From a8ca0d5daeae1ca733afe006953063626a3c22bf Mon Sep 17 00:00:00 2001 From: bangerth Date: Sun, 30 Oct 2011 08:28:04 +0000 Subject: [PATCH] Add a description of what a DoF is. git-svn-id: https://svn.dealii.org/trunk@24698 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/doc/doxygen/headers/glossary.h | 31 ++++++++++++++++++++++++++ 1 file changed, 31 insertions(+) diff --git a/deal.II/doc/doxygen/headers/glossary.h b/deal.II/doc/doxygen/headers/glossary.h index 57b501f98c..3ab82c87d6 100644 --- a/deal.II/doc/doxygen/headers/glossary.h +++ b/deal.II/doc/doxygen/headers/glossary.h @@ -334,6 +334,37 @@ * * * + *
@anchor GlossDoF Degree of freedom
+ * + *
The term "degree of freedom" (often abbreviated as "DoF") is commonly + * used in the finite element community to indicate two slightly different, + * but related things. The first is that we'd like to represent the finite + * element solution as a linear combination of shape function, in the form + * $u_h(\mathbf x) = \sum_{j=0}^{N-1} U_j \varphi_j(\mathbf x)$. Here, $U_j$ + * is a vector of expension coefficients. Because we don't know their values + * yet (we will compute them as the solution of a linear or nonlinear system), + * they are called "unknowns" or "degrees of freedom". The second meaning of + * the term con be explained as follows: A mathematical description of finite + * element problem is often to say that we are looking for a finite + * dimensional function $u_h \in V_h$ that satisfies some set of equations + * (e.g. $a(u_h,\varphi_h)=(f,\varphi_h)$ for all test functions $\varphi_h\in + * V_h$). In other words, all we say here that the solution needs to lie in + * some space $V_h$. However, to actually solve this problem on a computer we + * need to choose a basis of this space; this is the set of shape functions + * $\varphi_j(\mathbf x)$ we have used above in the expansion of $u_h(\mathbf + * x)$ with coefficients $U_j$. There are of course many bases of the space + * $V_h$, but we will specifically choose the one that is described by the + * finite element functions that are traditionally defined locally on the + * cells of the mesh. Describing "degrees of freedom" in this context requires + * us to simply enumerate the basis functions of the space $V_h$. For + * $Q_1$ elements this means simply enumerating the vertices of the mesh in + * some way, but for higher elements one also has to enumerate the shape + * functions that are associated with edges, faces, or cell interiors of the + * mesh. The class that provides this enumeration of the basis functions of + * $V_h$ is called DoFHandler. The process of enumerating degrees of freedom + * is referred to as "distributing DoFs" in deal.II. + *
+ * *
@anchor GlossDirectionFlag Direction flags
* *
The direction flag is used in triangulations embedded in a -- 2.39.5