Mixed Unfitted finite element approximation of band structures in photonic crystals

Abstract

Photonic crystals are periodic dielectric structures whose optical properties are characterized by Maxwell eigenvalue problems with discontinuous material coefficients. For arbitrarily shaped inclusions, their numerical solution remains challenging due to costly mesh generation and remeshing as the geometry changes. In this paper, we address this challenge by developing a mixed unfitted finite element method based on the Floquet-Bloch transform. Our method uses a uniform background mesh independent of the inclusion geometry, thereby avoiding interface-fitted meshing and allowing us to impose the periodic boundary condition through a direct identification of degrees of freedom. A key feature of our construction is that the resulting finite element pairs preserve the de Rham commuting relation for the shifted operator. This structure enables us to establish an inf-sup condition uniformly with respect to the cut position and, in particular, to rule out spurious modes. We further establish optimal convergence of the eigenvalues and eigenspaces, with constants independent of both the mesh size and the position of the interface. Finally, we present extensive numerical experiments in two and three dimensions to validate our theoretical results and demonstrate the robustness of the proposed method.