From 111d0cf267f4d4e4a45f3f656b110950e1e1cfe0 Mon Sep 17 00:00:00 2001 From: Wolfgang Bangerth Date: Fri, 12 Feb 2010 14:36:28 +0000 Subject: [PATCH] Use a picture instead of a text graphic. git-svn-id: https://svn.dealii.org/trunk@20570 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/deal.II/include/grid/tria.h | 66 ++++++++++++----------------- 1 file changed, 28 insertions(+), 38 deletions(-) diff --git a/deal.II/deal.II/include/grid/tria.h b/deal.II/deal.II/include/grid/tria.h index 966560291a..34714110dc 100644 --- a/deal.II/deal.II/include/grid/tria.h +++ b/deal.II/deal.II/include/grid/tria.h @@ -2,7 +2,7 @@ // $Id$ // Version: $Name$ // -// Copyright (C) 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009 by the deal.II authors +// Copyright (C) 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010 by the deal.II authors // // This file is subject to QPL and may not be distributed // without copyright and license information. Please refer @@ -732,41 +732,29 @@ namespace internal * It can be shown, that degradation of approximation occurs if the * triangulation contains vertices which are member of cells with levels * differing by more than one. One such example is the following: - * @verbatim - * | | | | - * x-----x-----x--x--x-- - * | | | | | - * | | x--x--x - * | | | | | - * x-----x-----x--x--x-- - * | | | - * | | | - * | | | - * | x-----x-- - * | | | - * | | | - * | | | - * x-----------x-----x-- - * @endverbatim - * It seems that in two space dimensions, the maximum jump in levels between - * cells sharing a common vertex is two (as in the example above). This is - * not true if more than four cells meet at a vertex. It is not uncommon - * that a coarse (initial) mesh contains vertices at which six or even eight - * cells meet, when small features of the domain have to be resolved even on - * the coarsest mesh. In that case, the maximum difference in levels is - * three or four, respectively. The problem gets even worse in three space - * dimensions. + * + * @image html limit_level_difference_at_vertices.png "" + * + * It would seem that in two space dimensions, the maximum jump in levels + * between cells sharing a common vertex is two (as in the example + * above). However, this is not true if more than four cells meet at a + * vertex. It is not uncommon that a coarse (initial) mesh contains + * vertices at which six or even eight cells meet, when small features of + * the domain have to be resolved even on the coarsest mesh. In that case, + * the maximum difference in levels is three or four, respectively. The + * problem gets even worse in three space dimensions. * * Looking at an interpolation of the second derivative of the finite * element solution (assuming bilinear finite elements), one sees that the - * numerical solution is almost totally wrong, compared with the true second - * derivative. Indeed, on regular meshes, there exist sharp estimations that - * the $H^2$-error is only $O(1)$, so we should not be surprised; however, the - * numerical solution may show a value for the second derivative which may - * be a factor of ten away from the true value. These problems are located - * on the small cell adjacent to the center vertex, where cells of - * non-subsequent levels meet, as well as on the upper and right neighbor - * of this cell (but with a less degree of deviation from the true value). + * numerical solution is almost totally wrong, compared with the true + * second derivative. Indeed, on regular meshes, there exist sharp + * estimations that the $H^2$-error is only $O(1)$, so we should not be + * surprised; however, the numerical solution may show a value for the + * second derivative which may be a factor of ten away from the true + * value. These problems are located on the small cell adjacent to the + * center vertex, where cells of non-subsequent levels meet, as well as on + * the upper and right neighbor of this cell (but with a less degree of + * deviation from the true value). * * If the smoothing indicator given to the constructor contains the bit for * @p limit_level_difference_at_vertices, situations as the above one are @@ -774,11 +762,13 @@ namespace internal * * In case of anisotropic refinement, the level of a cell is not linked to * the refinement of a cell as directly as in case of isotropic - * refinement. Furthermore, a cell can be strongly refined in one direction - * and not or at least much less refined in another. Therefore, it is very - * difficult to decide, which cases should be excluded from the refinement - * process. As a consequence, when using anisotropic refinement, the @p - * limit_level_difference_at_vertices flag must not be set. + * refinement. Furthermore, a cell can be strongly refined in one + * direction and not or at least much less refined in another. Therefore, + * it is very difficult to decide, which cases should be excluded from the + * refinement process. As a consequence, when using anisotropic + * refinement, the @p limit_level_difference_at_vertices flag must not be + * set. On the other hand, the implementation of multigrid methods in + * deal.II requires that this bit be set. * *
  • @p eliminate_unrefined_islands: * Single cells which are not refined and are surrounded by cells which are -- 2.39.5