From: wolf Date: Mon, 13 Feb 2006 00:35:15 +0000 (+0000) Subject: Show results. X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=14919b14e4cbeb3d64db87f02f3e4b52136f8197;p=dealii-svn.git Show results. git-svn-id: https://svn.dealii.org/trunk@12345 0785d39b-7218-0410-832d-ea1e28bc413d --- diff --git a/deal.II/doc/tutorial/chapter-2.step-by-step/step-20.data/results.html b/deal.II/doc/tutorial/chapter-2.step-by-step/step-20.data/results.html index abc80862ef..992ee6c614 100644 --- a/deal.II/doc/tutorial/chapter-2.step-by-step/step-20.data/results.html +++ b/deal.II/doc/tutorial/chapter-2.step-by-step/step-20.data/results.html @@ -1,8 +1,176 @@

Results

-convergence for h and p +

Output of the program and graphical visualization

+ +

+If we run the program as is, we get this output: +


+examples/step-20> make run
+============================ Remaking Makefile.dep
+==============debug========= step-20.cc
+============================ Linking step-20
+============================ Running step-20
+Number of active cells: 64
+Total number of cells: 85
+Number of degrees of freedom: 208 (144+64)
+10 CG Schur complement iterations to obtain convergence.
+Errors: ||e_p||_L2 = 0.178055,   ||e_u||_L2 = 0.0433435
+
+The fact that the number of iterations is so small, of course, is due to good +(but expensive!) preconditioner we have developed. To get confidence in the +solution, let us take a look at it. The following three images show (from left +to right) the x-velocity, the y-velocity, and the pressure (click on the images +for larger versions): + +

+ + + +

+ +

+Let us start with the pressure: it is highest at the left and lowest at the +right, so flow will be from left to right. In addition, though hardly visible +in the graph, we have chosen the pressure field such that the flow left-right +flow first channels towards the center and then outward again. Consequently, +the x-velocity has to increase to get the flow through the narrow part, +something that can easily be seen in the left image. The middle image +represents inward flow in y-direction at the left end of the domain, and +outward flow in y-directino at the right end of the domain. +

+ +

+As an additional remark, note how the x-velocity in the left image is only +continuous in x-direction, whereas the y-velocity is continuous in +y-direction. The flow fields are discontinuous in the other directions. This +very obviously reflects the continuity properties of the Raviart-Thomas +elements, which are, in fact, only in the space H(div) and not in the space +H1. Finally, the pressure field is completely discontinuous, but +that should not surprise given that we have chosen FE_DGQ(0) as +the finite element for that solution component. +

+ + +

Convergence

+ +

+The program offers two obvious places where playing and observing convergence +is in order: the degree of the finite elements used (passed to the constructor +of the MixedLaplaceProblem class from main()), and +the refinement level (determined in +MixedLaplaceProblem::make_grid_and_dofs). What one can do is to +change these values and observe the errors computed later on in the course of +the program run. +

+ +

+If one does this, one finds the following pattern for the L2 error +in the pressure variable: + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Finite element order
Refinement level012
0 1.45344 0.0831743 0.0235186
1 0.715099 0.0245341 0.00293983
2 0.356383 0.0063458 0.000367478
3 0.178055 0.00159944 4.59349e-05
4 0.0890105 0.000400669 5.74184e-06
5 0.0445032 0.000100218 7.17799e-07
6 0.0222513 2.50576e-05 9.0164e-08
O(h) O(h2 O(h3)
+ +The theoretically expected convergence orders are very nicely reflected by the +experimentally observed ones indicated in the last row of the table. +

+ +

+One can make the same experiment with the L2 error +in the velocity variables: + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Finite element order
Refinement level012
0 0.367423 0.127657 5.10388e-14
1 0.175891 0.0319142 9.04414e-15
2 0.0869402 0.00797856 1.23723e-14
3 0.0433435 0.00199464 1.86345e-07
4 0.0216559 0.00049866 2.72566e-07
5 0.010826 0.000124664 3.57141e-07
6 0.00541274 3.1166e-05 4.46124e-07
O(h) O(h2 O(h3)
+The result concerning the convergence order is the same here. +

+ try out preconditioner -try out different coefficient \ No newline at end of file +try out different coefficient + +different quadrature formula for errors