In the three dimensional (3D) cylindrical kind domain, the finite-element dimension splitting algorithm (DSA) of steady/unsteady Stokes equation based on a consistent splitting scheme is proposed in this paper. The numerical schemes of steady/unsteady Stokes equation based on the DSA are presented. The advantages of the DSA are that our numerical schemes can be easily solved in parallel with several processors, and only 2D meshes instead of 3D meshes are involved in the numerical implementation. A great number of numerical results show that the DSA algorithm can be up to the optimal convergence rates, and obtain more accurate approximate solution than the 3D Galerkin finite-element method with 4-node tetrahedron elements. Finally, the good speed-up and parallel efficiency of the DSA are obtained by a parallel solving technique.