We apply an analysis scheme combining propagation operator method (POM) and beam propagation method (BPM) to waveguide discontinuity problems. In this approach, finite element method (FEM) with higher adaptability to waveguide geometry is utilized for discretizing waveguide cross section. While BPM based on FEM (FE-BPM) can efficiently analyze waveguides with long propagation distance, it encounters serious degradation of accuracy when waveguides with discontinuous structure has to be analyzed. The presented method overcomes this defect by applying POM which can analyze discontinuous facets and sufficiently cover various mode coupling.