A stabilised characteristic finite element method for transient Navier–Stokes equations

Zhang, Tong; Si, Zhiyong; He, Ya‐Ling

doi:10.1080/10618562.2010.535791

Cited by 16 publications

(16 citation statements)

References 27 publications

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“…17 in Lemma 4.6 of ):

| (ι (x, t_{m + 1}) \frac{\partial {boldu}^{m + 1}}{\partial τ} - \frac{{boldu}^{m + 1} - {\overset{u}{true}}^{m}}{Δ t}, v_{h}) | \leq C Δ t^{\frac{1}{2}} {((), \int_{t_{m}}^{t_{m + 1}} {‖\frac{\partial^{2} boldu}{\partial τ^{2}}‖}_{0}^{2} italicds)}^{\frac{1}{2}} {(‖‖, {boldv}_{h})}_{H_{f}} .

By eq. 18 in Lemma 4.6 of and the Hölder inequality, we can bound the third term on the RHS of Equation as follows:

\begin{array}{l} | ((),, \frac{([], ((), u^{m + 1} - u_{h}^{m + 1}) - ((), {\overset{true‾}{boldu}}^{m} - {\overset{true^}{boldu}}_{h}^{m}))}{Δ t}, {boldv}_{h}) | \\ \leq | ((),, \frac{({boldu}^{m + 1} - {boldu}_{h}^{m + 1}) - ({boldu}^{m} - {boldu}_{h}^{m})}{Δ t}, {boldv}_{h}) | + | ((),, \frac{([], ((), u^{m} - u_{h}^{m}) - ((), {\overset{true‾}{boldu}}^{m} - {\overset{true^}{boldu}}_{h}^{m}))}{Δ t}, {boldv}_{h}) | \\ \leq \frac{C}{Δ t} ((), \int_{t_{m}}^{t_{m + 1}} ‖boldu<...>‖) \end{array}

…”

Section: The Error Analysismentioning

confidence: 84%

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Decoupled characteristic stabilized finite element method for time‐dependent Navier–Stokes/Darcy model

Jia

2018

Numerical Methods Partial

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In this article, we propose and analyze a new decoupled characteristic stabilized finite element method for the time‐dependent Navier–Stokes/Darcy model. The key idea lies in combining the characteristic method with the stabilized finite element method to solve the decoupled model by using the lowest‐order conforming finite element space. In this method, the original model is divided into two parts: one is the nonstationary Navier–Stokes equation, and the other one is the Darcy equation. To deal with the difficulty caused by the trilinear term with nonzero boundary condition, we use the characteristic method. Furthermore, as the lowest‐order finite element pair do not satisfy LBB (Ladyzhen‐Skaya‐Brezzi‐Babuska) condition, we adopt the stabilized technique to overcome this flaw. The stability of the numerical method is first proved, and the optimal error estimates are established. Finally, extensive numerical results are provided to justify the theoretical analysis.

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“…17 in Lemma 4.6 of ):

| (ι (x, t_{m + 1}) \frac{\partial {boldu}^{m + 1}}{\partial τ} - \frac{{boldu}^{m + 1} - {\overset{u}{true}}^{m}}{Δ t}, v_{h}) | \leq C Δ t^{\frac{1}{2}} {((), \int_{t_{m}}^{t_{m + 1}} {‖\frac{\partial^{2} boldu}{\partial τ^{2}}‖}_{0}^{2} italicds)}^{\frac{1}{2}} {(‖‖, {boldv}_{h})}_{H_{f}} .

By eq. 18 in Lemma 4.6 of and the Hölder inequality, we can bound the third term on the RHS of Equation as follows:

\begin{array}{l} | ((),, \frac{([], ((), u^{m + 1} - u_{h}^{m + 1}) - ((), {\overset{true‾}{boldu}}^{m} - {\overset{true^}{boldu}}_{h}^{m}))}{Δ t}, {boldv}_{h}) | \\ \leq | ((),, \frac{({boldu}^{m + 1} - {boldu}_{h}^{m + 1}) - ({boldu}^{m} - {boldu}_{h}^{m})}{Δ t}, {boldv}_{h}) | + | ((),, \frac{([], ((), u^{m} - u_{h}^{m}) - ((), {\overset{true‾}{boldu}}^{m} - {\overset{true^}{boldu}}_{h}^{m}))}{Δ t}, {boldv}_{h}) | \\ \leq \frac{C}{Δ t} ((), \int_{t_{m}}^{t_{m + 1}} ‖boldu<...>‖) \end{array}

…”

Section: The Error Analysismentioning

confidence: 84%

“…First, let us recall the following lemma.Lemma (Lemma 4.2 , ). For any

u \in H_{0}^{1} {((), {normalΩ}_{f})}^{2}

, we have

(\overset{u}{true}, \overset{u}{true}) - (boldu, boldu) \leq C Δ t (boldu, boldu),

where

\overset{true^}{boldu} = u (x - u (x, t)) .

.…”

Section: The Stability Analysismentioning

confidence: 99%

Decoupled characteristic stabilized finite element method for time‐dependent Navier–Stokes/Darcy model

Jia

2018

Numerical Methods Partial

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“…The choice of the stabilisation parameter d has been discussed by Becker and Hansbo (2008), Zhang et al (2010) and Feng et al (2011). Here, we choose d ¼ 10, which can lead to good results.…”

Section: Numerical Experimentsmentioning

confidence: 98%

A stabilised nonconforming finite element method for steady incompressible flows

Huang

Feng

Liu

2012

International Journal of Computational Fluid Dynamics

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A stabilised nonconforming finite element method for the steady incompressible flow problem with damping based on local Gauss integration is considered in this article. The method combines the nonconforming finite element method with the stabilised strategy. Moreover, the stability and error estimates are analysed. Finally, numerical results are shown to support the developed theory analysis. Compared with some classical, closely related mixed finite element methods, the results of the present method show its better performance than others.

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“…For instance, Russell considered the nonlinear coupled systems in [27], Süli studied the Navier-Stokes equations in [29]. The MMOC is based on the approximation of the material derivative term, that is, the time derivative term plus the convection term, and this scheme works well for convection dominant problem (see [35] and the reference therein).…”

Section: Introductionmentioning

confidence: 98%

Two-Grid Characteristic Finite Volume Methods for Nonlinear Parabolic Problem

Zhang¹

2013

JCM

Self Cite

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In this work, two-grid characteristic finite volume schemes for the nonlinear parabolic problem are considered. In our algorithms, the diffusion term is discretized by the finite volume method, while the temporal differentiation and advection terms are treated by the characteristic scheme. Under some conditions about the coefficients and exact solution, optimal error estimates for the numerical solution are obtained. Furthermore, the twogrid characteristic finite volume methods involve solving a nonlinear equation on coarse mesh with mesh size H, a large linear problem for the Oseen two-grid characteristic finite volume method on a fine mesh with mesh size h = O(H 2 ) or a large linear problem for the Newton two-grid characteristic finite volume method on a fine mesh with mesh size h = O(| log h| 1/2 H 3 ). These methods we studied provide the same convergence rate as that of the characteristic finite volume method, which involves solving one large nonlinear problem on a fine mesh with mesh size h. Some numerical results are presented to demonstrate the efficiency of the proposed methods.Mathematics subject classification: 35Q55, 65N30, 76D05.

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A stabilised characteristic finite element method for transient Navier–Stokes equations

Cited by 16 publications

References 27 publications

Decoupled characteristic stabilized finite element method for time‐dependent Navier–Stokes/Darcy model

Decoupled characteristic stabilized finite element method for time‐dependent Navier–Stokes/Darcy model

A stabilised nonconforming finite element method for steady incompressible flows

Two-Grid Characteristic Finite Volume Methods for Nonlinear Parabolic Problem

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