From: Wolfgang Bangerth <bangerth@colostate.edu>
Date: Tue, 1 May 2018 04:26:14 +0000 (-0600)
Subject: Minor edits to the step-51 introduction.
X-Git-Tag: v9.0.0-rc1~39^2
X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=0ad83f80f0cf8306f447949214fd2dad59ca45ea;p=dealii.git

Minor edits to the step-51 introduction.

Specifically, make sure 'FE_DGPMonomial' is spelled correctly. While there,
also address a couple of other things.
---

diff --git a/examples/step-51/doc/intro.dox b/examples/step-51/doc/intro.dox
index 9754c5170f..6a72e06e63 100644
--- a/examples/step-51/doc/intro.dox
+++ b/examples/step-51/doc/intro.dox
@@ -18,8 +18,7 @@ must solve in an implicit system.  This is because, unlike continuous finite
 elements, in typical discontinuous elements there is one degree of freedom at
 each vertex <i>for each of the adjacent elements</i>, rather than just one,
 and similarly for edges and faces.  As an example of how fast the number of
-unknowns grows,
-consider the <code>FE_DGP_Monomial</code> basis:  each
+unknowns grows, consider the FE_DGPMonomial basis: each
 scalar solution component is represented by polynomials of degree $p$
 with $(1/dim!)*\prod_{i=1}^{dim}(p+i)$ degrees of freedom per
 element. Typically, all degrees of freedom in an element are coupled
@@ -41,8 +40,15 @@ The HDG method achieves
 this goal by formulating the mathematical problem using Dirichlet-to-Neumann
 mappings.  The partial differential equations are first written as a first
 order system, and each field is then discretized via a DG method.  At this
-point the  single-valued "trace" values on the skeleton of the
+point, the  single-valued "trace" values on the skeleton of the
 mesh, i.e. element faces, are taken to be independent unknown quantities.
+This yields unknowns in the discrete formulation that fall into two categories:
+- Face unknowns that only couple with the unknowns from both sides of the face;
+- Cell unknowns that only couple with each other and the other unknowns
+  defined within the same cell. Crucially, no cell interior degree of freedom
+  on one cell ever couples to any interior cell degree of freedom of a
+  different cell.
+
 The Dirichlet-to-Neumann map concept then permits the following solution procedure:
 <ol>
   <li>  Use local element interior data to enforce a Neumann condition on the
@@ -56,8 +62,9 @@ solution process.
 
 The above procedure also has a linear algebra interpretation and is referred to
 as static condensation. Let us write the complete linear system associated to
-the HDG problem as a block system with the discrete DG variables $U$ as
-first block and the skeleton variables $\Lambda$ as the second block:
+the HDG problem as a block system with the discrete DG (cell interior)
+variables $U$ as first block and the skeleton (face) variables $\Lambda$ as the
+second block:
 @f{eqnarray*}
 \begin{pmatrix} A & B \\ C & D \end{pmatrix}
 \begin{pmatrix} U \\ \Lambda \end{pmatrix}