In this talk we consider Photonic Crystals which are expected to play a key role in future optical communication devices. Photonic Crystals consist of a periodic arrangement of dielectric materials at the nano-scale. Due to wave refraction and interference Photonic Crystals inhibit the propagation of light waves with certain frequencies. The so-called photonic band structure, which consists of countably many bands, mainly determines the optical properties of the Photonic Crystal. The frequencies for which the interference inside the crystal is destructive are called photonic band gaps. After having discussed the mathematical model for Photonic Crystals we will discuss the use of Fourier series in simulations. Since the material properties are modeled by discontinuous functions, oscillations at the locations of the material boundaries arise. We discuss attempts to circumvent these approximation problems.