From b31c5295d3cafcb8b041709cc94704facc4cb03e Mon Sep 17 00:00:00 2001 From: wolf Date: Mon, 29 Apr 2002 16:08:11 +0000 Subject: [PATCH] Enough for today. git-svn-id: https://svn.dealii.org/trunk@5752 0785d39b-7218-0410-832d-ea1e28bc413d --- .../step-14.data/results.html | 135 +++++++++++++++++- 1 file changed, 130 insertions(+), 5 deletions(-) diff --git a/deal.II/doc/tutorial/chapter-2.step-by-step/step-14.data/results.html b/deal.II/doc/tutorial/chapter-2.step-by-step/step-14.data/results.html index 60b09f5cab..7729ae91d7 100644 --- a/deal.II/doc/tutorial/chapter-2.step-by-step/step-14.data/results.html +++ b/deal.II/doc/tutorial/chapter-2.step-by-step/step-14.data/results.html @@ -252,6 +252,8 @@ Refinement cycle: 8 Refinement cycle: 9 Number of degrees of freedom=31438 Point x-derivative=-0.211258 + +

@@ -268,18 +270,101 @@ evaluation shows this: +This time, the grids in refinement cycles 0, 5, 6, 7, 8, and 9 look +like this: + + + -TODO + + + + + + + + + + + + + + +
+ + Grid 0 + + + + Grid 5 + +
+ + Grid 6 + + + + Grid 7 + +
+ + Grid 8 + + + + Grid 9 + +
+Note the assymetry of the grids compared with those we obtained for +the point evaluation, which is due to the directionality of the +x-derivative for which we tailored the refinement criterion.

Then, it is interesting to compare actually computed values of the quantity of interest (i.e. the x-derivative of the solution at one -point) with those values which we get from computing -on finer meshes. -TODO +point) with a reference value of -0.211289... plus or minus +0.000002. We get this reference value by computing on finer grid after +some more mesh refinements, with approximately 130,000 cells. +Recall that if the error is O(1/N) in the optimal case, then +taking a mesh with ten times more cells gives us one additional digit +in the result. +

+ +

+In the left part of the following chart, you again see the convergence +of the error towards this extrapolated value, while on the right you +see a comparison of true and estimated error: + + + + + + +
+ + Error in point value + + + + Error in point value + +
+It is obvious that here the error estimates are not as good as +previously, under-estimation the error by about a factor of 10. At +least the sign is correct, leading to a slight improvement in the +estimated values if we sum computed value and estimated error.

+TODO: explanation!

Step-13 revisited

@@ -290,9 +375,49 @@ If instead of the Exercise_2_3 data set, we choose computations of the previous example program, to compare whether the results obtained with the help of the dual weighted error estimator are better than those we had previously. -TODO

+

+First, the meshes after 9 and 10 adaptive refinement cycles, +respectively, look like this: + + + + + + +
+ Grid 9 + + + Grid 10 + +
+The features of the solution can still be seen slightly, but since the +solution is smooth, the roughness of the dual solution entirely +dominates the mesh refinement criterion, and leads to strongly +concentrated meshes. The solution after the seventh refinement step is +like so: + + + + +
+ Solution 7 + +
+Obviously, the solution is worse at some places, but the mesh +refinement process should have taken care that these places are not +important for computing the point value. +

+TODO +

Outlook

-- 2.39.5