Abstract: One- and two- dimensional discontinuous Galerkin finite-element (DGFE) operators for solving Maxwells equations are investigated in this paper. These operators are based on unconventional spaces of approximation functions. Unlike in the existing DGFE method where the field components E and H are expanded with the same family of basis functions, in our scheme, it is attempted to compose various discontinuous Galerkin operators by approximating each of the two components in different function spaces and employing center numerical fluxes. With the combination of various basis function spaces for E and H, we build a series of operators. Through the calculation of the resonant modes of one- and two- dimensional PEC cavities on regular and irregular meshes, the convergence and spuriousness-supporting properties of these operators are examined, and based on which a selection of the optimal spaces of basis functions is made. Resonant modes calculated in time-domain and frequency-domain agree well with each other. It is shown not only that the proposed scheme is both non-dissipative and spurious-free, but also that no additional auxiliary variables are required, thus providing a new way to develop high quality algorithm for the Maxwells equations and the corresponding electromagnetic field simulating software.