## 6 Error Estimates: Semi-Discrete (SD) Problems
### Error Estimates: SD Problem in \(\Omega_{FEM}\)  
Here the non-vanishing boundary data causes additional asymmetry (the first one: \(((\
abla\varepsilon)\cdot E,\
abla\cdot\mathbf{v})\) was tackled above), which may be symmetrized using a Nitsche-type approach of [46]. We circumvent [PERSON] invoking additional conditions on the Ritz projection at the boundary \(\partial\Omega_{FEM}\). The variational form and the finite element formulation in \(\Omega_{FEM}\) are  
\[ \begin{split}(\varepsilon\partial_{tt}E,\mathbf{v})_{\Omega_{FEM }}&+(\
abla E,\
abla\mathbf{v})_{\Omega_{FEM}}+((\
abla \varepsilon)\cdot E,\
abla\cdot\mathbf{v})_{\Omega_{FEM}}+((\varepsilon-1 )\
abla\cdot E,\
abla\cdot\mathbf{v})_{\Omega_{FEM}}\\ &-\langle\partial_{n}E,\mathbf{v}\rangle_{\partial\Omega_{FEM}}= 0,\qquad\forall\mathbf{v}\in\mathbf{W}^{E}(\Omega_{FEM}),\end{split}  \tag{59}\]  
and  
\[ \begin{split}(\varepsilon\partial_{tt}E_{h},\mathbf{v})_{\Omega_{ FEM}}&+(\
abla E_{h},\
abla\mathbf{v})_{\Omega_{FEM}}+((\
abla \varepsilon)\cdot E_{h},\
abla\cdot\mathbf{v})_{\Omega_{FEM}}+((\varepsilon -1)\
abla\cdot E_{h},\
abla\cdot\mathbf{v})_{\Omega_{FEM}}\\ &-\langle\partial_{n}E_{h},\mathbf{v}\rangle_{\partial\Omega_{FEM }}=0,\qquad\forall\mathbf{v}\in\mathbf{W}^{E}_{h}(\Omega_{FEM}),\end{split}  \tag{60}\]  
respectively. Hence, subtracting (60) from (59), where in (59) we restrict \(\mathbf{v}\in\mathbf{W}^{E}_{h}(\Omega_{FEM})\), we end up with the error equation for \(\varepsilon:=E-E_{h}\), in \(\Omega_{FEM}\):  
\[ \begin{split}(\varepsilon\partial_{tt}\tilde{e},\mathbf{v})_{ \Omega_{FEM}}&+(\
abla\tilde{e},\
abla\mathbf{v})_{\Omega_{FEM }}+((\
abla\varepsilon)\cdot\tilde{e},\
abla\cdot\mathbf{v})_{\Omega_{FEM }}+((\varepsilon-1)\
abla\cdot\tilde{e},\
abla\cdot\mathbf{v})_{\Omega_{FEM }}\\ &-\langle\partial_{n}\tilde{e},\mathbf{v}\rangle_{\partial\Omega_{ FEM}}=0,\qquad\forall\mathbf{v}\in\mathbf{W}^{E}_{h}(\Omega_{FEM}).\end{split}  \tag{61}\]  
To be concise, for \(\Omega_{FEM}\), we suppress the subscript \(\Omega_{FEM}\) from the inner products, and to distinguish from \(\Omega\), use \(\tilde{e}\), \(\tilde{B}\), and \(\tilde{a}\) for the error and the bilinear forms in \(\Omega_{FEM}\). Now we define a somewhat modified Ritz-Galerkin projection: \(\tilde{\mathcal{Q}}_{h}:H(div,\Omega_{FEM})\rightarrow\mathbf{W}^{E}_{h}( \Omega_{FEM})\) by  
\[\tilde{a}(\tilde{\mathcal{Q}}_{h}\mathbf{u},\mathbf{v})-\langle\partial_{n}( \tilde{\mathcal{Q}}_{h}\mathbf{u}),\mathbf{v}\rangle_{\partial\Omega_{FEM}}= \tilde{a}(\mathbf{u},\mathbf{v})-\langle\partial_{n}\mathbf{u},\mathbf{v} \rangle_{\partial\Omega_{FEM}}\quad\forall\mathbf{v}\in\mathbf{W}^{E}_{h}( \Omega_{FEM}), \tag{62}\]  
with \(\tilde{a}\) defined as \(a\), but now over \(\Omega_{FEM}\), and with the additional property of satisfying weak Neumann boundary condition that behaves like Galerkin approximation:  
\[\langle\partial_{n}(\tilde{\mathcal{Q}}_{h}\mathbf{u}),\mathbf{v}\rangle_{ \partial\Omega_{FEM}}=\langle\partial_{n}\mathbf{u}_{h},\mathbf{v}\rangle_{ \partial\Omega_{FEM}}\quad\forall\mathbf{v}\in\mathbf{W}^{E}_{h}(\Omega_{FEM }), \tag{63}\]  
Note that denoting \(\tilde{B}:=B_{\Omega_{FEM}}\), (61) yields a modified Galerkin orthogonality, viz.