<h1>Results</h1>
-<h2>Point values</h2>
+<h3>Point values</h3>
This program offers a lot of possibilities to play around. We can thus
</table>
-<h2>Comparing refinement criteria</h2>
+<h3>Comparing refinement criteria</h3>
Since we have accepted quite some effort when using the mesh
-<h2>Evaluation of point stresses</h2>
+<h3>Evaluation of point stresses</h3>
Besides evaluating the values of the solution at a certain point, the
-<h2>Step-13 revisited</h2>
+<h3>Step-13 revisited</h3>
If instead of the <code>Exercise_2_3</code> data set, we choose
-<h2>Conclusions and outlook</h2>
+<h3>Conclusions and outlook</h3>
The results here are not too clearly indicating the superiority of the
<a name="Intro"></a>
<h1>Introduction</h1>
-<h2>Overview</h2>
+<h3>Overview</h3>
This program does not introduce any new mathematical ideas; in fact, all it
does is to do the exact same computations that step-8
single machine if PETSc was configured without MPI.
-<h2>Parallelizing software with MPI</h2>
+<h3>Parallelizing software with MPI</h3>
Developing software to run in %parallel via MPI requires a bit of a change in
mindset because one typically has to split up all data structures so that
philosophy behind MPI as outlined above.
-<h2>What this program does</h2>
+<h3>What this program does</h3>
The techniques this program then demonstrates are:
- How to use the PETSc wrapper classes; this will already be visible in the
<h1>Results</h1>
-<h2>Running the problem</h2>
+<h3>Running the problem</h3>
The problem's input is parameterized by an input file <code>\step-36.prm</code>
which could, for example, contain the following text:
<h1>Results</h1>
-<h2>Logfile output</h2>
+<h3>Logfile output</h3>
First, the program produces the usual logfile here stored in <tt>deallog</tt>. It reads (with omission of intermediate steps)
@code
This log for instance shows that the number of conjugate gradient
iteration steps is constant at approximately 15.
-<h2>Postprocessing of the logfile</h2>
+<h3>Postprocessing of the logfile</h3>
<img src="https://www.dealii.org/images/steps/developer/step-39-convergence.svg" alt="">
Using the perl script <tt>postprocess.pl</tt>, we extract relevant
remainder.
-<h2> What's the issue? </h2>
+<h3> What's the issue? </h3>
The fundamental issue with the equation is that it takes four
derivatives of the solution. In the case of the Laplace equation
does not contain implementations of these shape functions.
-<h2> What to do instead? </h2>
+<h3> What to do instead? </h3>
So how does one approach solving such problems then? That depends a
bit on the boundary conditions. If one has the first set of boundary
the book chapter @cite Brenner2011 .
-<h2>The testcase</h2>
+<h3>The testcase</h3>
The last step that remains to describe is what this program solves
for. As always, a trigonometric function is both a good and a bad
uses as currently written.
-<h2> Possibilities for extensions </h2>
+<h3> Possibilities for extensions </h3>
There are a number of obvious extensions to this program that would
make sense:
provides resources to mesh generators.
-<h2>Where geometry and meshes intersect</h2>
+<h3>Where geometry and meshes intersect</h3>
Let us assume that you have a complex domain and that you already have a
coarse mesh that somehow represents the general features of the domain. Then
we will do in this program.
-<h2>The example case</h2>
+<h3>The example case</h3>
To illustrate how one describes geometries using charts in deal.II, we will
consider a case that originates in an application of the <a
in the program.
-<h2>Implementation</h2>
+<h3>Implementation</h3>
There are a number of issues we need to address in the program. At the largest scale,
we need to write a class that implements the interface of ChartManifold. This involves
and you would obtain exactly the same results as in test case 1 below.
-<h2> Test case 1: </h2>
+<h3> Test case 1: </h3>
For the default problem the value of $u$ on $\Gamma$ is set to the constant $1$:
this is like imposing a constant Dirichlet boundary condition on $\Gamma$, seen
To play around a little bit, we are going to complicate a little the fictitious
domain as well as the boundary conditions we impose on it.
-<h2> Test case 2 and 3: </h2>
+<h3> Test case 2 and 3: </h3>
If we use the following parameter file:
@code