[55] 1. In the case of metallic, homogeneous dielectric and composite objects with low permittivities and with no junctions, the *N* formulations give clearly faster converging iterative solutions than the *T* formulations. In all these cases, also with a high electric permittivity, the fastest convergence rate is obtained with the N-MFIE/mN-Müller formulation.

#### 7.1. Homogeneous Dielectric Objects

[60] Let us first consider the developed integral equation formulations in the case of a homogeneous dielectric object. Let us denote *n* = *n*_{1} = −*n*_{2}. The integral operators associated with the *T* formulations (*a*_{l} ≠ 0, *b*_{l} = 0, *c*_{l} = 0, *d*_{l} ≠ 0) can be written in the matrix form as follows

The corresponding operators of the *N* formulations (*a*_{l} = 0, *b*_{l} ≠ 0, *c*_{l} ≠ 0, *d*_{l} = 0) read

The equations in (42) and (43) are arranged so that the operators on the diagonal are well tested and the poorly tested operators appear on the off-diagonal blocks. As a consequence, the matrix equations will be diagonally dominant. In each of the CTF, CNF and JMCFIE formulations the matrix blocks on the diagonal are identical. Thus, also in this sense, the developed CTF, CNF and JMCFIE formulations are “optimal” EFIE-, MFIE- and CFIE-type formulations. Note that this is not generally true for the mN-Müller formulation.

[61] From (42) we find that a *T* formulation with *a*_{1}/*k*_{1} = −*a*_{2}/*k*_{2} and *d*_{1}/*k*_{1} = −*d*_{2}/*k*_{2} leads to a diagonally less dominant matrix equation, because in that case the main singularity of the _{tan} operator cancels. T-Müller (with coefficients *a*_{1} = ɛ_{1}η_{1}, *a*_{2} = −ɛ_{2}η_{2}, *d*_{1} = μ_{1}/η_{1} and *d*_{2} = −μ_{2}/η_{2}) is an example of an unstable *T* formulation [*Ylä-Oijala and Taskinen*, 2005b]. Similar result holds also for the *N* formulations. A *N* formulation with *b*_{1} = −*b*_{2} and *c*_{1} = −*c*_{2} leads to an integral equation of the first kind (N-PMCHWT formulation) which is very ill conditioned [*Lloyd et al.*, 2004] because the identity operator cancels and any of the operators is not well tested. In the mN-Müller (and N-Müller) the main singularity of the *n* × operator cancels and in the CNF the main singularity of the *n* × operator cancels. In the mN-Müller formulation the cancellation appears on the off-diagonal blocks and hence even a more diagonally dominant matrix equation is obtained. In the CNF this cancellation, however, takes place on the diagonal and most probably is the source of numerical problems observed with the CNF in the case of objects with high permittivity contrast. In that case also the off-diagonal blocks of the CNF become very unbalanced. However, for the low-contrast objects the CNF works well and in the case ɛ_{1} = ɛ_{2} and μ_{1} = μ_{2}, we have

and the solution of the CNF for the problem is *J* = *n* × *H*^{inc} and *M* = −*n* × *E*^{inc}, that is, the tangential components of the incident field.

[62] The worst convergence rate is always obtained with the T-PMCHWT formulation. The main problems of the PMCHWT formulation are that it is an integral equation of the first kind and that the matrix elements have clearly different magnitudes. The convergence of the T-PMCHWT formulation can be improved with the proper weighting coefficients as in the CTF.

[63] The *N* formulations, excluding N-PMCHWT, lead to integral equations of the second kind with a well tested identity operator. The *T* formulations, excluding T-PMCHWT, lead also to integral equations of the second kind, but the identity operator is not well tested. Since the *N* formulations usually lead to faster converging iterative solutions than the *T* formulations, the identity operator, and how it is tested, plays a crucial role in deriving integral equation formulations with good spectral properties and rapidly converging iterative solutions. Note that each of the CNF, JMCFIE and mN-Müller formulations are derived so that the identity operators of the coupled system have an equal coefficient. This property, which is a consequence of defining the equations so that they are mappings from the surface currents onto themselves, seems to be also important for formulations with good iterative properties.

[64] There are two reasons why the mN-Müller works so well with homogeneous dielectric objects. First, because of the cancellation of the main singularity of the *n* × operator, mN-Müller (and N-Müller) removes the low-frequency breakdown and leads to a well conditioned matrix equation on a wide frequency range [*Ylä-Oijala and Taskinen*, 2005b]. Second, the equations are scaled so that they are (approximately) in the current spaces. We remind that in the N-Müller formulation this is not the case.

#### 7.2. Inhomogeneous and Nonsmooth Objects

[65] The results of section 6 indicate that in the case of nonsmooth and inhomogeneous objects the *N* formulations do not work as well as they do in the case of smooth homogeneous objects. There are (at least) two reasons for that. First, the currents have singularities at the edges and junctions and in that case the RWG functions are not sufficiently smooth functions for expanding the unknowns and testing the equations in the MFIE-type *N* formulations [see also *Ergül and Gürel*, 2004]. Secondly the integral operators of the combined formulations have different performance at the junctions. As mentioned in the previous section, the main singularity of the *n* × operator cancels in the CNF and the main singularity of the *n* × operator cancels in the mN-Müller formulation. In the case of objects with junctions, however, these cancellations are not complete, and as a consequence, the mN-Müller formulation, for example, does not remove the low-frequency breakdown.

[66] Moreover, in the case of piecewise dielectric and composite objects the operators on the diagonal of the CTF, CNF and JMCFIE formulations are not generally identical. This also implies that the developed formulations are not as optimal as they are in the case of homogeneous objects.

#### 7.3. Other Frequencies

[67] The calculations of the previous section were performed at a fixed frequency. We finish this section by considering the same geometries as in sections 6.2, 6.4 and 6.5, by varying the frequency so that *k*_{0}*r* varies from 0.25 to 4.25. Here *r* is the radius of the sphere and *k*_{0} is the wavelength in vacuum. In the case of homogeneous dielectric objects (for both sphere and cube) all formulations give similar accuracy as in the case *k*_{0}*r* = 1 in section 6 if the number of unknowns is chosen so that the discretization is dense enough. In addition, in all formulations the number of iterations increases smoothly for increased frequency, as shown in the top left plot of Figure 10.

[68] In the case of piecewise dielectric and composite objects the situation, however, is different. Also in this case all formulations give similar accuracy as in the case *k*_{0}*r* = 1 (better accuracy is obtained at lower frequencies, because the number of unknowns was fixed), but the convergence rates of the *N* formulations essentially decrease after some threshold. This is clearly shown in the left-hand plots of Figure 10. This threshold seems to depend on the discretization and mesh density. Similar phenomenon can be observed also in Figures 7 and 9, where the number of iterations of the CNF and mN-Müller formulations increase more rapidly than that of the other formulations as the number of unknowns is increased. The *N* formulations show strange behavior also in the case of a PEC sphere with a dielectric coating at *k*_{0}*r* = 0.25 (bottom right plot of Figure 10), although in that case the surfaces are smooth. Note also that in most cases the T-PMCHWT formulation does not converge in 2000 iterations, and hence the results for the T-PMCHWT formulation are not shown in all cases. In Figure 10, the two vertical lines show for which size parameter *k*_{0}*r* the maximum edge length of the mesh is roughly λ/10 (on the left) and λ/5 (on the right), where λ is the wavelength of the region with the highest permittivity.