After we have created a grid in the previous example, we now show how
to define degrees of freedom on this mesh. For this example, we
-will use the lowest order (Q1) finite elements, for which the degrees
+will use the lowest order ($Q_1$) finite elements, for which the degrees
of freedom are associated with the vertices of the mesh. Later
examples will demonstrate higher order elements where degrees of freedom are
not necessarily associated with vertices any more, but can be associated
fe_values.JxW(q));
}
@endcode
-This very closely resembles the form we have originally written down the
-bilinear form and right hand side.
+This very closely resembles the form in which we have originally written down
+the bilinear form and right hand side.
There is one final term that we have to take care of: the right hand side
contained the term $(g,{\mathbf v}\cdot {\mathbf n})_{\partial\Omega}$, constituting the
problem laid out here, and will show how to solve it in parallel on a cluster
of computers. The resulting program will thus be able to solve this problem to
significantly higher accuracy, and more efficiently if this is
-required. In addition, in @ref step_20 "step-20", we will revisit some
+required. In addition, in @ref step_20 "step-20", @ref step_21 "step-21", and
+@ref step_22 "step-22", we will revisit some
vector-valued problems and show a few techniques that may make it
simpler to actually go through all the stuff shown above, with
<code>FiniteElement::system_to_component_index</code> etc.
@f}
For $d>2$, we extend $\beta$ and ${\mathbf x}_0$ by the same as the last
component. Regarding these functions, we have the following
-annotations:
+comments:
<ol>
<li> The advection field $\beta$ transports the solution roughly in
diagonal direction from lower left to upper right, but with a wiggle
structure superimposed.
<li> The right hand side adds to the field generated by the inflow
-boundary conditions a bulb in the lower left corner, which is then
+boundary conditions a blob in the lower left corner, which is then
transported along.
<li> The inflow boundary conditions impose a weighted sinusoidal
structure that is transported along with the flow field. Since