A penalty‐FEM for navier‐stokes type variational inequality with nonlinear damping term

Abstract

In this article, we consider a penalty finite element (FE) method for incompressible Navier‐Stokes type variational inequality with nonlinear damping term. First, we establish penalty variational formulation and prove the well‐posedness and convergence of this problem. Then we show the penalty FE scheme and derive some error estimates. Finally, we give some numerical results to verify the theoretical rate of convergence. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 918–940, 2017

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

“…In Li et al, the local projection stabilized MFEMs were proposed. However, for the problem , there were few numerical methods reported except Qiu et al, in which MFEMs were developed for the Navier‐Stokes–type variational inequality with damping.…”

“…In Li et al, 21 the local projection stabilized MFEMs were proposed. However, for the problem (1), there were few numerical methods reported except Qiu et al, 22,23 in which MFEMs were developed for the Navier-Stokes-type variational inequality with damping.On the other hand, the two-level method was introduced by Xu for solving nonlinear elliptic problems. 24,25 Its basic idea is solving one nonlinear system on a coarse mesh as an iterative initial value approximation of a fine mesh and then solving one linear system on the fine mesh.…”

Optimal $L^2$ error estimates of the penalty finite element method for the unsteady Navier-Stokes equations with nonsmooth initial data

“…Subsequently, many publications have been devoted to the study of the penalty method for the steady Stokes and NSEs, as well as for the unsteady NSEs, in continuous, semi-discrete and fully discrete cases, see, for example, [4,11,13,16,28] and reference, therein. Recent results on the penalty method can be categorised as follows: two-grid penalty method [3,15,2], iterative penalty method [7,14], methods based on different boundary conditions such as nonlinear slip boundary conditions [2,7,21], friction boundary conditions [22,24], slip boundary conditions [31], etc. Moreover, penalty method is used, very recently, for the stochastic 2-D incompressible NSEs [20], and for the incompressible NSEs with variable density [1].…”

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

“…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.…”