## 2 The FEM-DBCI Method
Another equation relating \(\mathbf{e}\) to \(\mathbf{e}_{\mathrm{F}}\) can be derived by means of the following representation formula:  
\[\frac{\gamma}{4\pi}\overline{\mathrm{E}}(\mathrm{P}_{\mathrm{F}})=\overline{ \mathrm{E}}_{\mathrm{ext}}(\mathrm{P}_{\mathrm{F}})\ +\frac{1}{4\pi}\cancel{ \iint_{\mathrm{T}}}\left(\frac{1}{r}\hat{\mathbf{n}}\times\
abla\times \overline{\mathrm{E}}+(\hat{\mathbf{n}}\times\overline{\mathrm{E}})\times \
abla\frac{1}{r}+\hat{\mathbf{n}}\cdot\overline{\mathrm{E}}\
abla\frac{1}{r} \right)\,\mathrm{d}\mathrm{S} \tag{7}\]  
where \(\mathrm{P}_{\mathrm{F}}\) is a point lying on \(\Gamma_{\mathrm{F}}\), \(\Gamma\) is another closed surface that includes the system (see Figure 1), \(\mathrm{r}\) is the distance between \(\mathrm{P}_{\mathrm{F}}\) and the generic point \(\mathrm{P}\) on \(\Gamma\), \(\overline{\mathrm{E}}_{\mathrm{ext}}\) is the electric field due to the coil currents external to \(\Gamma_{\mathrm{F}}\), and \(\gamma\) is a solid angle that is equal to \(4\pi\) if \(\mathrm{P}_{\mathrm{F}}\) is external to \(\Gamma\); if \(\mathrm{P}_{\mathrm{F}}\) lies on \(\Gamma\), \(\gamma\) is the solid angle at \(\mathrm{P}_{\mathrm{F}}\) of the unbounded free space external to \(\Gamma\). The physical meaning of Equation (7) is that if a vector field and its rotation are known on a closed surface \(\Gamma\), then the field is completely computable in the entire space outside that surface. The closed surface \(\Gamma\) is conveniently selected as constituted of the triangular faces of the tetrahedral elements. If the surface \(\Gamma\) is selected as strictly enclosed by \(\Gamma_{\mathrm{F}}\), the integrand function in Equation (7) is regular and the method is referred to as FEM-DBCI [21]. Otherwise, if the surface \(\Gamma\) is selected as coincident with \(\Gamma_{\mathrm{F}}\), the integrand function in (7) is singular and the method is called FEM-SDBCI (Singular DBCI) [22]. By substituting (7) in (5), the following integral equation is obtained:  
\[ \begin{array}{rl}\frac{\gamma}{4\pi}\mathrm{E}_{\mathrm{m}}&=\frac{\gamma}{ 4\pi\mathrm{L}_{\mathrm{m}}}\int_{\mathbf{e}_{\mathrm{m}}}\overline{\mathrm{E }}(\mathrm{P}_{\mathrm{F}})\cdot\hat{\mathbf{t}}_{\mathrm{m}}\,\mathrm{d} \mathrm{s}=\frac{1}{\mathrm{L}_{\mathrm{m}}}\int_{\mathbf{e}_{\mathrm{m}}} \overline{\mathrm{E}}_{\mathrm{ext}}(\mathrm{P}_{\mathrm{F}})\cdot\hat{ \mathbf{t}}_{\mathrm{m}}\,\mathrm{d}\mathrm{s}\ +\\ &\quad+\frac{1}{4\pi\mathrm{L}_{\mathrm{m}}}\sum\limits_{\mathrm{k}}\int \int_{\Gamma_{\mathrm{k}}}\int_{\mathrm{e}_{\mathrm{m}}}\left(\frac{1}{r}\hat{ \mathbf{n}}\times\
abla\times\overline{\mathrm{E}}+(\hat{\mathbf{n}}\times \overline{\mathrm{E}})\times\
abla\frac{1}{r}+\hat{\mathbf{n}}\cdot\overline{ \mathrm{E}}\
abla\frac{1}{r}\right)\cdot\hat{\mathbf{t}}_{\mathrm{m}}\mathrm{d} \mathrm{s}\,\mathrm{d}\mathrm{S}\end{array}  \tag{8}\]  
where \(\mathrm{T}_{\mathrm{k}}\) is the k-th triangular patch of the integration surface \(\Gamma\), resulting from the tetrahedral meshing of the bounded domain. In each patch, the electrical field is developed as in (3), where the six edges are those of the corresponding tetrahedron to which the triangular patch belongs. Finally, by developing the electrical field as in (3) and by applying (8) to all the edges \(\mathbf{e}_{\mathrm{m}}\) of the fictitious boundary, we have the following:  
\[\mathbf{He}_{\mathrm{F}}=\mathbf{e}_{\mathrm{ext}}+\mathbf{Ge} \tag{9}\]  
where \(\mathbf{e}_{\mathrm{ext}}\) is the vector of the mean tangent components of the electrical field along the edges of the fictitious boundary \(\Gamma_{\mathrm{F}}\) due to the external coils, and \(\mathbf{H}\) and \(\mathbf{G}\) are dense matrices collecting the geometrical coefficients in (8) relative to the edges on \(\Gamma_{\mathrm{F}}\) and to the internal edges, respectively. Note that not all internal edges appear in (8), but only those of the tetrahedra external to \(\Gamma\) and with a face lying on it.