<h1>Results</h1>
+We run the program with a right hand side that will produce the
+solution $u = \sin(\pi x) \sin(\pi y)$ and with clamped
+boundary conditions in the domain $\Omega = (0,1)^2$.
+We test this setup using $Q_2$, $Q_3$, and $Q_4$ elements, which one can
+change `fe\_degree` in `main()`. With mesh
+refinement, the $L_2$ convergence rates, $H_1$-seminorm convergence
+and $H_2$-seminorm convergence of $u$
+should then be around 2, 2, 1 for $Q_2$ , 4, 3, 2 for
+$Q_3$, and 5, 4, 3 for $Q_4$ separately.
+We use different penalties $\eta = 1$, $2$, and $p(p+1)$ where $p$
+is the degree of polynomials,
+and compare convergence rates of numerical solutions computed by these
+penalties.
+
+
+<h3>Test results on <i>Q<sub>2</sub></i> with <i>\eta = p(p+1)</i> </h3>
+
+<h4>Convergence table</h4>
+
+We run the code with differently refined meshes
+and get the following convergence rates.
+
+<table align="center" class="doxtable">
+ <tr>
+ <th>number of refinements </th><th> $\|u-u_h^\circ\|_{L_2}$ </th><th> Conv. rates </th><th> $|u-u_h|_{H_1}$ </th><th> Conv. rates </th><th> $|u-u_h|_{H_2}$ </th><th> Conv. rates </th>
+ </tr>
+ <tr>
+ <td> 2 </td><td> 1.539e-02 </td><td> </td><td> 8.528e-02 </td><td> </td><td> 1.602 </td><td> </td>
+ </tr>
+ <tr>
+ <td> 3 </td><td> 4.563e-03 </td><td> 1.75 </td><td> 2.408e-02 </td><td> 1.82 </td><td> 7.965e-01 </td><td> 1.00 </td>
+ </tr>
+ <tr>
+ <td> 4 </td><td> 1.250e-03 </td><td> 1.86 </td><td> 6.438e-03 </td><td> 1.90 </td><td> 3.969e-01 </td><td> 1.00 </td>
+ </tr>
+ <tr>
+ <td> 5 </td><td> 3.277e-04 </td><td> 1.93 </td><td> 1.666e-03 </td><td> 1.94 </td><td> 1.981e-01 </td><td> 1.00 </td>
+ </tr>
+</table>
+We can see that the $L_2$ convergence rates are around 2,
+$H_1$-seminorm convergence rates are around 2,
+and $H_2$-seminorm convergence rates are around 1.
+
+<h3>Test results on <i>Q<sub>3</sub></i> with <i>\eta = p(p+1)</i> </h3>
+
+<h4>Convergence table</h4>
+
+
+<table align="center" class="doxtable">
+ <tr>
+ <th>number of refinements </th><th> $\|u-u_h^\circ\|_{L_2}$ </th><th> Conv. rates </th><th> $|u-u_h|_{H_1}$ </th><th> Conv. rates </th><th> $|u-u_h|_{H_2}$ </th><th> Conv. rates </th>
+ </tr>
+ <tr>
+ <td> 2 </td><td> 2.187e-04 </td><td> </td><td> 4.46269e-03 </td><td> </td><td> 1.638e-01 </td><td> </td>
+ </tr>
+ <tr>
+ <td> 3 </td><td> 1.334e-05 </td><td> 4.03 </td><td> 5.54622e-04 </td><td> 3.00 </td><td> 4.095e-02 </td><td> 2.00 </td>
+ </tr>
+ <tr>
+ <td> 4 </td><td> 8.273e-07 </td><td> 4.01 </td><td> 6.90599e-05 </td><td> 3.00 </td><td> 1.023e-02 </td><td> 2.00 </td>
+ </tr>
+ <tr>
+ <td> 5 </td><td> 5.164e-08 </td><td> 4.00 </td><td> 8.62168e-06 </td><td> 3.00 </td><td> 2.558e-03 </td><td> 2.00 </td>
+ </tr>
+</table>
+We can see that the $L_2$ convergence rates are around 4,
+$H_1$-seminorm convergence rates are around 3,
+and $H_2$-seminorm convergence rates are around 2.
+This, of course, matches our theoretical expectations.
+
+<h3>Test results on <i>Q<sub>4</sub></i> with <i>\eta = p(p+1)</i> </h3>
+
+<h4>Convergence table</h4>
+
+
+<table align="center" class="doxtable">
+ <tr>
+ <th>number of refinements </th><th> $\|u-u_h^\circ\|_{L_2}$ </th><th> Conv. rates </th><th> $|u-u_h|_{H_1}$ </th><th> Conv. rates </th><th> $|u-u_h|_{H_2}$ </th><th> Conv. rates </th>
+ </tr>
+ <tr>
+ <td> 2 </td><td> 8.34446e-06 </td><td> </td><td> 0.000239323 </td><td> </td><td> 0.0109785 </td><td> </td>
+ </tr>
+ <tr>
+ <td> 3 </td><td> 2.98497e-07 </td><td> 4.80 </td><td> 1.63221e-05 </td><td> 3.87 </td><td> 0.0013551 </td><td> 3.01 </td>
+ </tr>
+ <tr>
+ <td> 4 </td><td> 9.87063e-09 </td><td> 4.91 </td><td> 1.06066e-06 </td><td> 3.94 </td><td> 0.000167898 </td><td> 3.01 </td>
+ </tr>
+ <tr>
+ <td> 5 </td><td> 7.88939e-10 </td><td> 3.64 </td><td> 6.75478e-08 </td><td> 3.97 </td><td> 2.08912e-05 </td><td> 3.00 </td>
+ </tr>
+</table>
+We can see that the $L_2$ norm convergence rates are around 5,
+$H_1$-seminorm convergence rates are around 4,
+and $H_2$-seminorm convergence rates are around 3.
+On the finest mesh, the $L_2$ norm convergence rate
+is much smaller than our theoretical expectations
+because the linear solver becomes the limiting factor due
+to round-off. But the $L_2$ error is pretty small in that case.
+
+<h3>Test results on <i>Q<sub>2</sub></i> with <i>\eta = 1</i> </h3>
+
+<h4>Convergence table</h4>
+
+
+<table align="center" class="doxtable">
+ <tr>
+ <th>number of refinements </th><th> $\|u-u_h^\circ\|_{L_2}$ </th><th> Conv. rates </th><th> $|u-u_h|_{H_1}$ </th><th> Conv. rates </th><th> $|u-u_h|_{H_2}$ </th><th> Conv. rates </th>
+ </tr>
+ <tr>
+ <td> 2 </td><td> 4.86048e-02 </td><td> </td><td> 3.30386e-01 </td><td> </td><td> 4.34917 </td><td> </td>
+ </tr>
+ <tr>
+ <td> 3 </td><td> 1.29921e-02 </td><td> 1.90 </td><td> 1.4852e-01 </td><td> 1.15 </td><td> 4.01192 </td><td> 0.116 </td>
+ </tr>
+ <tr>
+ <td> 4 </td><td> 3.33539e-03 </td><td> 1.96 </td><td> 7.20252e-02 </td><td> 1.04 </td><td> 3.96138 </td><td> 0.018 </td>
+ </tr>
+ <tr>
+ <td> 5 </td><td> 8.41058e-04 </td><td> 1.98 </td><td> 3.57705e-02 </td><td> 1.00 </td><td> 3.95719 </td><td> 0.001 </td>
+ </tr>
+</table>
+Although $L_2$ norm and $H_1$-seminorm convergence rates of $u$
+follow the theoretical expectations, $H_2$-seminorm does not converge.
+Comparing results from $\eta = 1$ and $\eta = p(p+1)$,
+$\eta = p(p+1)$ is a better penalty.
+
+<h3>Test results on <i>Q<sub>2</sub></i> with <i>\eta = 2</i> </h3>
+
+<h4>Convergence table</h4>
+
+
+<table align="center" class="doxtable">
+ <tr>
+ <th>number of refinements </th><th> $\|u-u_h^\circ\|_{L_2}$ </th><th> Conv. rates </th><th> $|u-u_h|_{H_1}$ </th><th> Conv. rates </th><th> $|u-u_h|_{H_2}$ </th><th> Conv. rates </th>
+ </tr>
+ <tr>
+ <td> 2 </td><td> 5.482e-03 </td><td> </td><td> 7.652e-02 </td><td> </td><td> 1.756e-00 </td><td> </td>
+ </tr>
+ <tr>
+ <td> 3 </td><td> 2.227e-02 </td><td> 1.29 </td><td> 2.177e-02 </td><td> 1.81 </td><td> 8.711e-01 </td><td> 1.01 </td>
+ </tr>
+ <tr>
+ <td> 4 </td><td> 9.088e-04 </td><td> 1.29 </td><td> 6.026e-03 </td><td> 1.85 </td><td> 4.196e-01 </td><td> 1.05 </td>
+ </tr>
+ <tr>
+ <td> 5 </td><td> 2.822e-04 </td><td> 1.68 </td><td> 1.605e-03 </td><td> 1.90 </td><td> 2.041e-01 </td><td> 1.03 </td>
+ </tr>
+</table>
+In the case, all convergence rates of $u$
+follow the theoretical expectations.
+But compared to the results from $\eta = p(p+1)$,
+it does not show a good convergence on $L_2$ errors.
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