A numerical study was conducted for the vortex-induced vibrations of an elastic circular cylinder at low Reynolds numbers. An Arbitrary Lagrangian-Eulerian (ALE) method was employed to deal with the fluid-structure interaction with an H-O type of non-staggered grids incorporating the domain decomposition method(DDM), which could save the computational CPU time due to re-meshing. The computational domain was divided into nine sub-domains including one ALE sub-domain and eight Eulerian sub-domains. The convection term and dissipation term in the N-S equations were discretized using the third-order upwind compact scheme and the fourth-order central compact scheme, respectively. The motion of the cylinder was modeled by a spring-damper-mass system and solved using the Runge-Kutta method. By simulating the non-linear fluid-structure interaction, the "lock-in", "beating" and "phase switch" phenomena were successfully captured, and the results agree with experimental data. Furthermore, the vortex structure, the unsteady lift and drag on the cylinder, and the cylinder displacement at various natural frequency of the cylinder for Re=200 were discussed in detail, by which a jump transition of the wake structure was captured.