The axisymmetric vortex breakdown of a laminar, incompressible, viscous, and swirling flow in a cylindrical tube is simulated. The numerical solutions of the time-dependent axisymmetric angular momentum and the vorticity transport equations are obtained by using the finite element method. Linear and quadratic isoparametric elements are used for the space discretization, while the leapfrog time integration scheme is employed for the time discretization. The iteration process is started with the numerical solution of the linearized equations as the Reynolds number approaches zero. As the Reynolds number is changed, the numerical solution proceeds from one steady state to another. The numerical solutions are obtained for different initial velocity profiles and a range of Reynolds numbers and swirl ratios. The numerical solutions ar, compared with experimental data and prior finite difference studies.