A new hybrid streamline upwind finite-element method, which we mentioned in the previous paper, is based on the finite analytic method developed recently in the finite difference method. Using an adjoint discrete del operator and a discrete del operator which are defined on a dual space, it can be anticipated that we can achieve both the reduction of required computer memory and the highly accurate calculation of the finite-element method. In the present paper, we will discuss a steady one-dimensional advection-diffusion problem, for which exist some exact solutions or other numerical solutions. In order to verify the above-mentioned hybrid streamline upwind finite-element method, an unsteady one-dimensional Burgers' equation, which is well known as a typical example of convection-dominated flows, is numerically calculated at the Reynolds numbers Re(=1/ν)=10, 102, 105. Furthermore, the Neumann's stability condition of the present scheme is also obtained for the one-dimensional advection-diffusion problem.