A finite-element method based on a function minimization techinique is developed for analyzing homogeneous waveguide problems. The continuous eigen-value operator as in the Ritz finite-difference method is replaced by a matrix operator. The resulting equations are, however, matrix eigen-value equations. Relevant properties of the matrices are discussed. It is demonstrated that the method has advantages in dealing with awkward boundaries and singularities. A feature of the method is that the error in the eigen-value is a monotonically decreasing function of successive sub-divisions of the cross-section. Minimization of a variational expression assures rapid convergence to the correct eigen-value. Three waveguides are analysed in some detail.