## 4 Derivation of Computational Schemes
### Finite Element Discretization in \(\Omega_{FEM}\)  
To solve the model problem (8) via the domain decomposition FE/FD method, we use the split \(\Omega=\Omega_{FEM}\cup\Omega_{FDM}\), see Figure 1. Thus, in \(\Omega_{FEM}\) we use \(FEM\) to solve the equation  
\[\begin{array}{ll}\varepsilon\partial_{tt}E-\Delta E+\
abla(\
abla\cdot(1- \varepsilon)E)=0&\text{ in }\Omega_{FEM}\times(0,T),\\ E(\cdot,0)=f_{0}(\cdot),\text{ and }\partial_{t}E(\cdot,0)=f_{1}(\cdot)&\text{ in }\Omega_{FEM},\\ E=(E|\alpha_{\Omega_{FEM}})=g,&\text{ on }\partial\Omega_{FEM}\times(0,T),\\ \
abla\cdot(\varepsilon E)=0&\text{ in }\Omega_{FEM}.\end{array} \tag{17}\]  
Here, \(g\) corresponds to the restriction of the exact solution to \(\partial\Omega_{FEM}\). Therefore test and trial functions are not vanishing at this boundary, hence the term corresponding to \(T_{5}\) in (11) will appear in the weak formulation for \(\Omega_{FEM}\).  
To formulate the finite element method for (17) in \(\Omega_{FEM}\), mimicking (11), we introduce the finite element space \(W^{E}_{h}(\Omega_{FEM})\) for each component of the electric field \(E\) defined by  
\[W^{E}_{h}(\Omega_{FEM}):=\{w\in H^{1}(\Omega_{FEM}):w|_{K}\in P_{1}(K),\quad \forall K\in\mathcal{T}_{h}\}.\]  
Setting \(\mathbf{W}^{E}_{h}(\Omega_{FEM}):=[W^{E}_{h}(\Omega_{FEM})]^{3}\) we define \(\mathcal{P}_{h}g\) to be the usual \(L_{2}\)-projection of \(g\) in \(\partial\Omega_{FEM}\). Then, similar to the FE scheme for \(\Omega\), we get the following finite element scheme for \(\Omega_{FEM}\): Given approximate values \(f_{0,h}\) and \(f_{1,h}\) of \(f_{0}\) and \(f_{1}\), respectively, and \(\mathcal{P}_{h}g\), find \(\tilde{E}_{h}\in\mathbf{W}^{E}_{h}(\Omega_{FEM})\) such that  
\[\left(\varepsilon\partial_{tt}\tilde{E}_{h},\mathbf{v}\right) +(\
abla\tilde{E}_{h},\
abla\mathbf{v})+(\
abla\cdot(\varepsilon \tilde{E}_{h}),\
abla\cdot\mathbf{v})-(\
abla\cdot\tilde{E}_{h},\
abla\cdot \mathbf{v})\] \[=\langle n\cdot(\
abla\cdot(\varepsilon-1)\mathcal{P}_{h}g), \mathbf{v}\rangle_{\partial\Omega_{FEM}}+\langle\partial_{n}(\mathcal{P}_{h}g), \mathbf{v}\rangle_{\partial\Omega_{FEM}}\] \[=\{\varepsilon\equiv 1\ \text{ on }\partial\Omega_{FEM}\}= \langle\partial_{n}(\mathcal{P}_{h}g),\mathbf{v}\rangle_{\partial\Omega_{FEM}}, \quad\forall\mathbf{v}\in\mathbf{W}^{E}_{h}(\Omega_{FEM}) \tag{18}\]  
\[\tilde{E}_{h}(\cdot,0)=f_{0,h}(\cdot)\quad\text{ and }\quad\partial_{t}\tilde{E}_{h}( \cdot,0)=f_{1,h}(\cdot)\quad\text{ in }\Omega_{FEM}\]  
where we set \(\tilde{E}_{h}|_{\partial\Omega_{FEM}}=E_{h}\Omega|_{\partial\Omega_{FEM}}\), i.e., the restriction of the finite element solution on whole \(\Omega\) to \(\partial\Omega_{FEM}\), (which, here, contrary to the whole domain where \(E_{h}|_{\partial\Omega}=0\), will be non-zero).  
A corresponding fully discrete problem in \(\Omega_{FEM_{T}}\) reads as follows:Given \(f_{0,h}\), \(f_{1,h}\), \(\mathcal{D}_{h}g\), \(E^{k}_{h}\), and \(E^{k-1}_{h}\); find \(E^{k+1}_{h}\) such that \(\forall\mathbf{v}\in\mathbf{W}^{E}_{h}(\Omega_{FEM_{T}})\),  
\[\begin{split}\left(\hat{E}^{k+1}_{h},\mathbf{v}\right)=& \left(2\hat{E}^{k}_{h},\mathbf{v}\right)-\left(\hat{E}^{k-1}_{h}, \mathbf{v}\right)-\tau^{2}(1/\varepsilon\
abla\hat{E}^{k}_{h},\
abla\mathbf{v })+\tau^{2}(1/\varepsilon\
abla\cdot\hat{E}^{k}_{h},\
abla\cdot\mathbf{v})\\ &-\tau^{2}(1/\varepsilon\
abla\cdot(\epsilon\hat{E}^{k}_{h}), \
abla\cdot\mathbf{v})+\tau^{2}\langle\partial_{n}(\mathcal{D}_{h}g)/ \varepsilon,\mathbf{v}\rangle_{\partial\Omega_{FEM}},\end{split} \tag{19}\]  
\[E^{0}_{h}=f_{0,h}\quad E^{1}_{h}=E^{0}_{h}+\tau f_{1,h}\quad\text{ in }\Omega_{FEM_{T}}.\]