Abstract: We apply here the second-order finite element method to the boundary-value problem of waveguides filled with inhomogeneous dielectrics. The coupling matrix of high-order element and other related formulas are developed and cast in a form that is convenient for computer programming. To determine the accuracy obtainable from the finite-element method of various order, the eigenvalues of a slab-loaded rectangular waveguide, which is amendable to an exact analysis, are computed by the first-and second-order method. It is shown that the accuracy of the second-order method is 10-10³ higher than that of the first-order method. The relative efficiency of the first-and second-order method is also discussed on the basis of numerical analysis by a computer.