A method for the solution of discontinuity problems in a planar dielectric waveguide is described. The approach is a combination of the finite element method and the analytical method. The finite element approach is used for the interior region enclosing the waveguide discontinuities, while the analytical approach is utilized for the exterior region. The singularities of Green's functions can be excluded by overlapping these regions. In order to minimize the details, the symmetric transverse electric mode excitation of a symmetric slab waveguide is considered. The extension to the antisymmetric case, the transverse magnetic case, and the asymmetric slab waveguide presents no difficulty.