From: Wolfgang Bangerth Date: Tue, 23 Apr 2019 22:16:14 +0000 (-0600) Subject: Avoid latex formulas in section headings. X-Git-Tag: v9.1.0-rc1~185^2 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=refs%2Fpull%2F7950%2Fhead;p=dealii.git Avoid latex formulas in section headings. --- diff --git a/examples/step-61/doc/intro.dox b/examples/step-61/doc/intro.dox index 6d3ed92ed8..a93d150da1 100644 --- a/examples/step-61/doc/intro.dox +++ b/examples/step-61/doc/intro.dox @@ -365,7 +365,7 @@ that we only need to use the cell-interior part $\varphi_i^\circ$ for each shape function $\varphi_i$. -

Post-processing and $L_2$-errors

+

Post-processing and L2-errors

The discussions in the previous sections have given us a linear system that we can solve for the numerical pressure $p_h$. We can use diff --git a/examples/step-61/doc/results.dox b/examples/step-61/doc/results.dox index 6d21bb519c..6b7efd1aab 100644 --- a/examples/step-61/doc/results.dox +++ b/examples/step-61/doc/results.dox @@ -2,7 +2,7 @@ We run the test example $p = \sin(\pi x) \sin(\pi y)$ with homogenous 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 $\mbox{WG}(Q_2,Q_2;RT_{[2]})$. -

Test results on $\mbox{WG}(Q_0,Q_0;RT_{[0]})$

+

Test results on WG(Q0,Q0;RT[0])

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. @@ -48,7 +48,7 @@ velocity and flux. We can see that the convergence rates of $\mbox{WG}(Q_0,Q_0;RT_{[0]})$ are around 1. -

Test results on $\mbox{WG}(Q_1,Q_1;RT_{[1]})$

+

Test results on WG(Q1,Q1;RT[1])

The following figures are interior pressures and face pressures implemented on $\mbox{WG}(Q_1,Q_1;RT_{[1]})$. 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) @@ -87,7 +87,7 @@ These are the convergence rates of pressure, velocity and flux on $\mbox{WG}(Q_1
The convergence rates of $WG(Q_1,Q_1;RT_{[1]})$ are around 2. -

Test results on $WG(Q_2,Q_2;RT_{[2]})$

+

Test results on WG(Q2,Q2;RT[2])

These are interior pressures and face pressures implemented on $WG(Q_2,Q_2;RT_{[2]})$, with mesh size $h = 1/32$.