## 4 Numerical Examples  
In this section, three experiments are constructed to analyze the performance of the proposed method with the vertical or horizontal dipole buried in media including sea, wet and other dissipative materials. The algorithm is implemented by MATLAB 2011b on a computer with CPU i3-4030U working at 1.9 GHz with 4 GB RAM and a Windows 10 operating system. The Low Frequency (LF) Domain Solver of CST Studio Suite in the same computer is used to simulate the models for comparison. We first consider a non-planar air-sea-sealed structure with a hemispherical depression at the seabed shown in Figure 3 for the application that the naval vessel receives signals from the transmitter burried in the seabed. The air permittivity and permeability are \(\varepsilon_{0}\)\(\mu_{0}\), respectively, which are the same as vacuum. Generally, the conductivity of the sea is \(\sigma_{1}=4\) S/m, and the permittivity and permeability of the sea are \(\varepsilon_{1}=74\varepsilon_{0}\) and \(\mu_{1}=\mu_{0}\), respectively. Conductivity of the seabed is \(\sigma_{2}=0.01\) S/m, with permittivity \(\varepsilon_{2}=13\varepsilon_{0}\) and permeability \(\mu_{2}=\mu_{0}\). Media interface \(S_{0}\) and \(S_{1}\) are both circular with radius \(r=50\) m, located at plane \(z=0\) m and \(z=-20\) m, respectively. The hemispherical depression is set at point \((0,0,-20)\) with radius \(r=10\) m. A vertical electric dipole excitation source is located at point \((0,0,-25)\), with dipole moment \(1\) A \(\cdot\) m and the frequency is \(100\) Hz. In order to reduce the mesh amount, we refine the mesh in the source areas and other part is sparse. The number of triangles on the air-sea interface \(S_{0}\) and the sea-seabed interface \(S_{1}\) are \(856\) and \(1528\). As a result, \(7088\) unknowns are generated. In numerical calculations, the processor takes \(49.4\) s and \(0.58\) GB to solve the matrix equation. However, the calculation of the same model with \(1,046,366\) unknowns using CST takes \(156\) s and \(2.3\) GB of memory. Figure 4 shows the \(x\)-direction electric field and the total electric field at point \((x,0,-60)\), \(0\leq x\leq 25\). The results calculated by CST are also plotted in Figure 4 for comparison. These results show clearly that the numerical results calculated by the proposed method are close to the results computed by CST, with most errors within a range of \(\pm 2\%\). However, from the results listed in Table 1, it can be observed that there are relatively large errors in the first and last column, which are larger than \(2\%\). As the electric field in \(E_{x}\) close to \(z\)-axis is a minimum value, non-uniform discretized mesh can introduce a relatively large error to the first column in numerical calculations. The sparse mesh far from source also causes large error in the last column. It can be also observed from the results in Table 2 that most of the results are within the error range of \(\pm 2\%\). However, there is relatively large error when field point is in the sparse mesh area. The comparisons indicate that the proposed method provides good performance in solving time and high accuracy results for the electric dipole in the two-layered dissipative structure. Figure 4: The total electric field and the electric field in \(x\)-direction at test point \((x,0,-60)\). SIE: surface integral equation. CST: Computer Simulation Technology. Figure 3: Air–sea–seabed layered media structure with a non-planar surface. For the second example, we consider the air-rock-earth-mine layered structure shown in Figure 5. It can be used in Through-The-Earth communication for mines, where communication devices communicate with other nodes by using horizontal electric dipoles located in the earth layer. The frequency of excitation source is 10 kHz. The conductivity of the rock layer is \(\sigma_{1}=0.001\) S/m, and the permittivity and permeability of the rock are \(\epsilon_{1}=3\epsilon_{0}\) and \(\mu_{1}=\mu_{0}\), respectively.