Stabilized mixed finite element methods for the Navier‐Stokes equations with damping

Abstract

In this paper, the stabilized mixed finite element methods are presented for the Navier‐Stokes equations with damping. The existence and uniqueness of the weak solutions are proven by use of the Brouwer fixed‐point theorem. Then, optimal error estimates for the H1‐norm and L2‐norm of the velocity and the L2‐norm of the pressure are derived. Moreover, on the basis of the optimal L2‐norm error estimate of the velocity, a stabilized two‐step method is proposed, which is more efficient than the usual stabilized methods. Finally, two numerical examples are implemented to confirm the theoretical analysis.

Two‐grid stabilized algorithms for the steady Navier–Stokes equations with damping

“…For problem (2.14), we further have the following stability and error estimate (see, e.g., other works [15][16][17] ).…”

“…Obviously, there are two appeal features in this paper. First, if we choose appropriate scalings between the coarse grid and fine grid, our proposed algorithms can yield an approximate solution with an accuracy comparable to that of the ones computed by one-grid and two-step stabilized algorithms 17 with a substantial decrease in CPU time. Second, compared with the stabilized P 1 − P 1 algorithm, our stabilized P 2 − P 2 method is much more efficient and takes less CPU time than the stabilized P 1 − P 1 one when both of them get nearly the same precision of the approximate solution.…”

“…5 Simultaneously, recent studies have paid more attention to devising some efficiently numerical approaches based on finite element approximations for the Stokes and Navier-Stokes equations with nonlinear damping term. We can refer to previous works [7][8][9][10][11][12][13][14][15][16][17] and the references therein. In the above methods, finite element methods are quite interesting from researchers in that the methods in approximating the solution domain are flexible, and their developments of theoretical analysis are perfect.…”

Unconditional Optimal Error Estimates for the Transient Navier-Stokes Equations with Damping

“…In [7], the superclose and superconvergence phenomenon of some stable MFEs were studied. In [8][9][10], the local projection stabilized MFEMs with the P 1 -P 1 element pair were proposed for the Stokes or Navier-Stokes equations with damping. In [11,12], the two-level and multi-level MFEMs were applied to the problem to save computation cost.…”

Weak Galerkin method for the Stokes equations with damping

“…For the Stokes equations with damping, there have been many relevant studies. In [11], the existence and uniqueness of the weak solution are proved and the conforming mixed finite element method (MFEM) is developed to discretize the model. In [19] the superclose and superconvergence results of MFEM for Stokes equations with damping are presented.…”