features is that it provides discrete approximations that, thanks to an appropriate choice of the finite-dimensional approximation spaces, indeed satisfy discrete analogous of the continuous Maxwell equations. In more technical terms, in the spirit of de Rham [@monk:2003], sequences of functional spaces which are closed and exact can be approximated by closed and exact sequences at a discrete level.

The combination of splines and Fourier bases in cylindrical coordinates effectively mitigates two problems. On the one hand, electrotechnical devices, particularly those based on high-frequency electromagnetic fields, are very sensitive to shape variations and are usually described by solutions of high regularity [@corno:2016], which favours splines-based field and geometry representations. On the other hand, tools rarely provide volumetric representations. Therefore, considering the cylindrical coordinate system also helps in bridging the gap between and simulation. Furthermore, the orthogonality of the Fourier representation allows to decouple the computation associated to different Fourier modes. Notice that, recently, the scaled boundary approach was proposed and demonstrated for Laplace-type problems in [@arioli:2019]. While not limited to axisymmetric domains, this approach does not exploit the orthogonality of the Fourier basis and does not display high convergence rates of the spectral method proposed here.

The paper is structured 