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Photonic crystals are materials with periodically modulated dielectric constant, through which certain frequencies of electromagnetic radiation cannot propagate; the luminary analogues of semiconductors. The modes admitted by photonic crystals can be investigated effectively using the finite element method with the assistance of the Bloch-Floquet theorem, by considering a unit cell of the material and imposing periodic boundary conditions. Along with the Dirichlet and metric matrices, a third type of elemental matrix emerges. The types of results that are of interest to photonic crystal manufacturers are introduced and presented; in this context, the benefits of using subspace iteration techniques to solve the eigensystems are discussed.