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{latex}
\begin{eqnarray*}
\nabla \times \left(\frac{1}{\mu} \nabla \times \vec{\mathbf E}\right) - k^2 \epsilon \vec{\mathbf E} & = 0 & on \quad \Omega \\
\vec{\mathbf n} \times \vec{\mathbf E} & = 0 & on \quad \Gamma_{E} \\
\vec{\mathbf n} \times \left( \frac{1}{\mu} \nabla \times \vec{\mathbf E} \right) & = 0 & on \quad \Gamma_{M} \\
\end{eqnarray*}
{latex}
{center} |
We use NeelecNedelec-type vector basis functions to discretize the electric field:
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\[
{\mathbf Kx} + i \sum_j \sqrt{k^2-k^2_{cj}} {\mathbf W}_j {\mathbf x} = k^2 {\mathbf Mx}
\]
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{center:class=myclass}
{latex}
\[
{\mathbf Kx} + i \sum_{m,n} \sqrt{k^2-(k^{TE}_{mn})^2} {\mathbf W}^{TE}_{mn} {\mathbf x} + i \sum_{m,n} \frac{k^2}{\sqrt{k^2-(k^{TM}_{mn})^2}} {\mathbf W}^{TM}_{mn} {\mathbf x} = k^2 {\mathbf Mx}
\]
{latex}{center} |
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{center:class=myclass}{latex}
\[
({\mathbf W}^{TE}_{mn})_{ij} = \int_{\Gamma} \vec{\mathbf e}^{TE}_{mn} \cdot \vec{\mathbf N}_i d\Gamma \int_{\Gamma} \vec{\mathbf e}^{TE}_{mn} \cdot \vec{\mathbf N}_j d\Gamma
\]
\[
({\mathbf W}^{TM}_{mn})_{ij} = \int_{\Gamma} \vec{\mathbf e}^{TM}_{tmn} \cdot \vec{\mathbf N}_i d\Gamma \int_{\Gamma} \vec{\mathbf e}^{TM}_{tmn} \cdot \vec{\mathbf N}_j d\Gamma
\]
{latex}{center} |
Numerical Methods
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