A Mixed-FEM for Navier–Stokes type variational inequality with nonlinear damping term

Qiu, Hailong; Mei, Liquan

doi:10.1016/j.camwa.2017.02.046

Cited by 16 publications

(12 citation statements)

References 13 publications

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“…Proof. Equations 15 and 17 can be found in Qiu et al 22 and Ciarlet, 45 and (18) is referred to Borggaard et al 46 We only need to prove (16). In fact, for any , ∈ R 2 , using Lagrange mean value theorem, we have…”

Section: Mixed Variational Formulationmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

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confidence: 99%

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

Shi

2018

Math Methods in App Sciences

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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.

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Section: Mixed Variational Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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

Shi

2018

Math Methods in App Sciences

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“…The next theorem gives the inf‐sup condition of (7), the proof can be found in Ref. . Theorem

\exists β > 0

satisfying

\inf_{χ \in M} \sup_{v \in V} \frac{d (v, χ)}{‖ v ‖_{V} ‖ χ ‖_{0, 2}} \geq β .

Theorem Assume

f \in L^{2} {(Ω)}^{d}, g \in L^{2} (S)

r

satisfy ( H0 ), and if

\frac{C_{*} {(κ (‖ f ‖ + ‖ g ‖_{L^{2} (S)}))}^{r - 2}}{μ_{0}^{r - 1}} < \frac{1}{2} a n d \frac{4 κ N (‖ f ‖ + ‖ g ‖_{L^{2} (S)})}{μ_{0}^{2}} < \frac{1}{2} h o l d,

with

C_{*} = α C_{0}^{r} \cdot \max {4^{r - 2} {\overset{C}{true}}_{1}, 2^{r - 1}},

then the variational inequality problem (7) admits a unique solution

u \in Σ_{0} : = {v \in V_{0} : ‖ v ‖_{V <...>}

…”

Section: Preliminariesmentioning

confidence: 99%

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

Qiu

Mei

Xue

2017

Numerical Methods Partial

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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

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“…The system (1.1) is a coupling system between Navier-Stokes equations and the deformation gradient with a damping term. When the damping term is ab- [24]. In this paper, we consider the global existence and L 2 -norm decay rates of the compressible viscoelastic flows with the term for 1 β = in H 3 framework.…”

Section: Introductionmentioning

confidence: 99%

Global Existence and Optimal Decay Rates for Three-Dimensional Compressible Viscoelastic Flows System with Damping

Xu¹

2018

JAMP

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In this paper, we consider the large time behavior of the strong solutions to the three dimensional compressible viscoelastic flows with damping. Based on the energy method and spectral analysis, we analyze the influences of the damping on the global existence and decay rates of compressible viscoelastic flows under some small assumptions in H 3-framework. Compared with the time decay rates of solutions to the compressible viscoelastic flows in [1], our results imply that the friction of the damping is stronger than the dissipation effect of the viscosities.

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A Mixed-FEM for Navier–Stokes type variational inequality with nonlinear damping term

Cited by 16 publications

References 13 publications

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

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

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

Global Existence and Optimal Decay Rates for Three-Dimensional Compressible Viscoelastic Flows System with Damping

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