We present an efficient method for solving a class of \(4\times 4\) block saddle-point systems arising from the finite element discretization of the generalized three-dimensional Stokes problem. The spectral properties of the preconditioned system are investigated, including the distribution of eigenvalues and the behavior of the associated eigenvectors. To efficiently handle multiple right-hand sides within the resulting subsystems, we propose the Preconditioned Global Conjugate Gradient (PGCG) method as a block iterative solver and establish new convergence results. Numerical experiments demonstrate that the proposed preconditioned iterative approach substantially improves the efficiency of solving the 3D Stokes problem.
