A semi-analytical dual edge element is proposed to solve the waveguide discontinuities. The governing equations for electromagnetic waveguide are converted to the Hamiltonian system, and the corresponding variational principle based on the dual variables is given. For waveguide sections which are homogeneous along the longitudinal direction, the dual edge element is employed to discretize the cross section, and a precise integration method based on the Riccati equations is used for the longitudinal integration to generate the export stiffness matrices. The whole waveguide discontinuity problems can be solved by combining the export stiffness matrices of homogeneous waveguide sections with the system matrices by conventional three-dimentional finite element method for inhomogneous waveguide sections. Numerical examples demonstrate the high accuracy and efficiency of this method for solving waveguide discontinuity problems.