Kernel Compensation Method for Maxwell Eigenproblem in Photonic Crystals With Mimetic Finite Difference Discretizations

We present a kernel compensation method for Maxwell eigenproblem in photonic crystals to avoid the infinite-dimensional kernels that cause many difficulties in the calculation of energy gaps. The quasi-periodic problem is first transformed into a periodic one on the cube by the Floquet-Bloch theory. Then the compensation operator is introduced in Maxwell's equation with the shifted curl operator. The discrete problem depends on the compatible discretization of the de Rham complex, which is implemented by the mimetic finite difference method in this paper. We prove that the compensation term of the discretization exactly fills up the kernel of the original discrete problem and avoids spurious eigenvalues. Also, we propose an efficient preconditioner and its FFT and multigrid solvers, which allow parallel computing. Numerical experiments for different three-dimensional lattices in photonic crystals are performed to validate the accuracy and effectiveness of the method.