  We propose an efficient finite-element analysis of the vector wave equation
in a class of relatively general curved polygons. The proposed method is
suitable for an accurate and efficient calculation of the propagation constants
of waveguides filled with pieces of homogeneous materials. To apply the method,
we first decompose the 2-D problem domain into a set of curved polygons of a
specific characteristic. Then we divide every polygon into a set of triangular
elements with two straight edges. Finally, we introduce a set of hierarchical
mixed-order curl-conforming vector basis functions inside every triangular
element to discretize the vector wave equation. The advantages of the method
are as follows. The curved boundaries of the elements are modeled exactly and
hence there is no approximation in the geometrical modeling. 2-D integrals of
the matrix elements are reduced to 1-D integrals. Therefore, the matrix filling
can be performed very fast. Total number of elements due to the discretization
of a given domain is very small, and hence the discretization of the problem
domain takes up a very small percentage of the total computational time. We
validate the method by comparing the results with those of Ansoft HFSS
simulator and investigate the accuracy and efficiency of the method through
some numerical examples.
