From: Wolfgang Bangerth <bangerth@colostate.edu>
Date: Mon, 16 Jan 2017 19:36:35 +0000 (-0700)
Subject: Document FETools::compute_node_matrix.
X-Git-Tag: v8.5.0-rc1~232^2
X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=refs%2Fpull%2F3809%2Fhead;p=dealii.git

Document FETools::compute_node_matrix.
---

diff --git a/include/deal.II/fe/fe_tools.h b/include/deal.II/fe/fe_tools.h
index 203013761a..5eb1dcb976 100644
--- a/include/deal.II/fe/fe_tools.h
+++ b/include/deal.II/fe/fe_tools.h
@@ -232,31 +232,65 @@ namespace FETools
                              FullMatrix<number> &matrix);
 
   /**
-   * Compute the matrix of nodal values of a finite element applied to all its
-   * shape functions.
-   *
-   * This function is supposed to help building finite elements from
-   * polynomial spaces and should be called inside the constructor of an
-   * element. Applied to a completely initialized finite element, the result
-   * should be the unit matrix by definition of the node values.
-   *
-   * Using this matrix allows the construction of the basis of shape functions
-   * in two steps.
-   *
-   * <ol>
-   *
-   * <li>Define the space of shape functions using an arbitrary basis
-   * <i>w<sub>j</sub></i> and compute the matrix <i>M</i> of node functionals
-   * <i>N<sub>i</sub></i> applied to these basis functions.
-   *
-   * <li>Compute the basis <i>v<sub>j</sub></i> of the finite element shape
-   * function space by applying <i>M<sup>-1</sup></i> to the basis
-   * <i>w<sub>j</sub></i>.
-   *
-   * </ol>
-   *
-   * @note The FiniteElement must provide generalized support points and and
-   * interpolation functions.
+   * This is a rather specialized function used during the construction of
+   * finite element objects. It is used to build the basis of shape functions
+   * for an element given a set of polynomials and interpolation points. The
+   * function is only implemented for finite elements with exactly @p dim
+   * vector components. In particular, this applies to classes derived from
+   * the FE_PolyTensor class.
+   *
+   * Specifically, the purpose of this function is as follows: FE_PolyTensor
+   * receives, from its derived classes, an argument that describes a polynomial
+   * space. This space may be parameterized in terms of monomials, or in some
+   * other way, but is in general not in the form that we use for finite
+   * elements where we typically want to use a basis that is derived from
+   * some kind of node functional (e.g., the interpolation at specific points).
+   * Concretely, assume that the basis used by the polynomial space is
+   * $\{\tilde\varphi_j(\mathbf x)\}_{j=1}^N$, and that the node functionals
+   * of the finite element are $\{\Psi_i\}_{i=1}^N$. We then want to compute a
+   * basis $\{\varphi_j(\mathbf x)\}_{j=1}^N$ for the finite element space so
+   * that $\Psi_i[\varphi_j] = \delta_{ij}$. To do this, we can set
+   * $\varphi_j(\mathbf x) = \sum_{k=1}^N c_{jk} \tilde\varphi_k(\mathbf x)$
+   * where we need to determine the expansion coefficients $c_{jk}$. We do this
+   * by applying $\Psi_i$ to both sides of the equation, to obtain
+   * @f{align*}
+   *   \Psi_i [\varphi_j] = \sum_{k=1}^N c_{jk} \Psi_i[\tilde\varphi_k],
+   * @f}
+   * and we know that the left hand side equals $\delta_{ij}$.
+   * If you think of this as a system of $N\times N$ equations for the
+   * elements of a matrix on the left and on the right, then this can be
+   * written as
+   * @f{align*}
+   *   I = C X^T
+   * @f}
+   * where $C$ is the matrix of coefficients $c_{jk}$ and
+   * $X_{ik} = \Psi_i[\tilde\varphi_k]$. Consequently, in order to compute
+   * the expansion coefficients $C=X^{-T}$, we need to apply the node
+   * functionals to all functions of the "raw" basis of the polynomial space.
+   *
+   * Until the finite element receives this matrix $X$ back, it describes its
+   * shape functions (e.g., in FiniteElement::shape_value()) in the form
+   * $\tilde\varphi_j$. After it calls this function, it has the expansion
+   * coefficients and can describe its shape functions as $\varphi_j$.
+   *
+   * This function therefore computes this matrix $X$, for the following
+   * specific circumstances:
+   * - That the node functionals $\Psi_i$ are point evaluations at points
+   *   $\mathbf x_i$ that the finite element in question describes via its
+   *   "generalized" support points (through
+   *   FiniteElement::get_generalized_support_points()). These point
+   *   evaluations need to necessarily evaluate the <i>value</i> of a shape
+   *   function at that point (the shape function may be vector-valued, and
+   *   so the functional may be a linear combination of the individual
+   *   components of the values); but, in particular, the nodal functions may
+   *   not be <i>integrals</i> over entire edges or faces,
+   *   or other non-local functionals. In other words, we assume that
+   *   $\Psi_i[\tilde\varphi_j] = f_j(\tilde\varphi_j(\mathbf x_i))$
+   *   where $f_j$ is a function of the (possibly vector-valued) argument
+   *   that returns a scalar.
+   * - That the finite element has exactly @p dim vector components.
+   * - That the function $f_j$ is given by whatever the element implements
+   *   through the FiniteElement::interpolate() function.
    */
   template <int dim, int spacedim>
   void compute_node_matrix(FullMatrix<double> &M,