From: Wolfgang Bangerth <bangerth@colostate.edu>
Date: Thu, 6 Apr 2017 15:21:27 +0000 (-0600)
Subject: Update a glossary entry on node functionals.
X-Git-Tag: v9.0.0-rc1~1732^2
X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=801d38ded34f383fcc79bca634e9de41019ff083;p=dealii.git

Update a glossary entry on node functionals.
---

diff --git a/doc/doxygen/headers/glossary.h b/doc/doxygen/headers/glossary.h
index 7353a9f1e9..36193541dc 100644
--- a/doc/doxygen/headers/glossary.h
+++ b/doc/doxygen/headers/glossary.h
@@ -1282,22 +1282,30 @@
  *
  * <dt class="glossary">@anchor GlossNodes <b>Node values or node functionals</b></dt>
  *
- * <dd>It is customary to define a FiniteElement as a pair consisting
- * of a local function space and a set of node values $N_i$ on the
- * mesh cells (usually defined on the @ref GlossReferenceCell
- * "reference cell"). Then, the basis of the local function space is
- * chosen such that $N_i(v_j) = \delta_{ij}$, the Kronecker delta.
- *
- * This splitting has several advantages, concerning analysis as well
+ * <dd>It is customary to define a finite element as a triple
+ * $(K,P,\Psi)$ where
+ * - $K$ is the cell, where in deal.II this is always a line segment,
+ *   quadrilateral, or hexahedron;
+ * - $P$ is a finite-dimensional space, e.g., a polynomial space mapped
+ *   from the @ref GlossReferenceCell "reference cell" to $K$;
+ * - $\Psi$ is a set of "node functionals", i.e., functionals
+ *   $\Psi_i : P \rightarrow {\mathbb R}$.
+ * The dimension of $P$ must be equal to the number of node functionals.
+ * With this definition, we can define a basis of the local function space,
+ * i.e., a set of "shape functions" $\varphi_j\in P$, by requiring that
+ * $\Psi_i(\varphi_j) = \delta_{ij}$, where $\delta$ is the Kronecker delta.
+ *
+ * This definition of what a finite element is has several advantages,
+ * concerning analysis as well
  * as implementation. For the analysis, it means that conformity with
  * certain spaces (FiniteElementData::Conformity), e.g. continuity, is
- * up to the node values. In deal.II, it helps simplifying the
+ * up to the node functionals. In deal.II, it helps simplifying the
  * implementation of more complex elements like FE_RaviartThomas
  * considerably.
  *
- * Examples for node functionals are values in @ref GlossSupport
- * "support points" and moments with respect to Legendre
- * polynomials. Let us give some examples:
+ * Examples for node functionals are values in
+ * @ref GlossSupport "support points" and moments with respect to Legendre
+ * polynomials. Examples:
  *
  * <table><tr>
  *   <th>Element</th>
@@ -1316,6 +1324,13 @@
  *     <td><i>Q<sub>k+1,k</sub> x Q<sub>k,k+1</sub></i></td>
  *     <td>Gauss points on edges(faces) and anisotropic Gauss points in the interior</td></tr>
  * </table>
+ *
+ * The construction of finite elements as outlined above allows writing
+ * code that describes a finite element simply by providing a polynomial
+ * space (without having to give it any particular basis -- whatever is convenient
+ * is entirely sufficient) and the nodal functionals. This is used, for example
+ * in the FiniteElement::convert_generalized_support_point_values_to_nodal_values()
+ * function.
  * </dd>
  *
  *
diff --git a/include/deal.II/fe/fe.h b/include/deal.II/fe/fe.h
index 1d7c58cbe0..1c8398f35c 100644
--- a/include/deal.II/fe/fe.h
+++ b/include/deal.II/fe/fe.h
@@ -1939,7 +1939,8 @@ public:
    * Given the values of a function $f(\mathbf x)$ at the (generalized) support
    * points, this function then computes what the nodal values of the element
    * are, i.e., $\Psi_i[f]$, where $\Psi_i$ are the node functionals of
-   * the element. The values $\Psi_i[f]$ are then the expansion coefficients
+   * the element (see also @ref GlossNodes "Node values or node functionals").
+   * The values $\Psi_i[f]$ are then the expansion coefficients
    * for the shape functions of the finite element function that
    * <i>interpolates</i> the given function $f(x)$, i.e.,
    * $ f_h(\mathbf x) = \sum_i \Psi_i[f] \varphi_i(\mathbf x)