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An efficient and accurate large-domain higher order two-dimensional (2-D) Galerkin-type technique based on the finite-element method (FEM) is proposed for analysis of arbitrary electromagnetic waveguides. The geometry of a waveguide cross section is approximated by a mesh of large Lagrangian generalized curvilinear quadrilateral patches of arbitrary geometrical orders (large domains). The fields over the elements are approximated by a set of hierarchical 2-D polynomial curl-conforming vector basis functions of arbitrarily high field-approximation orders. When compared to the conventional small-domain 2-D FEM techniques, the large-domain technique requires considerably fewer unknowns for the same (or higher) accuracy and offers a significantly faster convergence when the number of unknowns is increased. A comparative analysis of solutions using p- and h-refinements shows that the p-refinement represents a better choice for higher accuracy with lesser computation cost. In addition to increasing the field-approximation orders, the geometrical orders of elements (where needed) should also be set high for the improved accuracy of the solution without subdividing the elements. However, in general, an arbitrarily high accuracy cannot be achieved by performing the p-refinement in arbitrarily coarse meshes alone; instead, a combined hp-refinement should be utilized in order to obtain an optimal modeling performance.