The authors consider the approximate solution of the first-order hyperbolic differential equation \[ (\underline a.\underline\nabla)u= bu=f \] over a domain with a boundary condition \[ u= g. \] They extend the work of \textit{K. S. Bey} and \textit{J. T. Oden} [Comput. Methods Appl. Mech. Eng. 144, No. 3-4, 259-286 (1996; Zbl 0894.76036)] on the discontinuous Galerkin and the streamline-diffusion methods, and obtain error estimates in terms of \(h\) the mesh interval size and \(p\) the polynomial degree. They obtain bounds for \(u\) in terms of \(f\) and \(g\) and provide error estimates, and treat the problem of hanging nodes. Some numerical results obtained show that the methods discussed are satisfactory.