Finite Element Discretization

Writing the electric field as
\begin{equation} \label{eq:Efinite} E = \sum_{j}\tilde E_{j}\hat e_{j}(\hat x) \end{equation}
The finite elements $ \hat e_{j} $ vary linearly in the directions orthogonal to $ \hat e_{j} $ . For example, edges pointing in the x-direction vary linearly in the y and z directions but the value of the field remains constant along x. Below is a diagram of an element with the edge vectors shown.


A standard finite element discretization technique [see reference] is applied to the vector Helmholtz equation. Using expansion (1) we get

\begin{equation} \label{eq:Eigenprob} \sum_{j}A_{ij}E_{j} = \lambda \sum_{j}B_{ij}E_{j} \end{equation}

where $ \lambda \equiv \frac{\omega^{2}}{c^{2}} $ ,

\begin{equation} \label{eq:Aij} A_{ij} = \int dV (\nabla + i \hat k)^{*} \times \hat e_{i} \cdot (\nabla + i \hat k) \times \hat e_{j} \end{equation}


\begin{equation} \label{eq:Bij} B_{ij} = \int dV \hat e_{i} \cdot \epsilon \cdot \hat e_{j} \end{equation}
Both of these matrices (A and B) are sparse and the integrations are done over the unit cell. Also, recall that we could as easily replaced the E's in the above equations with H's (keeping in mind the different B.C.'s)
Ryan McClarren