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A three-dimensional vectorial finite element method (3-D-VFEM) for waveguide discontinuity problems is newly formulated and implemented. Through the use of the 3-D-VFEM, transmission characteristics of two types of crossing slot-waveguides with finite core-height are investigated for the first time. One is a direct crossing and the other is a crossing filling up the slots locally. Results show that three-dimensional analysis considering the finite core-height is indispensable.