The edge-based finite element method is introduced to systematically investigate the transmission characteristics in an antipodal overlapping finline loaded with right- and left-handed materials. Moreover, the influence of the thickness of metallic fins on the characteristics is also discussed. In addition, the paper discusses the influences of materials on the cutoff wavelength of the dominant mode, the single-mode bandwidth, the dispersion characteristics and field patterns of the dominant mode and the first higher-order mode. The results show that the cutoff wavelength of the dominant mode when the left-handed material is loaded is approximately less than that when the right-handed material is loaded. Moreover, the cutoff wavelength of the dominant mode and the single-mode bandwidth vary inversely with the expansion of the loading area, and the field line distribution of the dominant mode in the loading area is entirely different from that of the first high-order mode. However, the variation in the dispersion characteristics is similar. The numerical results provided herein are considered usefully in the design of new microwave and millimeter-wave devices.