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Merge pull request #9495 from bangerth/14
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Convert step-14 from using eps/gnuplot to svg/vtu.
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@kronbichler
kronbichler committed Feb 12, 2020
2 parents 00fed5c + 7ff9701 commit d4e101d
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86 changes: 49 additions & 37 deletions examples/step-14/doc/results.dox
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Original file line number Diff line number Diff line change
Expand Up @@ -47,56 +47,61 @@ Refinement cycle: 8
@endcode


First let's look what the program actually computed. On the fifth
grid, primal and dual numerical solutions look like this:
First let's look what the program actually computed. On the seventh
grid, primal and dual numerical solutions look like this (using a
color scheme intended to evoke the snow-capped mountains of
Colorado that the original author of this program now calls
home):
<table align="center">
<tr>
<td width="50%">
<img src="https://www.dealii.org/images/steps/developer/step-14.point-value.solution-5.png" alt="">
<img src="https://www.dealii.org/images/steps/developer/step-14.point-value.solution-7.9.2.png" alt="">
</td>
<td width="50%">
<img src="https://www.dealii.org/images/steps/developer/step-14.point-value.solution-5-dual.png" alt="">
<img src="https://www.dealii.org/images/steps/developer/step-14.point-value.solution-7-dual.9.2.png" alt="">
</td>
</tr>
</table>
Obviously, the region at the bottom left is so unimportant for the
Apparently, 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
unrefined there, even though the solution has singularities at the inner
corner of that cell! 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.
differently -- because we said that we really only care about a single
function value somewhere at the top right.



Looking at the grids that are produced in the course of subsequent
refinement, here are some of them:
Here are some of the meshes that are produced in refinement cycles 0,
2, 4 (top row), and 5, 7, and 8 (bottom row):

<table width="80%" align="center">
<tr>
<td><img src="https://www.dealii.org/images/steps/developer/step-14.point-value.grid-0.png" alt="" width="100%"></td>
<td><img src="https://www.dealii.org/images/steps/developer/step-14.point-value.grid-2.png" alt="" width="100%"></td>
<td><img src="https://www.dealii.org/images/steps/developer/step-14.point-value.grid-4.png" alt="" width="100%"></td>
<td><img src="https://www.dealii.org/images/steps/developer/step-14.point-value.grid-0.9.2.png" alt="" width="100%"></td>
<td><img src="https://www.dealii.org/images/steps/developer/step-14.point-value.grid-2.9.2.png" alt="" width="100%"></td>
<td><img src="https://www.dealii.org/images/steps/developer/step-14.point-value.grid-4.9.2.png" alt="" width="100%"></td>
</tr>
<tr>
<td><img src="https://www.dealii.org/images/steps/developer/step-14.point-value.grid-5.png" alt="" width="100%"></td>
<td><img src="https://www.dealii.org/images/steps/developer/step-14.point-value.grid-7.png" alt="" width="100%"></td>
<td><img src="https://www.dealii.org/images/steps/developer/step-14.point-value.grid-8.png" alt="" width="100%"></td>
<td><img src="https://www.dealii.org/images/steps/developer/step-14.point-value.grid-5.9.2.png" alt="" width="100%"></td>
<td><img src="https://www.dealii.org/images/steps/developer/step-14.point-value.grid-7.9.2.png" alt="" width="100%"></td>
<td><img src="https://www.dealii.org/images/steps/developer/step-14.point-value.grid-8.9.2.png" alt="" width="100%"></td>
</tr>
</table>

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
should 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
somewhere close to 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
Expand Down Expand Up @@ -227,28 +232,33 @@ evaluation shows this:
<table align="center">
<tr>
<td width="50%">
<img src="https://www.dealii.org/images/steps/developer/step-14.point-derivative.solution-5-dual.png" alt="">
<img src="https://www.dealii.org/images/steps/developer/step-14.point-derivative.solution-7-dual.png" alt="">
</td>

<td width="50%">
<img src="https://www.dealii.org/images/steps/developer/step-14.point-derivative.solution-7-dual-close-up.png" alt="">
</td>
</table>
This time, the grids in refinement cycles 0, 5, 6, 7, 8, and 9 look
like this:

<table align="center" width="80%">
<tr>
<td><img src="https://www.dealii.org/images/steps/developer/step-14.point-derivative.grid-0.png" alt="" width="100%"></td>
<td><img src="https://www.dealii.org/images/steps/developer/step-14.point-derivative.grid-5.png" alt="" width="100%"></td>
<td><img src="https://www.dealii.org/images/steps/developer/step-14.point-derivative.grid-6.png" alt="" width="100%"></td>
<td><img src="https://www.dealii.org/images/steps/developer/step-14.point-derivative.grid-0.9.2.png" alt="" width="100%"></td>
<td><img src="https://www.dealii.org/images/steps/developer/step-14.point-derivative.grid-5.9.2.png" alt="" width="100%"></td>
<td><img src="https://www.dealii.org/images/steps/developer/step-14.point-derivative.grid-6.9.2.png" alt="" width="100%"></td>
</tr>
<tr>
<td><img src="https://www.dealii.org/images/steps/developer/step-14.point-derivative.grid-7.png" alt="" width="100%"></td>
<td><img src="https://www.dealii.org/images/steps/developer/step-14.point-derivative.grid-8.png" alt="" width="100%"></td>
<td><img src="https://www.dealii.org/images/steps/developer/step-14.point-derivative.grid-9.png" alt="" width="100%"></td>
<td><img src="https://www.dealii.org/images/steps/developer/step-14.point-derivative.grid-7.9.2.png" alt="" width="100%"></td>
<td><img src="https://www.dealii.org/images/steps/developer/step-14.point-derivative.grid-8.9.2.png" alt="" width="100%"></td>
<td><img src="https://www.dealii.org/images/steps/developer/step-14.point-derivative.grid-9.9.2.png" alt="" width="100%"></td>
</tr>
</table>

Note the asymmetry 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.
the point evaluation. This is due to the fact that the domain and the primal
solution may be symmetric about the diagonal, but the $x$-derivative is
not, and the latter enters the refinement criterion.



Expand Down Expand Up @@ -285,32 +295,34 @@ computed value of $J(u_h)$.


If instead of the <code>Exercise_2_3</code> data set, we choose
<code>CurvedRidges</code> in the main function, we can redo the
<code>CurvedRidges</code> in the main function, and choose $(0.5,0.5)$
as the evaluation point, then 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:
First, the meshes after 9 adaptive refinement cycles obtained with
the point evaluation and derivative evaluation refinement
criteria, respectively, look like this:

<table width="80%" align="center">
<tr>
<td><img src="https://www.dealii.org/images/steps/developer/step-14.step-13.grid-9.png" alt="" width="100%"></td>
<td><img src="https://www.dealii.org/images/steps/developer/step-14.step-13.grid-10.png" alt="" width="100%"></td>
<td><img src="https://www.dealii.org/images/steps/developer/step-14.step-13.point-value.png" alt="" width="100%"></td>
<td><img src="https://www.dealii.org/images/steps/developer/step-14.step-13.point-derivative.png" alt="" width="100%"></td>
</tr>
</table>

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:
The features of the solution can still be seen in the mesh, but since the
solution is smooth, the singularities of the dual solution entirely
dominate the mesh refinement criterion, and lead to strongly
concentrated meshes. The solution after the seventh refinement step looks
like the following:

<table width="80%" align="center">
<tr>
<td><img src="https://www.dealii.org/images/steps/developer/step-14.step-13.solution-7.png" alt="" width="100%"></td>
<td><img src="https://www.dealii.org/images/steps/developer/step-14.step-13.solution-7.9.2.png" alt="" width="100%"></td>
</tr>
</table>

Expand Down
12 changes: 6 additions & 6 deletions examples/step-14/step-14.cc
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Original file line number Diff line number Diff line change
Expand Up @@ -325,8 +325,8 @@ namespace Step14
const Vector<double> & /*solution*/) const
{
std::ofstream out(output_name_base + "-" +
std::to_string(this->refinement_cycle) + ".eps");
GridOut().write_eps(dof_handler.get_triangulation(), out);
std::to_string(this->refinement_cycle) + ".svg");
GridOut().write_svg(dof_handler.get_triangulation(), out);
}
} // namespace Evaluation

Expand Down Expand Up @@ -750,8 +750,8 @@ namespace Step14
data_out.build_patches();

std::ofstream out("solution-" + std::to_string(this->refinement_cycle) +
".gnuplot");
data_out.write(out, DataOutBase::gnuplot);
".vtu");
data_out.write(out, DataOutBase::vtu);
}


Expand Down Expand Up @@ -2100,8 +2100,8 @@ namespace Step14
data_out.build_patches();

std::ofstream out("solution-" + std::to_string(this->refinement_cycle) +
".gnuplot");
data_out.write(out, DataOutBase::gnuplot);
".vtu");
data_out.write(out, DataOutBase::vtu);
}


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