In this study, the characteristics of two-dimensional dispersive dielectric photonic crystals are studied via a numerical approach that combines the finite-element method (FEM) with Newton's iterative method. For a dispersive photonic crystal whose permittivity is dependent on the frequency, the band structure problem is formulated as a nonlinear eigenvalue problem. In this study, first, a discrete variational formulation is derived from Maxwell's equations based on the FEM. Thereafter, by selecting approximate solutions as the initial values for the iteration, this nonlinear problem is solved for different values of the wave vector k based on Newton's iterative method. Consequently, the band structure of dispersive dielectric photonic crystals is obtained numerically. Several dispersive photonic crystals in the transverse electric (TE) and transverse magnetic (TM) modes are investigated. The numerical results reveal that the proposed method is effective for dispersive photonic crystals.