From: Wolfgang Bangerth <bangerth@colostate.edu>
Date: Thu, 2 May 2024 18:28:13 +0000 (+0530)
Subject: Restate the equation we solve at the beginning of step-6.
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Restate the equation we solve at the beginning of step-6.

The equation is the same as step-5, but that is never overly explicitly
said. It's also useful to just restate the equations for reference.

step-5 does not describe the coefficient we use in the introduction. Add that
too while there.
---

diff --git a/examples/step-5/doc/intro.dox b/examples/step-5/doc/intro.dox
index f7f52f99d7..319aae90b0 100644
--- a/examples/step-5/doc/intro.dox
+++ b/examples/step-5/doc/intro.dox
@@ -70,9 +70,17 @@ The equation to solve here is as follows:
 If $a(\mathbf x)$ was a constant coefficient, this would simply be the Poisson
 equation that we have already solved in step-3 and step-4. However, if it is
 indeed spatially variable, it is a more complex equation (sometimes referred
-to as the "Poisson equation with a coefficient"). Depending on
-what the variable $u$ refers to, it models a variety of situations with wide
-applicability:
+to as the "Poisson equation with a coefficient"). Specifically, we will here
+choose it as follows:
+@f{align*}{
+  a(\mathbf x) =
+  \begin{cases}
+    20 & \text{if}\ |\mathbf x|<0.5, \\
+    1  & \text{otherwise.}
+  \end{cases}
+@f}
+Depending on what the variable $u$ refers to, it models a variety of
+situations with wide applicability:
 
 - If $u$ is the electric potential, then $-a\nabla u$ is the electric current
   in a medium and the coefficient $a$ is the conductivity of the medium at any
diff --git a/examples/step-6/doc/intro.dox b/examples/step-6/doc/intro.dox
index c07d309273..408ac00727 100644
--- a/examples/step-6/doc/intro.dox
+++ b/examples/step-6/doc/intro.dox
@@ -8,7 +8,7 @@ the use of adaptively (locally) refined meshes. The program is still
 based on step-4 and step-5, and, as you will see, it does not actually
 take very much code to enable adaptivity. Indeed, while we do a great
 deal of explaining, adaptive meshes can be added to an existing program
-with barely a dozen lines of additional code. The program shows what
+with less than ten lines of additional code. The program shows what
 these lines are, as well as another important ingredient of adaptive
 mesh refinement (AMR): a criterion that can be used to determine whether
 it is necessary to refine a cell because the error is large on it,
@@ -16,6 +16,21 @@ whether the cell can be coarsened because the error is particularly
 small on it, or whether we should just leave the cell as it is. We
 will discuss all of these issues in the following.
 
+The program solves the same problem as step-5, that is, we solve the equation
+@f{align*}{
+  -\nabla \cdot a(\mathbf x) \nabla u(\mathbf x) &= 1 \qquad\qquad & \text{in}\ \Omega,
+  \\
+  u &= 0 \qquad\qquad & \text{on}\ \partial\Omega,
+@f}
+where $a(\mathbf x)$ is a spatially variable coefficient defined as
+@f{align*}{
+  a(\mathbf x) =
+  \begin{cases}
+    20 & \text{if}\ |\mathbf x|<0.5, \\
+    1  & \text{otherwise.}
+  \end{cases}
+@f}
+
 
 <h3> What adaptively refined meshes look like </h3>