## 2 The FEM-DBCI Method
So, the matrix \(\mathbf{H}\) is square by construction, while \(\mathbf{G}\), in general, is rectangular. Note that in the case of FEM-DBCI, matrix \(\mathbf{H}\) simplifies to the identity matrix and Equation (9) becomes explicit with respect to vector \(\mathbf{e}_{\mathrm{F}}\). Gauss quadratures are used to evaluate both the double integral on the triangle \(\mathrm{T}_{\mathrm{k}}\) and the line integral on the edge \(\mathbf{e}_{\mathrm{m}}\) in (8). In order to obtain a good tradeoff between accuracy and computing speed, the following rule is recommended. If \(\mathrm{L}_{\mathrm{max}}\) is the length of the longest edge of the triangle \(\mathrm{T}_{\mathrm{k}}\) on \(\Gamma\), \(\mathrm{L}_{\mathrm{m}}\) is the length of the edge on the fictitious boundary, \(\mathrm{L}=\mathrm{max}\) (\(\mathrm{L}_{\mathrm{max}}\), \(\mathrm{L}_{\mathrm{m}}\)), and \(\mathrm{d}\) is the distance between their centers, then for \(\mathrm{L}/\mathrm{d}\leq 0.2\), a one-point quadrature is used on both the triangle and the edge; for \(0.2<\mathrm{L}/\mathrm{d}\leq 1.1\), three Gauss points are used on the triangle and two points on the edge; otherwise, six points are used on the triangle and three points on the edge. In the case of FEM-SDBCI, the singularities arising in the integrand function in (8) are overcome by means of analytical formulas [24]. Some details are given in [22]. Combining the matrix Equations (6) and of (9), the global linear system of the FEM-DBCI (or FEM-SDBCI) method is obtained:  
\[ \begin{bmatrix}\mathbf{A}&\mathbf{A}_{\mathrm{F}}\\ -\mathbf{G}&\mathbf{H}\end{bmatrix}\begin{bmatrix}\mathbf{e}\\ \mathbf{e}_{\mathrm{F}}\end{bmatrix} = \begin{bmatrix}\mathbf{b}_{0}\\ \mathbf{e}_{\mathrm{ext}}\end{bmatrix}  \tag{10}\]  
The system matrix in (10) is partly sparse, partly full, and non-symmetric. Unfortunately, efficient solvers do not exist for this kind of linear algebraic system. However, a good solving strategy should treat the sparse subsystem coming from the FEM equations and the dense subsystem coming from the integral equations in different ways. In particular, this strategy should reduce the number of multiplications of the dense submatrices by vectors to a minimum and should fully exploit the sparsity and symmetry of matrix \(\mathbf{A}\). A strategy that satisfies the above requirements is the following two-block Gauss-Seidel algorithm [22]:  
1. select arbitrarily a first guess for \(\mathbf{e}_{\mathrm{F}}\), for example, \(\mathbf{e}_{\mathrm{F}}=\mathbf{0}\);
2. solve (6) for \(\mathbf{e}\) by means of the Complex Conjugate Gradient (CoCG) [25];
3. in the case of FEM-SDBCI decompose the square matrix \(\mathbf{H}\) is into the matrices \(\mathbf{L}\) and \(\mathbf{U}\); then Equation (9) is solved for \(\mathbf{e}_{\mathrm{F}}\); in the case of FEM-DBCI the vector \(\mathbf{e}_{\mathrm{F}}\) is computed directly as \(\mathbf{e}_{\mathrm{F}}=\mathbf{e}_{\mathrm{ext}}+\mathbf{G}\mathbf{e}\);
4. to measure the distance between the new guess for \(\mathbf{e}_{\mathrm{F}}\) and the old one, the convergence indicator \(\eta\) is computed: \[\eta=100\frac{\left\|\mathbf{e}_{\mathrm{F}}^{\mathrm{new}}-\mathbf{e}_{ \mathrm{F}}^{\mathrm{old}}\right\|_{2}}{\left\|\mathbf{e}_{\mathrm{F}}^{ \mathrm{new}}\right\|_{2}}\] (11) convergence is assumed if \(\eta<\eta_{0}\), where \(\eta_{0}\) is a user-selected end-iteration tolerance;
5. if convergence is not reached, go to step 2 by assuming a relaxed new guess for \(\mathbf{e}_{\mathrm{F}}\) such as: \[\mathbf{e}_{\mathrm{F}}^{\mathrm{(n)}}=\lambda\,\mathbf{e}_{\mathrm{F}}^{ \mathrm{new}}+(1-\lambda)\mathbf{e}_{\mathrm{F}}^{\mathrm{old}}\] (12) where \(\lambda\) is the relaxation coefficient. This iterative algorithm exhibits the following characteristics: 1. Since in each iteration, the first guess for the CoCG is the solution obtained in the previous iteration, the various solutions of the FEMI Equation (6) get faster as the iteration proceeds. 2.