permittivity and permeability, respectively. A first problem, which often arises in electromagnetics and that we will consider in Sections \[sec:pillbox\] and \[sec:tesla\_cavity\], amounts to determining the eigenmodes of a resonant cavity solving the following source problem: $$\label{eq:curlcurl_E}
\curlt \parT{\pmeab^{-1}\curlt\Ev} = \omega^2\pmitt\Ev~,$$ with suitable boundary conditions. The second type of problem we will consider arises in the stationary limit $\omega \rightarrow 0$, and amounts to determine the magnetic vector potential $\magvp$ regularized by the Coulomb gauge, solving the following equations: $$\label{eq:A_form_st}
\begin{aligned}
&\curlt \parT{\pmeab^{-1} \curlt \magvp} = \jv~, \\
&\divergt\parT{\pmitt\magvp} = 0~,
\end{aligned}$$ for a given source current density $\jv$ and with suitable boundary conditions.

The problems above will be solved on axisymmetric bounded Lipschitz domains $\voldomaxis \subset
\R^3$ represented in cylindrical coordinates. We will exploit the Fourier basis to define different de Rham complexes for each mode $m$, which will then be analysed and applied to some numerical examples in the following sections. In this work we will restrict our attention to the case $m\neq0$, which corresponds to functions that are not axisymmetric. Let us introduce the cylindrical coordinates $(\rho, \, z, \,
\theta)$, with the $z$-axis being the symmetry axis of the domain. The uncommon choice of placing the 