In the fields of ocean engineering and aerospace, the motion state of fluids has a significant impact on the performance and stability of systems. Stokes equations with damping terms are commonly used to describe the flow behavior of fluids under damping, such as the fluids in porous media. Two numerical iterative algorithms based on finite element discretization are proposed for the steady incompressible Stokes questions with the nonlinear damping term. The basic idea is to first use the finite element method to solve the Stokes problem and obtain the initial iterative solution. Secondly, it uses the Stokes iterative algorithm or Oseen iterative algorithm to solve the steady incompressible Stokes problem with nonlinear damping term and obtain approximate finite element solutions. Convergence and stability of the proposed algorithms are analyzed. Error estimates of the obtained approximate solutions are derived. Some numerical results are also given to show correctness of theoretical analysis and effectiveness of the algorithms. The results show that when the equation satisfies the stability condition, both numerical iterative algorithms are feasible.