 2 on 
-$\mbox{WG}(Q_1,Q_1;RT_{[1]})$, and 3 on $\mbox{WG}(Q_2,Q_2;RT_{[2]})$.
+We run the program with a right hand side that will produce the solution
+$p = \sin(\pi x) \sin(\pi y)$ and with homogeneous Dirichlet 
+boundary conditions in the domain $\Omega = (0,1)^2$. In addition, we choose
+the coefficient matrix in the differential operator $\mathbf{K}$ as the 
+identity matrix. We test this setup using $\mbox{WG}(Q_0,Q_0;RT_{[0]})$, 
+$\mbox{WG}(Q_1,Q_1;RT_{[1]})$ and $\mbox{WG}(Q_2,Q_2;RT_{[2]})$ element 
+combinations. We will then visualize pressure values in interiors of cells 
+and on faces. We want to see that the pressure maximum is around 1 and the minimum 
+is around 0. With mesh refinement, the convergence rates of pressure, 
+velocity and flux should then be around 1 for $\mbox{WG}(Q_0,Q_0;RT_{[0]})$ , 2 
+for $\mbox{WG}(Q_1,Q_1;RT_{[1]})$, and 3 for $\mbox{WG}(Q_2,Q_2;RT_{[2]})$.
 
 
 <h3>Test results on <i>WG(Q<sub>0</sub>,Q<sub>0</sub>;RT<sub>[0]</sub>)</i></h3>
 
-The following figures are interior pressures and face pressures implemented 
-on $\mbox{WG}(Q_0,Q_0;RT_{[0]})$. The mesh is refined 2 times and 4 times 
-separately. 
+The following figures show interior pressures and face pressures using the
+$\mbox{WG}(Q_0,Q_0;RT_{[0]})$ element. The mesh is refined 2 times (top)
+and 4 times (bottom), respectively. When the mesh is coarse, one can see
+the face pressures $p^\partial$ neatly between the values of the interior
+pressures $p^\circ$ on the two adjacent cells.
 
 <table align="center">
   <tr>
@@ -28,19 +33,20 @@ separately.
 </table>
 
 From the figures, we can see that with the mesh refinement, the maximum and 
-minimum are approaching to what we expect. 
-Since the mesh is a rectangular mesh and numbers of refinement are even, we 
-have symmetric solutions. From the 3d figures, we can see that on $\mbox{WG}(Q_0,Q_0;RT_{[0]})$, 
-pressure is a constant in the interior of the cell.
+minimum pressure values are approaching the values we expect. 
+Since the mesh is a rectangular mesh and numbers of cells in each direction is even, we 
+have symmetric solutions. From the 3d figures on the right, 
+we can see that on $\mbox{WG}(Q_0,Q_0;RT_{[0]})$, the pressure is a constant
+in the interior of the cell, as expected.
 
 <h4>Convergence table</h4>
 
 We run the code with finer meshes and get the following convergence rates of pressure, 
-velocity and flux.
+velocity, and flux (as defined in the introduction).
 
-<table align="center">
+<table align="center" class="doxtable">
   <tr>
-   <td>number of refinements </td><td>  $\|p-p_h^\circ\|$  </td><td>  $\|\mathbf{u}-\mathbf{u}_h\|$ </td><td> $\|(\mathbf{u}-\mathbf{u}_h) \cdot \mathbf{n}\|$ </td>
+   <th>number of refinements </th><th>  $\|p-p_h^\circ\|$  </th><th>  $\|\mathbf{u}-\mathbf{u}_h\|$ </th><th> $\|(\mathbf{u}-\mathbf{u}_h) \cdot \mathbf{n}\|$ </th>
   </tr>
   <tr>
    <td>   2                  </td><td>    1.587e-01        </td><td>        5.113e-01               </td><td>   7.062e-01 </td>
@@ -55,23 +61,25 @@ velocity and flux.
    <td>   5                  </td><td>    2.004e-02        </td><td>        6.297e-02               </td><td>   8.902e-02 </td>
   </tr>
   <tr>
-   <td>Conv.rate             </td><td>      1.00           </td><td>          1.00                  </td><td>      1.00   </td>
+   <th>Conv.rate             </th><th>      1.00           </th><th>          1.00                  </th><th>      1.00   </th>
   </tr>
 </table>
 
 We can see that the convergence rates of $\mbox{WG}(Q_0,Q_0;RT_{[0]})$ are around 1.
+This, of course, matches our theoretical expectations.
 
 
 <h3>Test results on <i>WG(Q<sub>1</sub>,Q<sub>1</sub>;RT<sub>[1]</sub>)</i></h3>
 
-The following figures are interior pressures and face pressures implemented on 
+We can repeat the experiment from above using the next higher polynomial
+degree:
+The following figures are interior pressures and face pressures implemented using 
 $\mbox{WG}(Q_1,Q_1;RT_{[1]})$. The mesh is refined 4 times.  Compared to the 
-previous figures on 
-$\mbox{WG}(Q_0,Q_0;RT_{[0]})$, on each cell, the result is not a constant. 
-Because we use higher order polynomials to do approximation. So there are 4 
-pressure values in one interior, 2 pressure values on each face. We use 
-data_out_face.build_patches (fe.degree)
-to divide each cell interior into 4 subcells.
+previous figures using
+$\mbox{WG}(Q_0,Q_0;RT_{[0]})$, on each cell, the solution is no longer constant
+on each cell, as we now use bilinear polynomials to do the approximation.
+Consequently, there are 4 pressure values in one interior, 2 pressure values on
+each face.
 
 <table align="center">
   <tr>
@@ -80,13 +88,20 @@ to divide each cell interior into 4 subcells.
   </tr>
 </table>
 
+Compared to the corresponding image for the $\mbox{WG}(Q_0,Q_0;RT_{[0]})$
+combination, the solution is now substantially more accurate and, in
+particular so close to being continuous at the interfaces that we can
+no longer distinguish the interface pressures $p^\partial$ from the
+interior pressures $p^\circ$ on the adjacent cells.
+
 <h4>Convergence table</h4>
 
-These are the convergence rates of pressure, velocity and flux on $\mbox{WG}(Q_1,Q_1;RT_{[1]})$
+The following are the convergence rates of pressure, velocity, and flux 
+we obtain from using the $\mbox{WG}(Q_1,Q_1;RT_{[1]})$ element combination:
 
-<table align="center">
+<table align="center" class="doxtable">
   <tr>
-   <td>number of refinements </td><td>  $\|p-p_h^\circ\|$  </td><td>  $\|\mathbf{u}-\mathbf{u}_h\|$ </td><td> $\|(\mathbf{u}-\mathbf{u}_h) \cdot \mathbf{n}\|$ </td>
+   <th>number of refinements </th><th>  $\|p-p_h^\circ\|$  </th><th>  $\|\mathbf{u}-\mathbf{u}_h\|$ </th><th> $\|(\mathbf{u}-\mathbf{u}_h) \cdot \mathbf{n}\|$ </th>
   </tr>
   <tr>
     <td>  2           </td><td>           1.613e-02      </td><td>          5.093e-02     </td><td>             7.167e-02   </td>
@@ -101,16 +116,24 @@ These are the convergence rates of pressure, velocity and flux on $\mbox{WG}(Q_1
     <td>  5           </td><td>           2.540e-04      </td><td>          7.979e-04     </td><td>             1.128e-03  </td>
   </tr>
   <tr>
-    <td>Conv.rate     </td><td>              2.00        </td><td>             2.00       </td><td>                 2.00    </td>
+    <th>Conv.rate     </th><th>              2.00        </th><th>             2.00       </th><th>                 2.00    </th>
   </tr>
 </table>
-The convergence rates of $WG(Q_1,Q_1;RT_{[1]})$ are around 2.
 
-<h3>Test results on <i>WG(Q<sub>2</sub>,Q<sub>2</sub>;RT<sub>[2]</sub>)</i></h3>
+The convergence rates of $WG(Q_1,Q_1;RT_{[1]})$ are around 2, as expected.
+
 
-These are interior pressures and face pressures implemented on 
-$WG(Q_2,Q_2;RT_{[2]})$, with mesh size $h = 1/32$.
 
+<h3>Test results on <i>WG(Q<sub>2</sub>,Q<sub>2</sub>;RT<sub>[2]</sub>)</i></h3>
+
+Let us go one polynomial degree higher.
+The following are interior pressures and face pressures implemented using 
+$WG(Q_2,Q_2;RT_{[2]})$, with mesh size $h = 1/32$ (i.e., 5 global mesh
+refinement steps). In the program, we use
+`data_out_face.build_patches(fe.degree)` when generating graphical output
+(see the documentation of DataOut::build_patches()), which here implies that
+we divide each 2d cell interior into 4 subcells in order to provide a better
+visualization of the quadratic polynomials.
 <table align="center">
   <tr>
     <td><img src="https://www.dealii.org/images/steps/developer/step-61.wg222_2d_5.png" alt=""></td>
@@ -118,14 +141,16 @@ $WG(Q_2,Q_2;RT_{[2]})$, with mesh size $h = 1/32$.
   </tr>
 </table>
 
+
 <h4>Convergence table</h4>
 
-This is the convergence table of $L_2$ errors of pressure, velocity and flux 
-on $\mbox{WG}(Q_2,Q_2;RT_{[2]})$
+As before, we can generate convergence data for the
+$L_2$ errors of pressure, velocity, and flux 
+using the $\mbox{WG}(Q_2,Q_2;RT_{[2]})$ combination:
 
-<table align="center">
+<table align="center" class="doxtable">
   <tr>
-   <td>number of refinements </td><td>  $\|p-p_h^\circ\|$  </td><td>  $\|\mathbf{u}-\mathbf{u}_h\|$ </td><td> $\|(\mathbf{u}-\mathbf{u}_h) \cdot \mathbf{n}\|$ </td>
+   <th>number of refinements </th><th>  $\|p-p_h^\circ\|$  </th><th>  $\|\mathbf{u}-\mathbf{u}_h\|$ </th><th> $\|(\mathbf{u}-\mathbf{u}_h) \cdot \mathbf{n}\|$ </th>
   </tr>
   <tr>
      <td>  2               </td><td>       1.072e-03       </td><td>         3.375e-03       </td><td>           4.762e-03   </td>
@@ -140,7 +165,9 @@ on $\mbox{WG}(Q_2,Q_2;RT_{[2]})$
     <td>   5               </td><td>       2.107e-06      </td><td>          6.620e-06       </td><td>           9.362e-06  </td>
   </tr>
   <tr>
-    <td>Conv.rate          </td><td>         3.00         </td><td>            3.00          </td><td>              3.00    </td>
+    <th>Conv.rate          </th><th>         3.00         </th><th>            3.00          </th><th>              3.00    </th>
   </tr>
 </table>
-The convergence rates of $\mbox{WG}(Q_2,Q_2;RT_{[2]})$ are around 3.
+
+Once more, the convergence rates of $\mbox{WG}(Q_2,Q_2;RT_{[2]})$ is
+as expected, with values around 3.