step-95.h

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, fe_degree);
 * @endcode
 * 
 * Then we do a second run with discontinuous elements (FE_DGQ).
 * 
 * @code
 *     poisson_solver.run(true /* is_dg */, fe_degree);
 *   }
 * @endcode
<a name="step_95-Results"></a><h1>Results</h1>
 
 
<a name="step_95-Convergencestudy"></a><h3>Convergence study</h3>
 
 
The results of the convergence studies are shown in the tables below.
 
For @f$p=1@f$ we get (for continuous elements, FE_Q):
 
| Cycle | Mesh size | @f$L^2@f$-Error | EOC  |
|:-----:|:---------:|:-----------:|:----:|
|   0   |  0.3025   |  8.3792e-02 |  -   |
|   1   |  0.1513   |  1.9073e-02 | 2.14 |
|   2   |  0.0756   |  4.2112e-03 | 2.18 |
|   3   |  0.0378   |  9.4586e-04 | 2.15 |
 
We see that the @f$L^2@f$ error decreases as we refine and that the estimated
order of convergence, EOC, is close to 2.
 
For @f$p=2@f$ we get (again for continuous elements):
 
| Cycle | Mesh size | @f$L^2@f$-Error | EOC  |
|:-----:|:---------:|:-----------:|:----:|
|   0   |  0.3025   |  1.2442e-04 |  -   |
|   1   |  0.1513   |  9.6514e-06 | 3.69 |
|   2   |  0.0756   |  8.3693e-07 | 3.53 |
|   3   |  0.0378   |  7.4062e-08 | 3.50 |
 
If we increase the degree of the finite element for both the level-set and
the solution we observe EOC close to 3.
 
We repeat the experiment for for discontinuous elements (FE_DGQ), where we get for @f$p=1@f$:
 
| Cycle | Mesh size | @f$L^2@f$-Error | EOC  |
|:-----:|:---------:|:-----------:|:----:|
|   0   |  0.3025   | 7.8706e-02  |  -   |
|   1   |  0.1513   | 1.8413e-02  | 2.10 |
|   2   |  0.0756   | 4.0863e-03  | 2.17 |
|   3   |  0.0378   | 9.3191e-04  | 2.13 |
 
For @f$p=2@f$ we get (again for discontinuous elements):
 
| Cycle | Mesh size | @f$L^2@f$-Error | EOC  |
|:-----:|:---------:|:-----------:|:----:|
|   0   |  0.3025   | 1.5622e-04  |  -   |
|   1   |  0.1513   | 1.0739e-05  | 3.86 |
|   2   |  0.0756   | 1.0044e-06  | 3.42 |
|   3   |  0.0378   | 7.9919e-08  | 3.65 |
 
We observe optimal order convergence also for DG.
 
<a name="step_95-Possibilitiesforextension"></a><h3>Possibilities for extension</h3>
 
 
This tutorial could be easily extended to time dependent problems like the heat equation by adding a mass operator and a
time integration scheme in the solver class.
 
Another option would be to implement a different ghost penalty like the volume based ghost penalty to allow
stabilization for arbitrary polynomial degree of the shape functions.
 
Also, the preconditioner could be extended to a more robust scheme like multigrid.
 
If you are interested in the mentioned possibilities for extensions we recommend our paper "High-performance matrix-free
unfitted finite element operator evaluation" @cite bergbauer2024 where volume ghost-penalty and multigrid preconditioning  already have been
addressed.
 *
 *
<a name="step_95-PlainProg"></a>
<h1> The plain program</h1>
@include "step-95.cc"
*/