## 4 Numerical Examples
Conductivity of the earth layer is \(\sigma_{2}=0.01\) S/m, with permittivity \(\epsilon_{2}=4\epsilon_{0}\) and permeability \(\mu_{2}=\mu_{0}\). The conductivity of the mine layer is \(\sigma_{3}=0.001\) S/m, and the permittivity and permeability of the mine are \(\epsilon_{3}=3\epsilon_{0}\) and \(\mu_{3}=\mu_{0}\), respectively. Media interfaces \(S_{0}\), \(S_{1}\) and \(S_{2}  
Figure 6: Electric field in the \(x\)-direction component at test point \((0,y,-500)\). Figure 7: Electric field in the \(z\)-direction component at test point \((x,0,-500)\). The proposed method is used to simulate the electromagnetic fields excited by horizontal electric dipole in a dissipative sphere, which is a closed structure coated by layered materials and is shown in Figure 9. The external sphere is located at origin and has a radius of 6 m with conductivity \(\sigma_{1}=0.001\) S/m, permittivity \(\varepsilon_{1}=3.1\varepsilon_{0}\) and permeability \(\mu_{1}=\mu_{0}\). The radius of the middle sphere is 3 m with conductivity \(\sigma_{2}=0.01\) S/m, permittivity \(\varepsilon_{2}=2\varepsilon_{0}\) and permeability \(\mu_{2}=\mu_{0}\), which is located at point \((-1,0,0)\). The internal sphere is located at origin and has a radius of 1 m, where the conductivity is \(\sigma_{3}=0.1\) S/m, permittivity is \(\varepsilon_{3}=2\varepsilon_{0}\), and permeability is \(\mu_{3}=\mu_{0}\). A horizontal electric dipole excitation source is located at origin with dipole moment of 1 A \(\cdot\cdot\) m along \(x\)-axis, and \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(y\)-Axis (m) & **45** & **95** & **695** & **745** & **845** \\ \hline CST(dB V/m) & \(-122.09\) & \(-123.97\) & \(-173.81\) & \(-178.06\) & \(-185.37\) \\ SIE(dB V/m) & \(-122.13\) & \(-124.10\) & \(-174.02\) & \(-177.79\) & \(-185.13\) \\ Error \% & \(-0.51\) & \(-1.4\) & \(-2.4\) & \(3.2\) & \(2.7\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Electric field in the \(x\)-direction component at test point \((0,y,-500)\). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(x\)-Axis (m) & **70** & **170** & **720** & **770** & **870** \\ \hline CST(dB A/m) & \(-141.25\) & \(-146.66\) & \(-187.65\) & \(-191.27\) & \(-198.01\) \\ SIE(dB A/m) & \(-141.34\) & \(-146.67\) & \(-187.76\) & \(-191.20\) & \(-198.05\) \\ Error \% & \(-1.1\) & \(0.1\) & \(-1.3\) & \(0.8\) & \(-0.4\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Magnetic field in the \(y\)-direction component at test point \((x,0,-500)\). Figure 8: Magnetic field in the \(y\)-direction component at test point \((x,0,-500)\). the frequency is 1 MHz. After discretization, the numbers of triangles on the external sphere \(S_{0}\), the middle sphere \(S_{1}\) and the internal sphere \(S_{2}\) are 1204, 604 and 626, respectively. As a result, 7302 unknowns are generated. It takes 54.3 s and 0.66 GB of memory to solve the matrix equation in numerical calculations. Compared with the proposed method, the calculation of the same model with \(923,446\) unknowns using CST takes 150 s and 1.9 GB of memory, which shows that the proposed method has a good performance in solving time and storage space. Figure 10 shows the near magnetic field component \(H_{z}\) as a function of the observation angle in plane \(z=0\), and Figure 11 illustrates the near electric field component \(E_{y}\), which are compared with CST. We can observe that the numerical results obtained from the proposed method are in good agreement with the results obtained by CST. Figure 12 gives the root-mean-square (RMS) error of different observation radius. The Root-mean-square (RMS) error is given by [28]:  
\[\text{RMS}(x)=\sqrt{\frac{\sum\limits_{i=1}^{N}\left|x^{\text{SIE}}(\mathbf{r})-x^ {\text{CST}}(\mathbf{r})\right|^{2}}{\sum\limits_{i=1}^{N}\left|x^{\text{CST}}(\mathbf{r })\right|^{2}}} \tag{40}\]  
where \(x^{\text{SIE}}\) is the calculated quantity and \(x^{\text{CST}}\) refers to the corresponding CST result. We can see that the \(H_{z}\) RMS error is smaller than that of \(E_{y}\) of each observation radius.