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In this talk, we will introduce how the Fast Fourier Transform (FFT) is used to solve the discrete Maxwell eigenvalue problem (MEP). We first take a one-dimensional Poisson equation as an example to introduce the relationship between Fourier transform and discrete operators. Then, aiming at the discrete Maxwell eigenvalue problem introduced by Yee’s finite difference method, we will show the role of Fourier transform in matrix decomposition of discrete curl operators. In our numerical experiments, the computation times for FFT-based matrix-vector multiplications with matrices of dimension 7 million are only 0.33 and 3.6 × 10 − 3 seconds using MATLAB with an Intel Xeon CPU and CUDA C++ programming with a single NVIDIA Tesla P100 GPU, respectively. Such multiplications significantly reduce the computational costs of the conjugate gradient method for solving linear systems. These results demonstrate the potential of our proposed algorithm to enable large-scale numerical simulations for novel physical discoveries and engineering applications of photonic crystals.