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Reading up on simulations of electromagnetic scattering with DG-FEM and trying some myself, I got stuck.

In some of papers I have read, a scattering field formulation is used, in which the total field is linearly decomposed in incident field and scattering field:

[itex] E^{T}=E^{S}+E^{I}[/itex]

And, the 2D equations for the scattering field in a lossless, isotropic medium are:

[itex] \epsilon_{r} \frac{\partial E^{S}}{\partial t} = \nabla \times H^{S} - (\epsilon_{r} - \epsilon_{r}^{I}) \frac{\partial E^{i}}{\partial t} [/itex]

[itex] \mu_{r} \frac{\partial H^{S}}{\partial t} = -\nabla \times E^{S} - (\mu_{r} - \mu_{r}^{I}) \frac{\partial H^{i}}{\partial t} [/itex]

My problem is in the interpretation of the "scattering field" and "incident field" in this context. In every use I see of this formulation [itex]\epsilon_{r}[/itex] is space dependent, while [itex]\epsilon_{r}^{I}[/itex] is a constant - specifically, the incident medium's permittivity (same for the permeability). How can this work for multi-substrate cases, where, if I am thinking correctly, the medium considered incident should change?

(I am quite confused with the affair in general, so any clarifications are quite welcome)

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# Scattering field formulation used in DG-FEM

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