--- /dev/null
+<a name="Intro"></a>
+<h1>Introduction</h1>
+
+This is the first example, where we actually use finite elements. We
+will solve a simple version of Laplace's equation witht zero boundary
+values, but a nonzero right hand side. This example is still quite
+simple, but it already shows the basic structure of most finite
+element programs, which are along the following lines:
+<ul>
+ <li> grid generation;
+ <li> assembling matrices and vectors of the discrete system;
+ <li> solving the linear system of equations;
+ <li> writing results to disk.
+</ul>
--- /dev/null
+<a name="Results"></a>
+<h1>Results</h1>
+
+The output of the program looks as follows:
+<pre>
+<code>
+Number of active cells: 1024
+Total number of cells: 1365
+Number of degrees of freedom: 1089
+DEAL:cg::Starting value 0.121094
+DEAL:cg::Starting value 0.323075
+DEAL:cg::Convergence step 47 value 5.33692e-13
+</code>
+</pre>
+
+The first three lines is what we wrote to <code>cout</code>. The last
+three lines were generated without our interaction by the CG
+solver. The first two lines state the residual at the start of the
+iteration, while the last line tells us that the solver needed 47
+iterations to bring the norm of the residual to 5.3e-13, i.e. below
+the threshold 1e-12 which we have set in the `solve' function. We will
+show in the next program how to suppress this output, which is
+sometimes useful for debugging purposes, but often clutters up the
+screen display.
+
+<p>
+
+Apart from the output shown above, the program generated the file
+<code>solution.gpl</code>, which is in GNUPLOT format. It can be
+viewed as follows: invoke GNUPLOT and enter the following sequence of
+commands at its prompt:
+<pre>
+<code>
+gnuplot> set data style lines
+gnuplot> splot "solution.gpl"
+</code>
+</pre>
+This produces the picture of the solution below left. Alternatively,
+you can order GNUPLOT to do some hidden line removal by the command
+<pre>
+<code>
+gnuplot> set hidden3d
+</code>
+</pre>
+to get the result at the right:
+<p>
+<TABLE WIDTH="100%">
+<tr>
+<td>
+<IMG SRC="step-3.data/solution-1.jpg" ALT="solution-1" WIDTH="400">
+</td>
+<td>
+<IMG SRC="step-3.data/solution-2.jpg" ALT="solution-2" WIDTH="400">
+</td>
+</tr>
+</table>
+<p>
<dl>
<dt><a href="step-1.html">Step 1</a></dt>
- <dd>What's new: everything. We create a grid and show a simple
+ <dd><strong>What's new:</strong> everything. We create a grid and show a simple
way to write it to a file.
</dd>
<dt><a href="step-2.html">Step 2</a></dt>
- <dd>What's new: Associate degrees of freedom to each vertex
+ <dd><strong>What's new:</strong> Associate degrees of freedom to each vertex
and compute the resulting sparsity pattern of matrices. Show
that renumbering reduces the bandwidth of matrices
significantly, i.e. clusters nonzero entries around the
diagonal.
</dd>
+
+ <dt><a href="step-3.html">Step 3</a></dt>
+ <dd><strong>What's new:</strong> Actually solve Laplace's
+ problem. Object-orientation. Assembling matrices and
+ vectors. Boundary values.
+ </dd>
</dl>
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