## 3 Results
### System Tests
e^{-\hat{\mu}_{\mathbf{\xi}}\mathbf{k}_{\mathbf{p}}\cdot\mathbf{r}} \tag{21}\]  
where \(\mathbf{r}\) is the position vector, \(\mathbf{k}_{\mathbf{p}}=-\sin(\theta)\cos(\phi)\hat{\mathbf{x}}-\sin(\theta)\sin(\phi) \hat{\mathbf{y}}-\cos(\theta)\hat{\mathbf{z}}\), with \(\hat{\mathbf{x}}\), \(\hat{\mathbf{y}}\), \(\hat{\mathbf{z}}\) the unit vectors in the Cartesian axis. The working frequency is set to 50 MHz. The results shown in the document correspond to \((\theta,\phi)=(\frac{\pi}{2},\frac{\pi}{4})\). Similar results, i.e., the same slope of the error, are obtained with different angles of incidence. In Table 3, we show the relation between the unknowns, the number of elements in the mesh, the maximum discretization size, and the value for \(\mathbf{e}_{\text{field}}\) in two setups of the grid, i.e., \(1\times 1\) and \(1\times 2\). As expected, the error is the same for both cases, as we shouldexpect from Table 1. Figure 6 shows the evolution of the error in log-log scale. The slope obtained is 1.9582 and 1.9141 for \(\mathbf{e}_{\text{field}}\) and \(\mathbf{e}_{\text{rot}}\), respectively, with both slopes being very close to the predicted result of 2. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Monomials** & **Full Mesh** & **9 Compl.** & **1 Compl.** \\ \hline \([1,0,0]\) & \(5.24\times 10^{-12}\) & \(5.32\times 10^{-12}\) & \(4.64\times 10^{-12}\) \\ \([0,1,0]\) & \(4.74\times 10^{-12}\) & \(4.52\times 10^{-12}\) & \(4.56\times 10^{-12}\) \\ \([0,0,1]\) & \(5.34\times 10^{-12}\) & \(4.91\times 10^{-12}\) & \(5.01\times 10^{-12}\) \\ \([x,0,0]\) & \(5.33\times 10^{-12}\) & \(5.28\times 10^{-12}\) & \(4.90\times 10^{-12}\) \\ \([y,0,0]\) & \(5.27\times 10^{-12}\) & \(5.37\times 10^{-12}\) & \(5.14\times 10^{-12}\) \\ \([z,0,0]\) & \(5.30\times 10^{-12}\) & \(4.58\times 10^{-12}\) & \(4.72\times 10^{-12}\) \\ \([0,x,0]\) & \(5.24\times 10^{-12}\) & \(5.22\times 10^{-12}\) & \(4.87\times 10^{-12}\) \\ \([0,y,0]\) & \(5.13\times 10^{-12}\) & \(5.03\times 10^{-12}\) & \(4.53\times 10^{-12}\) \\ \([0,z,0]\) & \(4.63\times 10^{-12}\) & \(5.05\times 10^{-12}\) & \(5.26\times 10^{-12}\) \\ \([0,0,x]\) & \(5.45\times 10^{-12}\) & \(4.74\times 10^{-12}\) & \(4.85\times 10^{-12}\) \\ \([0,0,y]\) & \(4.96\times 10^{-12}\) & \(4.58\times 10^{-12}\) & \(4.51\times 10^{-12}\) \\ \([0,0,z]\) & \(5.29\times 10^{-12}\) & \(5.39\times 10^{-12}\) & \(5.38\times 10^{-12}\) \\ \([y^{2},-xy,0]\) & \(5.37\times 10^{-12}\) & \(4.75\times 10^{-12}\) & \(5.32\times 10^{-12}\) \\ \([0,-yz,y^{2}]\) & \(5.48\times 10^{-12}\) & \(4.79\times 10^{-12}\) & \(4.67\times 10^{-12}\) \\ \([xy,-x^{2},0]\) & \(4.67\times 10^{-12}\) & \(4.75\times 10^{-12}\) & \(5.41\times 10^{-12}\) \\ \([xz,0,-x^{2}]\) & \(4.81\times 10^{-12}\) & \(4.55\times 10^{-12}\) & \(5.42\times 10^{-12}\) \\ \([z^{2},0,-xz]\) & \(4.83\times 10^{-12}\) & \(5.15\times 10^{-12}\) & \(4.52\times 10^{-12}\) \\ \([0,z^{2},-xz]\) & \(4.54\times 10^{-12}\) & \(5.24\times 10^{-12}\) & \(4.62\times 10^{-12}\) \\ \([yz,-xz,0]\) & \(5.11\times 10^{-12}\) & \(4.68\times 10^{-12}\) & \(5.41\times 10^{-12}\) \\ \([0,xz,-xy]\) & \(5.31\times 10^{-12}\) & \(4.88\times 10^{-12}\) & \(4.77\times 10^{-12}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Value of \(\mathbf{e}_{\text{field}}\) for all the monomials within the space of second-order tetrahedra basis functions. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Number of Elements** & **Degrees of Freedom** & **Discretization Size (m)** & \(\mathbf{1\times 1}\) & \(\mathbf{1\times 2}\) \\ \hline
617 & 4414 & 0.5 & \(3.44\cdot 10^{-3}\) & \(3.44\cdot 10^{-3}\) \\
1236 & 8644 & 0.4 & \(2.38\cdot 10^{-3}\) & \(2.38\cdot 10^{-3}\) \\
9773 & 65,122 & 0.2 & \(5.85\cdot 10^{-4}\) & \(5.85\cdot 10^{-4}\) \\
23,452 & 154,544 & 0.15 & \(3.24\cdot 10^{-4}\) & \(3.24\cdot 10^{-4}\) \\
36,060 & 236,300 & 0.13 & \(2.45\cdot 10^{-4}\) & \(2.45\cdot 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: \(\mathbf{e}_{\text{field}}\) for the plane wave using five meshes.