# Finite element method for an equation requiring switch between spectral and temporal domain

Some equations (such as the non-linear schrödinger equation for pulse propagation) are more easily solved in the spectral form, but still need a representation in the temporal domain to calculate physical parameters such as the energy of the electric field. This results usually in the following steps (with $$E$$ the electric field in the temporal domain, and $$\hat{E}$$ the electric field in the spectral domain): $$E_0\underbrace{\rightarrow}_{\text{FFT}}\hat{E}_0\underbrace{\rightarrow}_{\text{Propagation}}\hat{E}_1\underbrace{\rightarrow}_{\text{IFFT}}E_1$$ which requires two different "grids". Is such a calculation possible using FEM, and if yes, how (after requiring two different grids)?

Background to this question is that I have additional equations coupled to that problem which preferably can be solved using FEM and do not require the transformations from the temporal domain to the spectral domain and back (after they only depend on $$E_x$$, not on $$\hat{E}_x$$). For this coupling I'd like to avoid having to have two different grids (one for the non-linear schrödinger equation, one for the FEM-covered equation) which I have to interpolate data in between.

• If you use a DFT matrix $W$, could you form a coupled system of equations (a block matrix expression) by substituting $\hat{E_x}=WE_x$? en.wikipedia.org/wiki/DFT_matrix – Charlie S Mar 17 at 14:26
• I think that you could use the same grid, that's the spatial discretization. What changes is the time discretization, isn't it? – nicoguaro Mar 18 at 15:17
• People do this sort of thing all the time in the geodynamo community. – Wolfgang Bangerth Mar 18 at 16:29
• @WolfgangBangerth: Do you have examples for that, by chance? I'm completely unfamiliar with this community. – arc_lupus Mar 19 at 9:03
• I'm not familiar with the literature in that arena either, but you could start by looking at the Rayleigh and Calypso codes and the literatures that surround them. Both are available from geodynamics.org. I bet you can also find papers by Gary Glatzmeier on the topic. – Wolfgang Bangerth Mar 19 at 13:37