From cf5743756b87fa0d5681762ec72a17cf6057104e Mon Sep 17 00:00:00 2001 From: bangerth Date: Sun, 14 Aug 2011 23:59:13 +0000 Subject: [PATCH] Minor addition. git-svn-id: https://svn.dealii.org/trunk@24062 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-2/doc/intro.dox | 31 ++++++++++++++++----------- 1 file changed, 19 insertions(+), 12 deletions(-) diff --git a/deal.II/examples/step-2/doc/intro.dox b/deal.II/examples/step-2/doc/intro.dox index 46cacf2709..5b0b69b33d 100644 --- a/deal.II/examples/step-2/doc/intro.dox +++ b/deal.II/examples/step-2/doc/intro.dox @@ -23,16 +23,22 @@ how big your system matrix should be, and when copying the contributions of a single cell into the global matrix. 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. +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. (To be more precise, we say a discretization leads to a +sparse matrix if the number of nonzero entries per row in the matrix is +bounded by a number that is independent of the overall number of degrees of +freedom. For example, the simple 5-point stencil of a finite difference +approximation of the Laplace equation leads to a sparse matrix since the +number of nonzero entries per row is five, and therefore independent of the +total size of the matrix.) Sparsity is one of the distinguishing feature of +the finite element method compared to, say, approximating the solution of a +partial differential equation using a Taylor expension and matching +coefficients, or using a Fourier basis. 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 @@ -41,7 +47,8 @@ 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 +some algorithms, especially for some linear solvers and preconditioners, 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 -so. The results are written to a file and visualized using GNUPLOT. +so. The results are written to a file and visualized using GNUPLOT; you get to +see the outcome in the results section below. -- 2.39.5