Update formatting of step-41.

Specifically:
* Use the same numbering scheme for enumerations as below. The style
  [(1)] likely originated from using an `\item[(1)]` in the original
  latex version of the introduction.
* Add a sentence after the bullet point list at the end of the introduction
  to help doxygen understand that it should not indent the following
  section heading (from the commented program) the same as the bullet
  point list.

diff --git a/examples/step-41/doc/intro.dox b/examples/step-41/doc/intro.dox
index 066e6c776880c9b820287f811661d13794e94799..9dcd0db0a111f63306d9b03b25405793b40ea50e 100644 (file)
--- a/examples/step-41/doc/intro.dox
+++ b/examples/step-41/doc/intro.dox
@@ -239,10 +239,10 @@ pp. 865-888.
 
 The algorithm for the primal-dual active set method works as follows (NOTE: $B = B^T$):
 
-- [(0)] Initialize $\mathcal{A}_k$ and $\mathcal{F}_k$, such that
+1. Initialize $\mathcal{A}_k$ and $\mathcal{F}_k$, such that
  $\mathcal{S}=\mathcal{A}_k\cup\mathcal{F}_k$ and
  $\mathcal{A}_k\cap\mathcal{F}_k=\emptyset$ and set $k=1$.
-- [(1)] Find the primal-dual pair $(U^k,\Lambda^k)$ that satisfies
+2. Find the primal-dual pair $(U^k,\Lambda^k)$ that satisfies
  @f{align*}
   AU^k + B\Lambda^k &= F,\\
   [BU^k]_i &= G_i\quad&&\forall i\in\mathcal{A}_k,\\
@@ -251,16 +251,16 @@ The algorithm for the primal-dual active set method works as follows (NOTE: $B =
  Note that the second and third conditions imply that exactly $|S|$ unknowns
  are fixed, with the first condition yielding the remaining $|S|$ equations
  necessary to determine both $U$ and $\Lambda$.
-- [(2)] Define the new active and inactive sets by
+3. Define the new active and inactive sets by
  @f{equation*}
  \begin{split}
   \mathcal{A}_{k+1}:=\lbrace i\in\mathcal{S}:\Lambda^k_i + c([BU^k]_i - G_i)< 0\rbrace,\\
   \mathcal{F}_{k+1}:=\lbrace i\in\mathcal{S}:\Lambda^k_i + c([BU^k]_i - G_i)\geq 0\rbrace.
  \end{split}
  @f}
-- [(3)] If $\mathcal{A}_{k+1}=\mathcal{A}_k$ (and then, obviously, also
+4. If $\mathcal{A}_{k+1}=\mathcal{A}_k$ (and then, obviously, also
  $\mathcal{F}_{k+1}=\mathcal{F}_k$) then stop, else set $k=k+1$ and go to step
- (1).
+ (2).
 
 The method is called "primal-dual" because it uses both primal (the
 displacement $U$) as well as dual variables (the Lagrange multiplier
@@ -270,10 +270,10 @@ At the end of this section, let us add two observations. First,
 for any primal-dual pair $(U^k,\Lambda^k)$ that satisfies these
 condition, we can distinguish the following cases:
 
-- [1.] $\Lambda^k_i + c([BU^k]_i - G_i) < 0$ (i active):
+1. $\Lambda^k_i + c([BU^k]_i - G_i) < 0$ (i active):
   <br>
   Then either $[BU^k]_i<G_i$ and $\Lambda^k_i=0$ (penetration) or $\Lambda^k_i<0$ and $[BU^k]_i=G_i$ (pressing load).
-- [2.] $\Lambda^k_i + c([BU^k]_i - G_i)\geq 0$ (i inactive):
+2. $\Lambda^k_i + c([BU^k]_i - G_i)\geq 0$ (i inactive):
   <br>
   Then either $[BU^k]_i\geq G_i$ and $\Lambda^k_i=0$ (no contact) or $\Lambda^k_i\geq0$ and $[BU^k]_i=G_i$ (unpressing load).
 
@@ -416,3 +416,5 @@ follows step-4 with minor differences:
 - We change the preconditioner for the solver.
 
 
+You may want to read up on step-4 if you want to understand the
+current program.