## 3 Three-Dimensional Finite Element Method Forward Modeling Theory
### Validation Against Semi-Analytic Solution  
In this paper, the semi-analytical solution of a 100 \(\Omega\)-m full-space model was employed to verify the accuracy of our FE presented here. The transmitter consisted of a 100 m-long wire oriented along the x-direction, with the receivers positioned 100 m away from the source along its central axis. The forward modeling encompassed a total of 270,270 tetrahedral elements, comprising 316,559 edges, thereby defining a model with 316,559 degrees of freedom (DOFs). The extended boundary domain spanned \(200\;\mathrm{km}\times 200\;\mathrm{km}\times 200\;\mathrm{km}\) in all three directions. The forward calculation involved 1100 time channels, increasing in a stepwise manner. For interpolation, 46 time channels between \(10^{-5}\) s and \(10^{-2}\) s were selected to present the computed results of the electric field and induced electromotive force (EMF), which were then compared with 1D semi-analytical solutions for accuracy. The results are displayed in Figure 3. Both the electric field and EMF exhibited good agreement with the semi-analytical solutions. The relative errors for the EMF parameters were all below 2%. However, the electric field parameters exhibited a sign change between 0.1166 \(\times\)\(10^{-4}\) s and 0.1359 \(\times\)\(10^{-4}\) s, resulting in larger errors near the sign transition. Beyond 2.512 \(\times\)\(10^{-5}\) s, the relative errors also remained within 2%. It is evident that the FE presented in this paper demonstrated high numerical accuracy. To better illustrate the computational efficiency, Table 1 lists the computational parameters for the EMF parameters, with all computation times and memory usage falling within reasonable limits. Thus, the results of the forward modeling study for the borehole TEM method using grounded-wire sources, conducted with the algorithms and codes presented in this paper, are reliable.  
Figure 3: The forward modeling responses of the full-space model along with the relative errors compared to the 1D semi-analytical solution, where (**a**,**b**) are the results for the induced electromotive force (EMF), while (**c**,**d**) are those of the electric field.  
Taking the half-space model as an example, we considered a scenario where the resistivity of the air was \(10^{-8}\)\(\Omega\cdot\)m and the background was \(10^{-2}\)\(\Omega\cdot\)m. We compared our results with those obtained using the established SLDMEM code using the finite-difference (FD) method, which was based on the work by [PERSON] and [PERSON] [44] and was executed on servers. The results and parameters computed by the two algorithms are presented in Figure 4 and Table 2, respectively. It can be observed that the relative errors between the FE method presented in this paper and the semi-analytical solution for the half-space model were all below 3%, which were superior to those from the FD method. By further comparing the computational parameters of the two codes, we conclude that our FE code could achieve solutions with higher accuracy under similar numbers of DOFs while maintaining comparable computation times.