A finite element algorithm for solving the Navier‐Stokes equations is presented for the analysis of high‐speed viscous flows. The algorithm uses triangular elements. The unsteady equations are integrated to steady state with a Runge‐Kutta time‐marching scheme. A postprocessing artificial dissipation term is introduced to stabilize the computations and to dampen dissipation errors. Numerical results are compared with the calculation of uniform flow on a rectangular region which encounters an embedded oblique shock. A shock/turbulent boundary layer problem is also solved and results are compared with experimental data. It is shown that the postprocessing smoothing term and boundary conditions similar to the finite difference method work well in the present numerical studies. Copyright © 1991 John Wiley & Sons, Ltd