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.

Accuracy of the approximate velocity of the steady incompressible Navier-Stokes equations computed by the standard mixed finite element methods is often affected by the pressure. In order to circumvent or weaken the influence of pressure on the accuracy of the computed velocity, by combining grad-div stabilized method with two-level finite element method, this paper presents a kind of two-level grad-div stabilized finite element methods for solving the steady incompressible Navier-Stokes equations numerically. The basic idea of the methods is to first solve a grad-div stabilized nonlinear Navier-Stokes problems on a coarse grid, and then solve, respectively, Stokes-linearized, Newton-linearized and Oseen-linearized Navier-Stokes problem with grad-div stabilization on a fine grid. Numerical examples are given to verify the high efficiency of the two-level grad-div stabilized finite element methods.

Based on two-grid discretizations and domain decomposition techniques, this paper presents three parallel finite element algorithms for numerically solving the steady Navier-Stokes equations with damping term. The basic idea of the present algorithms is to first solve a fully nonlinear problem on a coarse grid to get a coarse grid solution, then solve Stokes, Oseen, and Newton linearized residual problems in parallel in overlapping local fine grid subdomains, and finally update the approximate solution in non-overlapping fine grid subdomains. The effectiveness of the proposed algorithms is demonstrated by some numerical examples.

In numerical solution of unsteady Navier-Stokes equations with standard finite element method, errors of computed velocity are usually affected by pressure errors, where smaller viscosity coefficients lead to greater velocity errors. To improve pressurerobustness, in this paper, we introduce a grad-div stabilization term to improve accuracy of approximate solutions. We present parallel two-level grad-div stabilized finite element algorithms for unsteady Navier-Stokes equations, where implicit Euler scheme and Galerkin finite element methods are used for temporal and spatial discretizations, respectively. At each time step, firstly we solve a nonlinear Navier-Stokes problem with grad-div stabilization on a coarse grid, and then linearized and grad-div stabilized problems are solved with Stokes, Oseen and Newton iterations on overlapping fine grid subdomains in a parallel manner to correct the coarse grid solution. Finally, numerical experiments are given to verify correctness of theoretical predictions and demonstrate effectiveness of the algorithms.

A parallel two-level Grad-div stabilized finite element algorithm for steady incompressible Stokes equations is proposed. Basic idea of the algorithm is to solve global Grad-div stabilized solution in a coarse mesh firstly, and then correct it in parallel on overlapping fine mesh subdomains. With reasonable selection of stabilization parameters and mesh sizes, an optimal convergence rate can be obtained. Numerical results verify efficiency of the algorithm.