## 5 Stability  
### Stability Estimate for Continuous Problem in \(\Omega\)  
In this section, we derive stability estimates for the semi-discrete approximations. For stability in \(\Omega\), these estimates are extensions of the stability approach derived for the continuous problem in [4]. As for the stability in \(\Omega_{FEM}\), we get slightly different norms involving contributions corresponding to the reflexive boundary, \(\partial\Omega_{FEM}\). We define the discrete version of a triple norm induced by the weak variational formulation of (8), where we also use the relation (12) (in \(\Omega_{FDM},\ \varepsilon=1\), therefore, the second term on the right-hand side in (12) is not present in here). Note that in the continuous case \(\
abla\cdot(\varepsilon E)=0\), (whereas, in general \(\
abla\cdot(\varepsilon E_{h})\
eq 0\)), and the continuous variational formulation in whole \(\Omega\), reads as:  
_Find \(E\in\mathbf{W}^{E}_{0}(\Omega)\) such that for all \(\mathbf{v}\in\mathbf{W}^{E}_{0}(\Omega)\)_  
\[\left\{ \begin{array}{c}(\varepsilon\partial_{tt}E,\mathbf{v})+(\
abla E, \
abla\mathbf{v})+((\
abla\varepsilon)\cdot E,\
abla\cdot\mathbf{v})+(( \varepsilon-1)\
abla\cdot E,\
abla\cdot\mathbf{v})=0,\\ \\ E(\cdot,0)=f_{0}(\cdot)\quad\text{ and }\ \ \partial_{t}E(\cdot,0)=f_{1}( \cdot)\quad\text{ in }\Omega.\end{array} \right. \tag{25}\]  
**Remark 3**. : _We emphasize that the bilinear form induced by (25), containing the time derivative term, is not coercive (whereas, without the \(\varepsilon\partial_{tt}E\) term, both (25) and its spatially discrete version are coercive). Further \(H^{1}(\Omega)\)-conforming finite element may result in spurious oscillatory solutions. A remedy is through modifying the equation by adding a gauge constraint of Coulomb-type, see, e.g., [36; 43]. This is supplied by the \"zero\"-term: \(\
abla\cdot(\varepsilon E)=0\), in (8), which we add in the continuous variational formulation in (11). This, however, is not necessarily true in the discrete forms, e.g., in (18), where most likely \(\
abla\cdot(\varepsilon E_{h})\
eq 0\)._  
Taking \(\mathbf{v}=\partial_{t}E\) in (25), (with the boundary condition \(E=0\) on \(\partial\Omega\)), yields\[(\varepsilon\partial_{tt}E,\partial_{t}E) +(\
abla E,\
abla\partial_{t}E)+((\
abla\varepsilon)\cdot E,\
abla \cdot\partial_{t}E) \tag{26}\] \[+\big{(}(\varepsilon-1)\
abla\cdot E,\
abla\cdot\partial_{t}E \big{)}\equiv 0,\]  
which, due to the fact that \(\varepsilon\) is independent of \(t\), can be rewritten as  
\[ \begin{split}\frac{1}{2}\frac{d}{dt}\Big{(}\varepsilon\partial_{t }E,\partial_{t}E\Big{)}&+\frac{1}{2}\frac{d}{dt}\Big{(}\
abla E, \
abla E\Big{)}+\Big{(}(\
abla\varepsilon)\cdot E,\
abla\cdot\partial_{t}E \Big{)}\\ &+\frac{1}{2}\frac{d}{dt}\Big{(}(\varepsilon-1)\
abla\cdot E, \
abla\cdot E\Big{)}\equiv I_{1}+I_{2}+I_{3}+I_{4}=0.\end{split}  \tag{27}\]  
Proposition 1: _Let \(\Omega\subset\mathbb{R}^{n}\), \(n=2,3\) be a bounded domain with piecewise linear boundary \(\partial\Omega\). Let Assumption 1 for function \(\varepsilon\) hold. Assume that there exists a solution \(E\in H^{2}(\Omega_{T})\) of problem (8). Then, Equation (8) has a unique solution \(E\in H^{2}(\Omega_{T})\). Further let \(f_{1}\in L_{\varepsilon}^{2}(\Omega)\) and \(f_{0}\in L_{2}(\Omega)\cap H^{1}_{|\
abla\varepsilon|+\varepsilon-1}(\Omega)\), then there is a constant \(C\), independent of \(\varepsilon\) and \(t\), such that, \(\forall t\in(0,T]\), the following stability estimate holds true_  
\[ \begin{split}||E||_{\varepsilon}^{2}(t):=&\| \partial_{t}E\|_{\varepsilon}^{2}(t)+\|\
abla E\|^{2}(t)+\|\
abla\cdot E\|_{ |\
abla\varepsilon|+\varepsilon-1}^{2}(t)+\|E\|_{|\
abla\varepsilon|}^{2}(t) \\ &+\int_{0}^{t}\|E_{t}\|_{|\
abla\varepsilon|}^{2}\,ds+\int_{0}^{t} \|\
abla\cdot E\|_{|\
abla\varepsilon|}^{2}\,ds\\ \leq& C\Big{(}\|f_{1}\|_{\varepsilon}^{2}+\|\
abla f _{0}\|^{2}+\|\
abla\cdot f_{0}\|_{|\
abla\varepsilon|+\varepsilon-1}^{2}+\|f_ {0}\|_{|\
abla\varepsilon|}^{2}\Big{)}.\end{split}