From a6b9c7bf7f226a8b549c53d634759a4441f54e06 Mon Sep 17 00:00:00 2001 From: Wolfgang Bangerth Date: Wed, 10 Feb 2010 21:56:36 +0000 Subject: [PATCH] Write a couple more words. git-svn-id: https://svn.dealii.org/trunk@20548 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-16/doc/results.dox | 70 +++++++++++++++++++++--- 1 file changed, 62 insertions(+), 8 deletions(-) diff --git a/deal.II/examples/step-16/doc/results.dox b/deal.II/examples/step-16/doc/results.dox index 72dea07a22..2e43c49882 100644 --- a/deal.II/examples/step-16/doc/results.dox +++ b/deal.II/examples/step-16/doc/results.dox @@ -1,24 +1,78 @@

Results

The output that this program generates is, of course, the same as that -of @ref step_6 "step-6", so you may see there for more results. +of @ref step_6 "step-6", so you may see there for more results. On the +other hand, since no tutorial program is a good one unless it has at +least one colorful picture, here is, again, the solution: +

+ @image html step-16.solution.png +

+When run, the output of this program is +
+Cycle 0:
+   Number of active cells:       20
+   Number of degrees of freedom: 25 (by level: 8, 25)
+   6 CG iterations needed to obtain convergence.
+Cycle 1:
+   Number of active cells:       44
+   Number of degrees of freedom: 57 (by level: 8, 25, 48)
+   7 CG iterations needed to obtain convergence.
+Cycle 2:
+   Number of active cells:       92
+   Number of degrees of freedom: 117 (by level: 8, 25, 80, 60)
+   8 CG iterations needed to obtain convergence.
+Cycle 3:
+   Number of active cells:       188
+   Number of degrees of freedom: 221 (by level: 8, 25, 80, 200)
+   11 CG iterations needed to obtain convergence.
+Cycle 4:
+   Number of active cells:       416
+   Number of degrees of freedom: 485 (by level: 8, 25, 89, 288, 280)
+   11 CG iterations needed to obtain convergence.
+Cycle 5:
+   Number of active cells:       800
+   Number of degrees of freedom: 925 (by level: 8, 25, 89, 288, 784, 132)
+   13 CG iterations needed to obtain convergence.
+Cycle 6:
+   Number of active cells:       1604
+   Number of degrees of freedom: 1833 (by level: 8, 25, 89, 304, 1000, 1116, 72)
+   13 CG iterations needed to obtain convergence.
+Cycle 7:
+   Number of active cells:       3164
+   Number of degrees of freedom: 3581 (by level: 8, 25, 89, 328, 1032, 2176, 1392)
+   15 CG iterations needed to obtain convergence.
+
+That's not perfect — we would have hoped for a constant number +of iterations rather than one that increases as we get more and more +degrees of freedom — but it is also not far away. The reason for +this is easy enough to understand, however: since we have a strongly +varying coefficient, the operators that we assembly by quadrature on +the lower levels become worse and worse approximations of the operator +on the finest level. Consequently, even if we had perfect solvers on +the coarser levels, they would not be good preconditioners on the +finest level. This theory is easily tested by comparing results when +we use a constant coefficient: in that case, the number of iterations +remains constant at 9 after the first three or four refinement steps. Let us compare the iteration steps needed to obtain convergence for the two methods: - + - - - + + +
cellsstep-5step-16
20 13 6
80 18 8
80 17 7
320 29 9
1280 52 9
5120 95 9
2048018210
1280 51 10
5120 94 11
2048018013
-This isn't only less iterations than in @ref step_5 "step-5" (each of which +This isn't only fewer iterations than in @ref step_5 "step-5" (each of which is, however, much more expensive) but more importantly, the number of -iterations stays roughly constant under mesh refinement. Therefore, whenever -possible, multigrid methods should be used for second order problems. +iterations also grows much more slowly under mesh refinement (again, +it would be almost constant if the coefficient was constant rather +than strongly varying as chosen here). This justifies the common +observation that, whenever possible, multigrid methods should be used +for second order problems. The output produced by this program is the following: -- 2.39.5