h2. Introduction 
Omega3P is a parallel finite-element electrogmagnetic code for high-fidelity modeling of cavities. It calculates the resonant frequencies and other rf parameters of a cavity, as well as damping effects due to external couplings. It uses Nedelec-type hierarchical vector basis up to 6th order with quadratic (10-points) tetrahedral elements for improved solution accuracy. 

h2. Mathematical Modeling
Maxwell's equations in the frequency domain for a perfectly conducting cavity can be written as the following PDE,
{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}

We use Neelec-type vector basis functions to descretize the electric field {latex}\[\vec{\mathbf E}=\sum_i x_i \vec{\mathbf N}_i\]{latex} and obtain the following generalized eigenvalue problem:
{latex}
\begin{eqnarray*}
{\mathbf Kx}  = k^2 {\mathbf Mx} \quad where & \\
{\mathbf K}_{ij} =  \int_{\Omega} \left( \nabla \times \vec{\mathbf N}_i \right) \cdot \frac{1}{\mu} \left( \nabla \times \vec{\mathbf N}_j \right) d\Omega & \\
{\mathbf M}_{ij} =  \int_{\Omega} \vec{\mathbf N}_i  \cdot \epsilon \vec{\mathbf N}_j  d\Omega & \\
\end{eqnarray*}
{latex}