## 2 Description of the Numerical Technique
### Iterative Domain Decomposition Analysis  
With the aforementioned considerations regarding domain separation as well as the application of MLFMA and CBFM, the iterative approach for the electromagnetic analysis can be described by the following steps, where n indicates the iteration number; \(\left[V_{a}^{(n)}\right]^{\prime}\) and \(\left[V_{r}^{(n)}\right]^{\prime}\) refer to the excitations on antenna and radome at the \(n\)-th iteration, respectively; and, analogously, \(\left[J_{a}^{(n)}\right]^{\prime}\) and \(\left[J_{r}^{(n)}\right]^{\prime}\) denote the currents on both domains for such an iteration:  
1. Obtain the excitation over the antenna domain: \[\left[V_{a}^{(1)}\right]^{\prime}=\left[V_{a}\right]^{\prime}\] (8) \[\left[V_{a}^{(n)}\right]^{\prime}=\left[V_{a}\right]^{\prime}-\left[Z_{D_{a} D_{r}D_{r}}\right]^{\prime\left[J_{r}^{(n-1)}\right]}\text{, if }n\ >\ 1\] (9)
2. Obtain the current \(\left[J_{a}^{(n)}\right]^{\prime}\) on the antenna domain by solving iteratively (applying MLFMA): \[\left[Z_{D_{a}D_{a}}\right]^{\prime}\left[J_{a}^{(n)}\right]^{\prime}=\left[V_ {a}^{(n)}\right]^{\prime}\] (10)
3. Obtain the field impressed on the radome domain \(\left[V_{r}^{(n)}\right]^{\prime}\) due to \(\left[J_{a}^{(n)}\right]^{\prime}\) as follows: \[\left[Z_{D_{r}D_{a}}\right]^{\prime}\left[J_{a}^{(n)}\right]^{\prime}=\left[V_ {r}^{(n)}\right]^{\prime}\] (11)
4. Retrieve the current on the radome domain \(\left[J_{r}^{(n)}\right]^{\prime}\) by solving the following: \[\left[Z_{D_{r}D_{r}}\right]^{\prime}\left[J_{r}^{(n)}\right]^{\prime}=\left[V_ {r}\right]^{\prime}-\left[V_{r}^{(n)}\right]^{\prime}\] (12)  
Figure 3: Scheme illustrating the procedure followed for the generation of the CBFs (charac-teristic basis functions).  
5. The total current distribution for the \(n\)-th iteration, as indicated by Equation (3) and expressed in terms of CBFs, is thus the following: \[\left[J^{\left(n\right)}\right]^{\prime}=\left[\begin{array}{c}\left[J^{ \left(n\right)}_{a}\right]^{\prime}\\ \left[J^{\left(n\right)}_{r}\right]\end{array}\right]\] (13)  
Note that the use of a number of dielectric layers composing the radome, as well as the embedded FSS elements, can introduce bad convergence properties, which makes it desirable to consider techniques to counterbalance such a bottleneck. In the results presented in this work, a SPAI preconditioner specifically tailored to the CBFM [31] is applied. Equations (10) and (12) require solving iteratively two separate systems of equations, and the use of two separate SPAI preconditioning matrices previously computed. Additionally, an approximate variation of the method presented above can be derived by considering that the CBFs contained within each block only interact with those close to them, either in the same block (self-coupling approach) or in neighboring blocks in addition to the same block (neighboring coupling approach). The accuracy of these possible variations will be studied in the next section, which shows an example of the application of this technique, including the results for different interactions between both domains. We have found that three iterations yield very good accuracy for all the test cases carried out in the development of this work.