From 7ae213622fbedcbee6d0051e6bca92f1a07b8682 Mon Sep 17 00:00:00 2001 From: bangerth Date: Thu, 2 Aug 2007 04:08:36 +0000 Subject: [PATCH] A few more words git-svn-id: https://svn.dealii.org/trunk@14877 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-27/doc/intro.dox | 39 ++++++++++++++++++++++++++ 1 file changed, 39 insertions(+) diff --git a/deal.II/examples/step-27/doc/intro.dox b/deal.II/examples/step-27/doc/intro.dox index 6a8852962d..0b03b44f8a 100644 --- a/deal.II/examples/step-27/doc/intro.dox +++ b/deal.II/examples/step-27/doc/intro.dox @@ -7,6 +7,45 @@ few tutorial programs, in particular on @ref step_4 "step-4" for dimension independent programming and @ref step_6 "step-6" for adaptive mesh refinement. +The $hp$ finite element method was proposed in the early 1980s by +Babuska and Guo as an alternative to either +(i) mesh refinement (i.e. decreasing the mesh parameter $h$ in a finite +element computation) or (ii) increasing the polynomial degree $p$ used for +shape functions. It is based on the observation that increasing the polynomial +degree of the shape functions reduces the approximation error if the solution +is sufficiently smooth. On the other hand, it is well known +that even for the generally well-behaved class of elliptic problems, higher +degrees of regularity can not be guaranteed in the vicinity of boundaries, +corners, or where coefficients are discontinuous; consequently, the +approximation can not be improved in these areas by increasing the polynomial +degree $p$ but only by refining the mesh, i.e. by reducing the mesh size +$h$. These differing means to reduce the +error have led to the notion of $hp$ finite elements, where the approximating +finite element spaces are adapted to have a high polynomial degree $p$ +wherever the solution is sufficiently smooth, while the mesh width $h$ is +reduced at places wherever the solution lacks regularity. It was +already realized in the first papers on this method that $hp$ finite elements +can be a powerful tool that can guarantee that the error is reduced not only +with some negative power of the number of degrees of freedom, but in fact +exponentially. + +In order to implement this method, we need several things above and beyond +what a usual finite element program needs, and in particular above what we +have introduced in the tutorial programs leading up to step-6. In particular, +we will have to discuss the following aspects: + +

A simple indicator for smoothness

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