is $h_K$, but for $Q_2$ elements is $h_K/2$).
+<h5>Results for 3d</h5>
+
+One can repeat these experiments in 3d and find the optimal time step
+for each value of $\beta$ and find the best value of $\beta$. What one
+finds is that for the same $\beta$ already used in 2d, the time steps
+needs to be a bit small, by around a factor of 1.2 or so. This is
+easily explained: the time step restriction is
+$k=\min_K \frac{ch_K}{\|\mathbf{u}\|_{L^\infty(K)}}$ where $h_K$ is
+the <i>diameter</i> of the cell. However, what is really needed is the
+distance between mesh points, which is $\frac{h_K}{\sqrt{d}}$. So a
+more appropriate form would be
+$k=\min_K \frac{ch_K}{\|\mathbf{u}\|_{L^\infty(K)}\sqrt{d}}$.
+
+The second find is that one needs to choose $\beta$ slightly bigger
+(about $\beta=0.05$ or so). This then again reduces the time step we
+can take.
+
+
+
+
<h5>Conclusions</h5>
Concluding, $\beta=0.03$ appears to be a good choice for the
-stabilization parameter, and then
+stabilization parameter in 2d, and $\beta=0.05$ in 3d. In a dimension
+independent way, we can model this as $\beta=0.015d$. As we have seen
+in the sections above, in 2d
$k=\frac 14 \frac 1{q_T}\frac{h_K}{\|\mathbf{u}\|_{L^\infty(K)}}$
is an appropriate time step, where $q_T$ is the polynomial degree of
the temperature shape functions (in the program, this corresponds to
-the variable <code>temperature_degree</code>).
+the variable <code>temperature_degree</code>). To reconcile this with
+the findings in 3d for the same $\beta$, we could write this as
+$k=\frac 1{2\sqrt{2}\sqrt{d}} \frac
+1{q_T}\frac{h_K}{\|\mathbf{u}\|_{L^\infty(K)}}$
+but this doesn't take into account that we also have to increase
+$\beta$ in 3d. The final form that takes all these factors in reads as
+follows:
+@f{eqnarray*}
+ k =
+ \frac 1{2\sqrt{2}} \frac 1{\sqrt{d}}
+ \frac 2d
+ \frac 1{q_T}
+ \frac{h_K}{\|\mathbf{u}\|_{L^\infty(K)}}
+ =
+ \frac 1{d\sqrt{2}\sqrt{d}}
+ \frac 1{q_T}
+ \frac{h_K}{\|\mathbf{u}\|_{L^\infty(K)}}.
+@f}
+In the first form (in the center of the equation), $\frac
+1{2\sqrt{2}}$ is a universal constant, $\frac 1{\sqrt{d}}$
+is the factor that accounts for the difference between cell diameter
+and grid point separation,
+$\frac 2d$ accounts for the increase in $\beta$ with space dimension,
+$\frac 1{q_T}$ accounts for the distance between grid points for
+higher order elements, and $\frac{h_K}{\|\mathbf{u}\|_{L^\infty(K)}}$
+for the local speed of transport relative to the cell size. This is
+the formula that we use in the program.
As for the question of whether to use $Q_1$ or $Q_2$ elements for the
temperature, the following considerations may be useful: First,