The finite element method has been used for the modelling of light propagation in periodic photonic nanostructures. The two-dimensional vectorial wave equation has been solved for photonic crystals of different lattice types with a linear defect at frequencies lying within or outside of the photonic bandgap. Moreover, the transmission of electromagnetic radiation through a double metallic grating with a sub-wavelength pitch at one of its sides was also studied. Computations were performed for an incident wavelength of 400 nm for a series of structures consisting of either silver, gold, chromium or a perfect conductor.