## 5 Stability
### Stability of the Semi-Discrete Problem in \(\Omega_{FEM}\)  
The stability of the semi-discrete problem in \(\Omega_{FEM}\), relying on the variational formulation (18), and due to the appearance of the data function \(g\), is slightly different from (32). We reformulate (18), in view of (12), and with \(\mathbf{v}=\mathbf{v}_{h}=\partial_{t}E_{h}\) as: given \(E_{h}(\cdot,0)=f_{0h}\), \(\partial_{t}E_{h}(\cdot,0)=f_{1h}\), and \(\mathcal{P}_{h}g\) (the \(L_{2}\)-projection of the boundary data \(g\) in (17)), find \(E_{h}\in\mathbf{W}_{h}^{E}(\Omega_{FEM})\) such that  
\[ \begin{split}(\varepsilon\partial_{tt}E_{h},\partial_{t}E_{h})& +(\
abla E_{h},\
abla\partial_{t}E_{h})+((\
abla\varepsilon) \cdot E_{h},\
abla\cdot\partial_{t}E_{h})+((\varepsilon-1)\
abla\cdot E_{h}, \
abla\cdot\partial_{t}E_{h})\\ &=\langle\partial_{n}(\mathcal{P}_{h}g),\partial_{t}E_{h}\rangle _{\partial\Omega_{FEM}},\end{split}  \tag{34}\]  
Note that \(\varepsilon=1\) at the boundary of \(\Omega_{FEM}\), hence the boundary term with the coefficient \(\varepsilon-1\) does not appear in hear. To deal with the \((\
abla\varepsilon)\cdot E_{h}\)-term, we use the discrete version of the \(I_{3}\)-estimate above and get  
\[ \begin{split}\int_{0}^{t}l_{h3}&\,ds=-\int_{ \Omega_{FEM}}\Big{(}\int_{0}^{t}((\
abla\varepsilon)\cdot\partial_{t}E_{h}) (\
abla\cdot E_{h})\,ds-[(\
abla\varepsilon\cdot E_{h})(\
abla\cdot E_{h}) ]_{0}^{t}\Big{)}\\ &:=I_{h3,1}+I_{h3,2}.\end{split}  \tag{35}\]  
Then  
\[I_{h3,1}\leq\frac{1}{2}\int_{0}^{t}\|\partial_{t}E_{h}\|_{|\
abla\varepsilon|,\Omega_{FEM}}^{2}ds+\frac{1}{2}\int_{0}^{t}\|\
abla\cdot E_{h}\|_{|\
abla \varepsilon|,\Omega_{FEM}}^{2}ds, \tag{36}\]  
and  
\[ \begin{split} I_{h3,2}\leq&\frac{1}{2}\Big{(}\|E_{h} \|_{|\
abla\varepsilon|,\Omega_{FEM}}^{2}+\|\
abla\cdot E_{h}\|_{|\
abla \varepsilon|,\Omega_{FEM}}^{2}\Big{)}(t)\\ &+\frac{1}{2}\Big{(}\|E_{h}\|_{|\
abla\varepsilon|,\Omega_{FEM} }^{2}+\|\
abla\cdot E_{h}\|_{|\
abla\varepsilon|,\Omega_{FEM}}^{2}\Big{)}(0). \end{split}  \tag{37}\]  
As for the boundary term on the right-hand side of (34), with \(\mathcal{P}_{h}g=E_{h,FDM}\sim E_{h}^{\Omega}|_{\partial\Omega_{FEM}}\), we have that  
\[\langle\partial_{n}(\mathcal{P}_{h}g),\partial_{t}\mathcal{P}_{h}g\rangle \alpha_{\Omega_{FEM}}\leq\frac{1}{2}\|\partial_{n}(\mathcal{P}_{h}g)\|_{ \partial\Omega_{FEM}}^{2}+\frac{1}{2}\|\partial_{t}(\mathcal{P}_{h}g)\|_{ \partial\Omega_{FEM}}^{2}. \tag{38}\]  
Now, in view of (34)-(38), and taking the time integral, as in the case of stability in \(\Omega\), we end up with the following \(L_{2}(\Omega_{FEM})\)-type stability estimate:  
\[ \begin{split}|||E_{h}|||_{\varepsilon}^{2,FEM}(t):=& \Big{(}\|\partial_{t}E_{h}\|_{\varepsilon}^{2}+\|\
abla E_{h}\|^{2}+\|E_{h}\|_{| \
abla\varepsilon|}^{2}+\|\
abla\cdot E_{h}\|_{|\
abla\varepsilon|+ \varepsilon-1}^{2}\Big{)}(t)\\ &+\int_{0}^{t}\|\partial_{t}E_{h}\|_{|\
abla\varepsilon|,\Omega_ {FEM}}^{2}ds+\int_{0}^{t}\|\
abla\cdot E_{h}\|_{|\
abla\varepsilon|,\Omega_ {FEM}}^{2}ds\\ \leq& C_{1}\Big{(}\|\partial_{t}E_{h}\|_{\varepsilon }^{2}+\|\
abla E_{h}\|^{2}+\|E_{h}\|_{|\
abla\varepsilon|}^{2}+\|\
abla\cdot E_{ h}\|_{|\
abla\varepsilon|+\varepsilon-1}^{2}\Big{)}(0)\\ &+C_{2}\int_{0}^{t}\|\partial_{n}(\mathcal{P}_{h}g)\|_{ \partial\Omega_{FEM}}^{2}ds+C_{3}\int_{0}^{t}\|\partial_{t}(\mathcal{P}_{h}g )\|_{\partial\Omega_{FEM}}^{2}.\end{split}  \tag{39}\]  
Note the appearance of the time integrals of spatial norms of the two boundary terms on the right-hand side of the estimate for \(|||E_{h}|||_{\varepsilon}^{2 FEM}(t)\) in (39). Summing up, we have the following stability estimate for the semi-discrete problem in \(\Omega_{FEM}\):  
**Lemma 2**.