From 4c11ccd9538703383c3f5ac135c1c97353f9dc33 Mon Sep 17 00:00:00 2001 From: bangerth Date: Sat, 22 Jul 2006 01:26:33 +0000 Subject: [PATCH] Remove data now found or generated from examples/step-XX/doc git-svn-id: https://svn.dealii.org/trunk@13408 0785d39b-7218-0410-832d-ea1e28bc413d --- .../point_derivative/error-estimation.eps | 621 -------------- .../point_derivative/error-estimation.gif | Bin 4670 -> 0 bytes .../step-14.data/point_derivative/error.eps | 768 ------------------ .../step-14.data/point_derivative/error.gif | Bin 5323 -> 0 bytes .../step-14.data/point_derivative/grid-0.gif | Bin 103590 -> 0 bytes .../step-14.data/point_derivative/grid-5.gif | Bin 103590 -> 0 bytes .../step-14.data/point_derivative/grid-6.gif | Bin 103590 -> 0 bytes .../step-14.data/point_derivative/grid-7.gif | Bin 103590 -> 0 bytes .../step-14.data/point_derivative/grid-8.gif | Bin 103590 -> 0 bytes .../step-14.data/point_derivative/grid-9.gif | Bin 103590 -> 0 bytes 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S7 stroke grestore } bind def -/D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def -/D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def -/D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def -/D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def -/D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def -/D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def -/D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def -/D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def -/DiaE { stroke [] 0 setdash vpt add M - hpt neg vpt neg V hpt vpt neg V - hpt vpt V hpt neg vpt V closepath stroke } def -/BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M - 0 vpt2 neg V hpt2 0 V 0 vpt2 V - hpt2 neg 0 V closepath stroke } def -/TriUE { stroke [] 0 setdash vpt 1.12 mul add M - hpt neg vpt -1.62 mul V - hpt 2 mul 0 V - hpt neg vpt 1.62 mul V closepath stroke } def -/TriDE { stroke [] 0 setdash vpt 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Point values

- -

-This program offers a lot of possibilities to play around. We can thus -only show a small part of all possible results that can be obtained -with the help of this program. However, you are encouraged to just try -it out, by changing the settings in the main program. Here, we start -by simply letting it run, unmodified: - -

-Refinement cycle: 0
-   Number of degrees of freedom=72
-   Point value=0.03243
-   Estimated error=0.000702385
-Refinement cycle: 1
-   Number of degrees of freedom=67
-   Point value=0.0324827
-   Estimated error=0.000888953
-Refinement cycle: 2
-   Number of degrees of freedom=130
-   Point value=0.0329619
-   Estimated error=0.000454606
-Refinement cycle: 3
-   Number of degrees of freedom=307
-   Point value=0.0331934
-   Estimated error=0.000241254
-Refinement cycle: 4
-   Number of degrees of freedom=718
-   Point value=0.0333675
-   Estimated error=7.4912e-05
-Refinement cycle: 5
-   Number of degrees of freedom=1683
-   Point value=0.0334087
-   Estimated error=3.64279e-05
-Refinement cycle: 6
-   Number of degrees of freedom=3940
-   Point value=0.0334313
-   Estimated error=1.51562e-05
-Refinement cycle: 7
-   Number of degrees of freedom=8672
-   Point value=0.0334405
-   Estimated error=6.46583e-06
-Refinement cycle: 8
-   Number of degrees of freedom=21118
-   Point value=0.0334443
-
- -

- -

-First let's look what the program actually computed. On the fifth -grid, primal and dual numerical solutions look like this: - - - - - - -
- Primal solution - - - Dual solution - -
-Obviously, the region at the bottom left is so unimportant for the -point value evaluation at the top right that the grid is left entirely -unrefined there, even though the solution has singularities there! Due -to the symmetry in right hand side and domain, the solution should -actually look like at the top right in all four corners, but the mesh -refinement criterion involving the dual solution chose to refine them -differently. -

- -

-Looking at the grids that are produced in the course of subsequent -refinement, these are some of them: - - - - - - - - - - - - - - - - - - -
- - Grid 0 - - - - Grid 2 - -
- - Grid 4 - - - - Grid 5 - -
- - Grid 7 - - - - Grid 8 - -
-Note the subtle interplay between resolving the corner singularities, -and resolving around the point of evaluation. It will be rather -difficult to generate such a mesh by hand, as this would involve to -judge quantitatively how much which of the four corner singularities -shall be resolved, and to set the weight compared to the vicinity of -the evaluation point. -

- -

-The program prints the point value and the estimated error in this -quantity. From extrapolating it, we can guess that the exact value is -somewhat like 0.0334473, plus or minus 0.0000001 (note that we get -almost 6 valid digits from only 22,000 (primal) degrees of -freedom. This number cannot be obtained from the value of the -functional alone, but I have used the assumption that the error -estimator is mostly exact, and extrapolated the computed value plus -the estimated error, to get an approximation of the true -value. Computing with more degrees of freedom shows that this -assumption is indeed valid. -

- -

-From the computed results, we can generate two graphs: one that shows -the convergence of the error J(u)-J(uh) (taking the -extrapolated value as correct) in the point value, and the value that -we get by adding up computed value J(uh) and estimated -error eta (if the error estimator eta were exact, then the value -J(uh)+eta would equal the exact point value, and the error -in this quantity would always be zero; however, since the error -estimator is only a - good - approximation to the true error, we can -by this only reduce the size of the error). In this graph, we also -indicate the complexity O(1/N) to show that mesh refinement -acts optimal in this case. The second chart compares -true and estimated error, and shows that the two are actually very -close to each other, even for such a complicated quantity as the point -value: -

- - - - - - - -
- - Error in point value - - - - Error in point value - -
- - -

Comparing refinement criteria

- -

-Since we have accepted quite some effort when using the mesh -refinement driven by the dual weighted error estimator (for solving -the dual problem, and for evaluating the error representation), it is -worth while asking whether that effort was successful. To this end, we -first compare the achieved error levels for different mesh refinement -criteria. To generate this data, simply change the value of the mesh -refinement criterion variable in the main program. The results are -thus (for the weight in the Kelly indicator, we have chosen the -function 1/(r2+0.12), where r -is the distance to the evaluation point; it can be shown that this is -the optimal weight if we neglect the effects of boundaries): -

-

- Error comparison - -

- -

-Checking these numbers, we see that for global refinement, the error -is proportional to O(1/(sqrt(N) log(N))), and for the dual -estimator O(1/N). Generally speaking, we see that the dual -weighted error estimator is better than the other refinement -indicators, at least when compared with those that have a similarly -regular behavior. The Kelly indicator produces smaller errors, but -jumps about the picture rather irregularly, with the error also -changing signs sometimes. Therefore, its behavior does not allow to -extrapolate the results to larger values of N. Furthermore, if we -trust the error estimates of the dual weighted error estimator, the -results can be improved by adding the estimated error to the computed -values. In terms of reliability, the weighted estimator is thus better -than the Kelly indicator, although the latter sometimes produces -smaller errors. -

- - -

Evaluation of point stresses

- -

-Besides evaluating the values of the solution at a certain point, the -program also offers the possibility to evaluate the x-derivatives at a -certain point, and also to tailor mesh refinement for this. To let the -program compute these quantities, simply replace the two occurences of -PointValueEvaluation in the main function by -PointXDerivativeEvaluation, and let the program run: - -

-Refinement cycle: 0
-   Number of degrees of freedom=72
-   Point x-derivative=-0.0719397
-   Estimated error=-0.0126173
-Refinement cycle: 1
-   Number of degrees of freedom=61
-   Point x-derivative=-0.0707956
-   Estimated error=-0.00774316
-Refinement cycle: 2
-   Number of degrees of freedom=131
-   Point x-derivative=-0.0568671
-   Estimated error=-0.00313426
-Refinement cycle: 3
-   Number of degrees of freedom=247
-   Point x-derivative=-0.053033
-   Estimated error=-0.00136114
-Refinement cycle: 4
-   Number of degrees of freedom=541
-   Point x-derivative=-0.0526461
-   Estimated error=-0.000555479
-Refinement cycle: 5
-   Number of degrees of freedom=1286
-   Point x-derivative=-0.0526896
-   Estimated error=-0.0002261
-Refinement cycle: 6
-   Number of degrees of freedom=2924
-   Point x-derivative=-0.0527503
-   Estimated error=-9.38035e-05
-Refinement cycle: 7
-   Number of degrees of freedom=6578
-   Point x-derivative=-0.0527877
-   Estimated error=-3.94139e-05
-Refinement cycle: 8
-   Number of degrees of freedom=14780
-   Point x-derivative=-0.0528047
-   Estimated error=-1.85456e-05
-Refinement cycle: 9
-   Number of degrees of freedom=31438
-   Point x-derivative=-0.0528145
-
- -

- -

-The solution looks roughly the same as before (the exact solution of -course is the same, only the grid changed a little), but the -dual solution is now different. A close-up around the point of -evaluation shows this: - - - -
- Dual solution - -
-This time, the grids in refinement cycles 0, 5, 6, 7, 8, and 9 look -like this: - - - - - - - - - - - - - - - - - - -
- - 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 a reference value of -0.0528223... plus or minus -0.0000005. 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 derivative - - - - Error in point derivative - -
-After an initial phase where the true error changes its sign, the -estimated error matches it quite well, again. Also note the dramatic -improvement in the error when using the estimated error to correct the -computed value of J(uh). -

- - -

Step-13 revisited

- -

-If instead of the Exercise_2_3 data set, we choose -CurvedRidges in the main function, we can redo the -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. -

- -

-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. -

- - -

-The next point is to compare the new (duality based) mesh refinement -criterion with the old ones. These are the results: -

-

- Error comparison - -

- -

-The results are, well, somewhat mixed. First, the Kelly indicator -disqualifies itself by its unsteady behavior, changing the sign of the -error several times, and with increasing errors under mesh -refinement. The dual weighted error estimator has a monotone decrease -in the error, and is better than the weighted Kelly and global -refinement, but the margin is not as large as expected. This is, here, -due to the fact the global refinement can exploit the regular -structure of the meshes around the point of evaluation, which leads to -a better order of convergence for the point error. However, if we had -a mesh that is not locally rectangular, for example because we had to -approximate curved boundaries, or if the coefficients were not -constant, then this advantage of globally refinement meshes would -vanish, while the good performance of the duality based estimator -would remain. -

- - - -

Conclusions and outlook

- -

-The results here are not too clearly indicating the superiority of the -dual weighted error estimation approach for mesh refinement over other -mesh refinement criteria, such as the Kelly indicator. This is due to -the relative simplicity of the shown applications. If you are not -convinced yet that this approach is indeed superior, you are invited -to browse through the literature indicated in the introduction, where -plenty of examples are provided where the dual weighted approach can -reduce the necessary numerical work by orders of magnitude, making -this the only way to compute certain quantities to reasonable -accuracies at all. -

- -

-Besides the objections you may raise against its use as a mesh -refinement criterion, consider that accurate knowledge of the error in -the quantity one might want to compute is of great use, since we can -stop computations when we are satisfied with the accuracy. Using more -traditional approaches, it is very difficult to get accurate estimates -for arbitrary quantities, except for, maybe, the error in the energy -norm, and we will then have no guarantee that the result we computed -satisfies any requirements on its accuracy. Also, as was shown for the -evaluation of point values and derivatives, the error estimate can be -used to extrapolate the results, yielding much higher accuracy in the -quantity we want to know. -

- -

-Leaving these mathematical considerations, we tried to write the -program in a modular way, such that implementing another test case, or -another evaluation and dual functional is simple. You are encouraged -to take the program as a basis for your own experiments, and to play a -little. -

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