## 2 The FEM-DBCI Method
In the case of FEM-SDBCI, the LU decomposition of matrix \(\mathbf{H}\) is performed only once at the beginning of the iterative procedure; round-off errors in the LU decomposition are treated by selecting an appropriate (double precision) accuracy in computing and storing matrices \(\mathbf{H}\) and \(\mathbf{G}\). 3. The whole iterative procedure is convergent if an appropriate relaxation coefficient \(\lambda\) is selected, which, unfortunately, is not known a priori; divergence may occur if an unsuitable coefficient \(\lambda\) is used. 4. Consequently, the integral Equation (9) is used only a few times, compared to its use in an iterative CG-like solver for the whole non-symmetric system (10). Another solving strategy has been devised by considering the following reduced algebraic system: \[\mathbf{M}\mathbf{e}_{\mathrm{F}}=\mathbf{b}\] (13) where \[\mathbf{M}=\mathbf{H}+\mathbf{G}\mathbf{A}^{-1}\mathbf{A}_{\mathrm{F}}\] (14) \[\mathbf{b}=\mathbf{e}_{\mathrm{ext}}+\mathbf{G}\mathbf{A}^{-1}\mathbf{b}_{0}\] (15)Of course, the definitions of matrix \(\mathbf{M}\) in (14) and vector \(\mathbf{b}\) in (15) are only formal because the inverse matrix \(\mathbf{A}^{-1}\) cannot be really obtained when the number of unknowns is large. However, these arrays can be used, since vector \(\mathbf{b}\) is directly computable, while matrix \(\mathbf{M}\) can be virtually used to perform matrix multiplications by vectors, as explained below. In order to compute vector \(\mathbf{b}\), the following must be performed: solve the FEM Equation (6) with \(\mathbf{e}_{\text{F}}=\mathbf{0}\) by means of the CoCG solver to obtain \(\mathbf{e}=\mathbf{A}^{-1}\mathbf{b}_{0}\), and then compute \(\mathbf{b}=\mathbf{e}_{\text{ext}}+\mathbf{G}\mathbf{e}\). Similarly, in order to compute the product \(\mathbf{M}\mathbf{e}_{\text{F}}\) of matrix \(\mathbf{M}\) by a given vector \(\mathbf{e}_{\text{F}}\), solve FEM Equation (6) with \(\mathbf{b}_{0}=\mathbf{0}\) using the CoCG solver to obtain \(\mathbf{e}=\mathbf{A}^{-1}\mathbf{A}_{\text{F}}\mathbf{e}_{\text{F}}\), and then compute \(\mathbf{M}\mathbf{e}_{\text{F}}=\mathbf{H}\mathbf{e}_{\text{F}}+\mathbf{G} \mathbf{e}\). Note that the iterative algorithm described above can be seen as the application of the Richardson method to solve (13). However, since this solution method is known to be weak and possibly non-convergent, one should look for better solvers. Due to the virtual availability of matrix \(\mathbf{M}\), it is possible to use various non-stationary solvers, and in particular, solvers that are polynomial accelerations of the Richardson method for non-symmetric matrices. Among these, the Bicorjugate Gradient (BiCG), the Quasi Minimal Residual (QMR), the Conjugate Gradient Squared (CGS), the Biconjugate Gradient Stabilized (BiCGstab), and the Generalized Minimal Residual (GMRES) are the most popular [25]. Since, concerning system (13), the matrix-vector multiplications are much more expensive than in a system where the coefficient matrix is directly available, it is crucial to reduce the number of iterations. Then, GMRES should be preferred, because it minimizes the number of iterations by performing a true minimization of the residual [26]. Moreover, note that the residual can be computed directly with the approximate solution (but this requires a further matrix-vector multiplication) or by using the orthonormal basis of the Krylov subspace [26]. The latter option is preferable. In fact, in general, the greater drawbacks of GMRES are the computing time and memory required to compute and store the orthonormal basis, which increases linearly with the number of iterations, but in this context, the unknowns in system (13) are the values of the electric field along the edges of the fictitious boundary, the number of which is relatively low compared to the total number of edges. Another important point regards the GMRES restating procedure. It is convenient to use long restarts, which generally result in a full GMRES due to the quick convergence characteristic of the simple iterative procedure.