## 2 Formulation of the Hybrid Method
### Scattered Field Generated by the DGTD Region  
The primary scattering contribution from the DGTD region is computed by using the DGTD method. Let \(\Omega\subset\mathrm{R}^{3}\) be the calculation area enclosing a scatterer, and \(n\) the unitary outward normal to its boundary \(\partial\Omega\). Define \(\Omega_{h}\) as a discretization of \(\Omega\); \(\Omega_{h}\) consists of non-overlapping tetrahedral elements, and \(\Omega_{h}=\cup_{m=1}^{N}\Omega_{m}\), where \(N\in\mathrm{N}^{*}\) is the number of mesh elements. As shown in the Figure 2, the inner surfaces of the discretization are denoted by \(\partial\Omega_{mk}=\partial\Omega_{m}\cap\partial\Omega_{k}\); \(\Omega_{k}\) are adjacent cells of \(\Omega_{m}\), and \(\overset{\rightarrow}{n}_{mk}\) is defined as the unit vector normal to the surface \(\partial\Omega_{mk}\) oriented from \(\Omega_{m}\) toward \(\Omega_{k}\). Applying the discontinuous Galerkin testing to Maxwell curl equations in \(\Omega_{m}\) yields [20]:  
\[ \begin{array}{l}\underset{\Omega_{m}}{\iint}\mu\frac{\partial H}{\partial t }\cdot N_{q}^{m}d\upsilon+\underset{\Omega_{m}}{\iint}\sigma_{h}H_{m}\cdot N _{q}^{m}d\upsilon+\underset{\Omega_{m}}{\iint}(\
abla\times E_{m})\cdot N_{q }^{m}d\upsilon\\ =\underset{\Omega_{m}}{\iint}\sigma_{h}\underset{\Omega_{m}}{\iint}k_{m}^{m }[n\times(E_{m}-E_{k})-M_{s}]\cdot N_{q}^{m}ds+\underset{\Omega_{m}}{\iint} \sigma_{k}\underset{\Omega_{m}}{\iint}\upsilon_{h}^{m}\{n\times[n\times(H_{k} -H_{m})-J_{s}]\}\cdot N_{q}^{m}ds\\ \underset{\Omega_{m}}{\iint}e^{\frac{\mathrm{d}\tau}{\mathrm{d}\tau}}\cdot N_ {q}^{m}d\upsilon+\underset{\Omega_{m}}{\iint}\sigma_{e}E_{m}\cdot N_{q}^{m}d \upsilon+\underset{\Omega_{m}}{\iint}(\
abla\times H_{m})\cdot N_{q}^{m}d \upsilon\\ =\underset{\partial\Omega_{m}}{\iint}k_{h}^{m}[n\times(H_{k}-H_{m})-J_{s}] \cdot N_{q}^{m}ds+\underset{\Omega_{m}}{\iint}\sigma_{h}\underset{\Omega_{ m}}{\iint}\upsilon_{e}^{m}\{n\times[n\times(E_{m}-E_{k})-M_{s}]\}\cdot N_{q}^{m}ds \end{array}  \tag{3}\]  
where \(\varepsilon\), \(\mu\) and \(\sigma\) are the permittivity, permeability and conductivity of the \(\Omega\) respectively. In addition, \(J_{s}\) and \(M_{s}\) are surface electric current and magnetic current at the interface of adjacent elements, respectively. Moreover, \(N_{q}^{m}\) is the edge basis function of the \(m\)-th tetrahedral element. Meanwhile, the terms \(k_{\varepsilon}^{m}\), \(k_{h}^{m}\), \(\upsilon_{\varepsilon}^{m}\) and \(\upsilon_{h}^{m}\) are defined as the numerical flux factors in Table 1, and the subscript \(k\) represents elements adjacent to \(m\) elements. Numerical flux is led into at the common interface of adjacent elements--that is, the tangential component of electromagnetic field on the shared surface of two adjacent elements is no longer continuous. We assume that a multi-scale structure is divided into subdomains; the semi-discretized system of equations by the DGTD method will be [20]:  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Numerical Flux Factors** & \(\mathbf{k_{\varepsilon}^{m}}\) & \(\mathbf{k_{h}^{m}}\) & \(\upsilon_{h}^{m}\) & \(\upsilon_{\varepsilon}^{m}\) \\ \hline Centered numerical flux & \(1/2\) & \(1/2\) & \(0\) & \(0\) \\ Upwind numerical flux & \(\frac{\gamma^{m}}{\gamma^{m}+\frac{1}{2}\gamma^{m}}\) & \(\frac{\gamma^{m}}{Z^{m}+Z^{m}}\) & \(\frac{1}{\gamma^{m}+\gamma^{m}}\) & \(\frac{1}{Z^{m}+Z^{m}}\) \\ Partially penalized numerical flux & \(\frac{\gamma^{m}}{\gamma^{m}+\gamma^{m}}\) & \(\frac{2^{m}}{Z^{m}+Z^{m}}\) & \(\frac{\gamma}{Z^{m}+Y^{m}}\\[\mu\{M\}\frac{dH_{m}}{dt}+\sigma_{h}\{M\}H_{m}+\{S\}E_{m}+\Big{\{}F_{ ke}^{k}\Big{\}}E_{m}-\{F_{ke}\}E_{m}+\Big{\{}G_{vh}^{k}\Big{\}}H_{k}-\{G_{vh}\}H_{m}+M_{ sk}-J_{sv}=0 \tag{4}\] \[\varepsilon\{M\}\frac{dE_{m}}{dt}+\sigma_{e}\{M\}E_{m}-\{S\}H_{m} -\Big{\{}F_{bh}^{k}\Big{\}}H_{k}+\{F_{kh}\}H_{m}+\Big{\{}G_{ve}^{k}\Big{\}}E_{k} -\{G_{ve}\}E_{m}+M_{sv}+J_{sk}=0 \tag{5}\]  
\[\left\{