## 3 Numerical Results
The sparse approximate inverse (SPAI) preconditioner [31] was applied in all the simulations performed, as well as the restarted generalized minimal residual method (GMRES) [33] to iteratively solve the systems of equations. Figure 6 shows the results for 181 observation directions along the \(\varphi=0^{\circ}\) angular cut, ranging from \(\theta=0^{\circ}\) to \(\theta=180^{\circ}\). Some difference can be appreciated, especially at the lower field levels, as the number of iterations change. In the same figure, it is possible to see a comparison of the proposed approach with three iterations between domains and the conventional MoM-MLFMA, showing a very good agreement in the resulting radiation pattern. Figure 5: (**a**) Geometrical scheme of the FSS (frequency selective surfaces) unit cell (planar); (**b**) re-flection and transmission coefficients on an infinite planar layout for a frequency sweep between 4 GHz and 24 GHz. Figure 7 shows the results considering three iterations between domains, and the variations given using only the self-coupling blocks (left) and neighboring blocks (right). It is possible to see that, in this case, the accuracy is not good when only the coupling terms inside each block are used, while the results improve noticeably when introducing the coupling between adjacent blocks, although some differences at lower levels can be seen compared with the base, full-coupled algorithm with three iterations. The coupling distance has a noticeable effect in these simulations. While only considering the coupling between CBFs located in the same block presents noticeable disagreements, as seen in Figure 6(a), the inclusion of adjacent blocks improves the accuracy to a degree that can be considered acceptable in many cases, with disagreements appearing about 30 dB under the maximum level. Some results comparing the analysis of the radome covering the antenna and the scenario containing only the antenna are provided in Figure 8. The presented radome introduces small insertion losses and boresight error and mainly affects the secondary lobes. Figure 7(b) shows, in turn, a frequency sweep of the gain results close to the resonance at 14 GHz, where it is possible to see that the FSS introduces the desired effect of serving as  
Figure 6: Radiation pattern results for the \(\varphi=0^{\circ}\) angular cut: (**a**) comparison of the proposed ap-proach considering one, two, and three iterations between domains; (**b**) comparison between the proposed approach with three iterations between domains and MoM-MLFMA (method of moments with the multilevel fast multipole algorithm). Figure 7: Radiation pattern results for the \(\varphi=0^{\circ}\) angular cut: (**a**) considering the proposed approach with only self-block coupling and three iterations; (**b**) considering the proposed approach with self and neighboring block coupling and three iterations. a passband filter centered at the working frequency. It shall be noted that the differences shown in Figure 8 are due to the effect of the placement of the radome, which mainly affects the width of the secondary lobe, introducing small insertion losses and a bandpass filtering effect as designed with the embedded FSS. Table 1 contains the CPU-time required to compute the analyses of this test case with MoM-MLFMA and the proposed approach with one, two, and three iterations between domains, and considering the workstation described at the beginning of this section. The times are broken down into the pre-processing stage (which in this case is considered to be the time required before the iterative solution process), the iteration stage, and the total simulation time. It can be seen that the extra pre-processing time required to compute the CBFs is compensated during the solution process when comparing MoM-MLFMM and the proposed approach. Additionally, the iteration time increases quite linearly as the number of iterations between both domains increases.