## 6 Error Estimates: Semi-Discrete (SD) Problems  
In what follows, assuming certain regularity of the data set, \(E\in\mathbf{W}^{E}(\tilde{\Omega})\cap H^{s}(\tilde{\Omega})\), and with the spectral polynomial order \(p\) chosen as the assumed regularity degree \(s\geq 1\) we can prove semi-discrete spatial error estimates of the form  
\[|||E-E_{h}|||_{\tilde{\Omega}}\leq Ch^{p}\sim Ch^{s},\qquad\text{for}\quad \tilde{\Omega}=\Omega\quad\text{or}\quad\tilde{\Omega}=\Omega_{FEM}. \tag{41}\]  
Below we use a very similar argument to derive (41) for the spatial domains \(\Omega\) and \(\Omega_{FEM}\). To do so, we rely on an approach (see, e.g., [46]) based on a split of the bilinear form \(B(\cdot,\cdot)\) to a temporal term, plus a time-harmonic sub-bilinear form below denoted \(a(\cdot,\cdot)\).  
**Splitting of \(B\)**. Here we extract from \(B\) the bilinear form \(a(\cdot,\cdot)\), by suppressing the time-derivative term, and establish well-posedness for \(a\). Then we deal with the convergence of the original, time-dependent scheme of \(B\). Thus  
\[a(E,\mathbf{v}):=(\
abla E,\
abla\mathbf{v})+((\
abla\varepsilon)\cdot E, \
abla\cdot\mathbf{v})+((\varepsilon-1)\
abla\cdot E,\
abla\cdot\mathbf{v}), \tag{42}\]  
Due to the homogeneous Dirichlet boundary condition, the inherited Galerkin orthogonality yields:  
\[a(E-E_{h},\chi)=0,\qquad\forall\chi\in\mathbf{W}^{E}_{0,h}. \tag{43}\]  
Now we can write the equation (25) as Find \(E\in\mathbf{W}^{E}(\Omega)\) such that for all \(\mathbf{v}\in\mathbf{W}^{E}_{0,h}\),  
\[(\varepsilon\partial_{tt}E,\mathbf{v})+a(E,\mathbf{v})=0. \tag{44}\]  
The **consistency** of \(a(\cdot,\cdot)\) is a consequence of (44).  
We also define the norm associated to \(a(\cdot,\cdot)\):  
\[\big{[}\|E\|\big{]}^{2}:=\|E\|^{2}+\|\
abla E\|^{2}+\|\
abla\cdot E\|^{2}_{ \varepsilon-1+|\
abla\varepsilon|}, \tag{45}\]  
which, as an immediate concern, requires the condition below  
\[|\
abla\varepsilon|<\min(1/2,\varepsilon-1). \tag{46}\]  
The **coercivity** of \(a(\cdot,\cdot)\) is due to vanishing boundary terms and assuming (for simplicity) that \(|\Omega|=1\), we use the Poincare inequality \(\|E\|\leq\|\
abla E\|\), and the second condition in (46) to write \[a(E,E)= (\
abla E,\
abla E)+((\
abla\varepsilon)\cdot E,\
abla\cdot E)+(( \varepsilon-1)\
abla\cdot E,\
abla\cdot E) \tag{47}\] \[\geq \frac{1}{2}\|E\|^{2}+\frac{1}{2}\|\
abla E\|^{2}-\frac{1}{2}\|E\|^ {2}_{|\
abla\varepsilon|}-\frac{1}{2}\|\
abla\cdot E\|^{2}_{|\
abla \varepsilon|}\] \[+\frac{1}{4}\|\
abla\cdot E\|^{2}_{\varepsilon-1}+\|\
abla\cdot E \|^{2}_{|\
abla\varepsilon|}\] \[\geq \frac{1}{2}\|E\|^{2}+\frac{1}{4}\|\
abla E\|^{2}+\frac{1}{4}\| \
abla\cdot E\|_{(\varepsilon-1+2|\
abla\varepsilon|)}\] \[\geq \frac{1}{4}\Big{(}\|E\|^{2}+\|\
abla E\|^{2}+\|\
abla\cdot E\|^{2 }_{\varepsilon-1+|\
abla\varepsilon|}\Big{)}=\frac{1}{4}\{\|E\|^{2},\]  
where the second inequality is a consequence of reusing Poincare inequality for the term \(-\frac{1}{2}\|E\|^{2}_{|\
abla\varepsilon|}\), and we have a desired coercivity for \(a(\cdot,\cdot)\).  
The **continuity** of \(a(\cdot,\cdot)\) is evident from the coercivity (47) and (46), which yields  
\[\max\Big{(}\|E\|,\|\
abla E\|,\|\
abla\cdot E\|_{\varepsilon-1+| \
abla\varepsilon|}\Big{)}\leq\frac{1}{2}[\|E\|] \tag{48}\] \[a(\eta,\xi) =(\
abla\eta,\
abla\xi)+((\
abla\varepsilon)\cdot\eta,\
abla \cdot\xi)+((\varepsilon-1)\
abla\cdot\eta,\
abla\cdot\xi)\] (49) \[\leq\|\
abla\eta\|\|\
abla\xi\|+\|\eta\|_{|\
abla\varepsilon|}\| \
abla\cdot\xi\|_{|\
abla\varepsilon|}+\|\
abla\cdot\eta\|_{\varepsilon-1}\| \
abla\cdot\xi\|_{\varepsilon-1}\] \[\leq\frac{1}{4}[\|\eta\|][\|\xi\|]+\frac{1}{4}[\|\eta\|][\|\xi\|]+ \frac{1}{4}[\|\eta\|][\|\xi\|]\] \[=\frac{3}{4}[\|\eta\|][\|\xi\|]\leq[\|\eta\|][\|\xi\|].\]