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.
