## Convergence analysis of the FEM coupled with Fourier-mode expansion for the electromagnetic scattering by biperiodic structures

Guanghui Hu and Andreas Rathsfeld

### Abstract

Scattering of time-harmonic electromagnetic plane waves by a doubly periodic surface structure in $R^3$ can be simulated by a boundary value problem of the time-harmonic curl-curl equation. For a truncated FEM domain, non-local boundary conditions are required in order to satisfy the radiation conditions for the upper and lower half spaces. As an alternative to boundary integral formulations, to approximate radiation conditions and absorbing boundary methods, Huber et al. [SIAM J. Sci. Comput., 31 (2009), pp. 1500–1517] have proposed a coupling method based on an idea of Nitsche. In the case of profile gratings with perfectly conducting substrate, the authors have shown previously that a slightly modified variational equation can be proven to be equivalent to the boundary value problem and to be uniquely solvable. Now it is shown that this result can be used to prove convergence for the FEM coupled by truncated wave mode expansion. This result covers transmission gratings and gratings bounded by additional multi-layer systems.

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### Key words

electromagnetic scattering, diffraction gratings, convergence analysis, finite element methods, mortar technique

### AMS subject classifications

78A45, 78M10, 65N30, 35J20

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