From: Wolfgang Bangerth Date: Tue, 8 Feb 2011 21:23:51 +0000 (+0000) Subject: More of an intro than just a sentence. X-Git-Tag: v8.0.0~4392 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=708cfa5c2f2a646fd72410e78ca38990663d7d72;p=dealii.git More of an intro than just a sentence. git-svn-id: https://svn.dealii.org/trunk@23306 0785d39b-7218-0410-832d-ea1e28bc413d --- diff --git a/deal.II/examples/step-2/doc/intro.dox b/deal.II/examples/step-2/doc/intro.dox index ebd20f45d5..d12f73392e 100644 --- a/deal.II/examples/step-2/doc/intro.dox +++ b/deal.II/examples/step-2/doc/intro.dox @@ -10,7 +10,30 @@ not necessarily associated with vertices any more, but can be associated with edges, faces, or cells. Defining degrees of freedom ("DoF"s in short) on a mesh is a rather -simple task, since the library does all the work for you. However, for +simple task, since the library does all the work for you. Essentially, +all you have to do is create a finite element object (from one of the +many finite element classes deal.II already has, see for example the +@ref fe documentation) and give it to a DoFHandler object. + +The next step would then be to compute a matrix and right hand side +corresponding to a particular differential equation using this finite +element and mesh. We will keep this step for the step-3 program and +rather talk about one practical aspect of a finite element program, +namely that finite element matrices are almost always very sparse, +i.e. almost all entries in these matrices are zero. This is one of the +distinguishing feature of the finite element method compared to, say, +approximating the solution of a partial differential equation using +for example a Taylor expension and matching coefficients. In practical +terms, it is the sparsity of matrices that enables us to solve +problems with millions or billions of unknowns. + +The sparsity is generated by the fact that finite element shape +functions are defined locally on individual cells, rather than +globally, and that the local differential operators in the bilinear +form only couple shape functions that have some overlap. By default, +the DoFHandler class enumerates degrees of freedom on a mesh in a +rather random way; consequently, the sparsity pattern is also not +optimized for any particular purpose. However, for some algorithms, especially for some linear solvers, it is advantageous to have the degrees of freedom numbered in a certain order, and we will use the algorithm of Cuthill and McKee to do