Stabilized finite element methods: II. The incompressible Navier-Stokes equations

Franca, Leopoldo P.; Frey, Sérgio

doi:10.1016/0045-7825(92)90041-h

Cited by 662 publications

(508 citation statements)

References 26 publications

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“…2 shows the approximate pressure p h for P1-P1 elements after one time step. As the time step decreases from (∆t) 3 to (∆t) 6 , p h appears to converge to a state that is not an accurate approximation of the true solution (21). We recall that for linear elements, the term − u h in (18) vanishes and the stabilizing contribution of the spatial term (20) …”

Section: Motivating Computational Experimentsmentioning

confidence: 99%

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On stabilized finite element methods for the Stokes problem in the small time step limit

Bochev

Gunzburger

Lehoucq

2006

Numerical Methods in Fluids

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SUMMARYRecent studies indicate that consistently stabilized methods for unsteady incompressible flows, obtained by a method of lines approach, may experience difficulty when the time step is small relative to the spatial grid size. Using as a model problem the unsteady Stokes equations, we show that the semi-discrete pressure operator associated with such methods is not uniformly coercive. We prove that for sufficiently large (relative to the square of the spatial grid size) time steps, implicit time discretizations contribute terms that stabilize this operator. However, we also prove that if the time step is sufficiently small, then the fully discrete problem necessarily leads to unstable pressure approximations. The semi-discrete pressure operator studied in the paper also arises in pressureprojection methods, thereby making our results potentially useful in other settings.

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Section: Motivating Computational Experimentsmentioning

confidence: 99%

“…Stabilized methods for time-dependent problems are commonly defined by using a method of lines approach whereby the spatial and temporal discretization steps are separated; see [21,27,28,32]. Stabilization terms are introduced in the semi-discrete (in space) equation by using residuals of the time-dependent partial differential equation.…”

Section: Introductionmentioning

confidence: 99%

On stabilized finite element methods for the Stokes problem in the small time step limit

Bochev

Gunzburger

Lehoucq

2006

Numerical Methods in Fluids

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“…Besides the usual Galerkin terms, equations (4.1) and (4.2) involve two terms, which additionally penalize the violation of the div-free constraint by the FE velocity and vorticity. Including such terms is often a part of Petrov-Galerkin FE formulations for the Navier-Stokes equations [14] and is well-known as the grad-div stabilization [29]; γ 1 ≥ 0 and γ 2 ≥ 0 are user-defined stabilization parameters. The motivation for including the grad-div stabilization with O(1) stabilization parameter in the velocity equation comes from recent work in [22,28], where its use in similar (rotational form) schemes was found effective at relieving the velocity error of an undesired scaling with the large error associated with the Bernoulli pressure.…”

Section: Algorithm 41 (Vvh1)mentioning

confidence: 99%

Velocity–vorticity–helicity formulation and a solver for the Navier–Stokes equations

Olshanskii

Rebholz

2010

Journal of Computational Physics

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For the three-dimensional incompressible Navier-Stokes equations, we present a formulation featuring velocity, vorticity and helical density as independent variables. We find the helical density can be observed as a Lagrange multiplier corresponding to the divergence-free constraint on the vorticity variable, similar to the pressure in the case of the incompressibility condition for velocity. As one possible practical application of this new formulation, we consider a time-splitting numerical scheme based on an alternating procedure between vorticity -helical density and velocity -Bernoulli pressure systems of equations. Results of numerical experiments include a comparison with some well-known schemes based on pressure-velocity formulation and illustrate the competitiveness on the new scheme as well as the soundness of the new formulation.

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“…The Navier-Stokes and scalar transport equations are solved using a streamline-upwind PetrovGalerkin (SUPG) method [23,24]. This method contains additional stabilization terms providing smooth solutions to convection dominating flows.…”

Section: Finite Element Solutionmentioning

confidence: 99%

Numerical simulation of the galvanizing process during GA to GI transition

Ilinca

Ajersch

Baril³

et al. 2006

Numerical Methods in Fluids

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This paper presents the application of a three‐dimensional finite element solution algorithm for the prediction of the velocity, temperature and species concentration fields in an industrial continuous galvanizing bath. Simulations were carried out using a parallel CFD software developed at IMI‐NRC. The turbulent flow, heat and mass transfer has been solved using a high Reynolds number k–ε model. Simulations were carried out for the case when the density of the molten metal depends only on the temperature and also for the case when both temperature and Al concentration affect the density. When considering the buoyancy effect of the Al concentration, differences are especially apparent during the melting of ingots with high Al content. Otherwise, thermal effects are dominant. The continuous monitoring of the temperature and the Al and Fe content in an industrial bath was used to validate the flow, temperature and compositional variations. A period of three hours, corresponding to three different ingot additions, was simulated successfully, resulting in a good agreement of the temperature and compositional variations. Copyright © 2006 Crown in the right of Canada. Published by John Wiley & Sons, Ltd.

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Stabilized finite element methods: II. The incompressible Navier-Stokes equations

Cited by 662 publications

References 26 publications

On stabilized finite element methods for the Stokes problem in the small time step limit

On stabilized finite element methods for the Stokes problem in the small time step limit

Velocity–vorticity–helicity formulation and a solver for the Navier–Stokes equations

Numerical simulation of the galvanizing process during GA to GI transition

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