On the parameter choice in grad-div stabilization for the Stokes equations

Jenkins, Eleanor W.; John, Volker; Linke, Alexander; Rebholz, Leo G.

doi:10.1007/s10444-013-9316-1

Cited by 103 publications

(110 citation statements)

References 27 publications

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“…Numerical analysis, e.g., in [90,117] shows that for finite element discretizations satisfying (9) the choice of the stabilization parameter τ c = O(1) with respect to the mesh width leads to optimal error estimates. However, a good choice of τ c depends usually on (unknown) norms of the solution (u, p) of (59) and on whether or not the sequence of weakly divergence-free subspaces contained in the discretely divergence-free spaces X h,div has an optimal approximation property.…”

Section: Numerical Analysismentioning

confidence: 99%

A Review of Variational Multiscale Methods for the Simulation of Turbulent Incompressible Flows

Ahmed

Rebollo

John

et al. 2015

Arch Computat Methods Eng

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Various realizations of variational multiscale (VMS) methods for simulating turbulent incompressible flows have been proposed in the past fifteen years. All of these realizations obey the basic principles of VMS methods: They are based on the variational formulation of the incompressible Navier-Stokes equations and the scale separation is defined by projections. However, apart from these common basic features, the various VMS methods look quite different. In this review, the derivation of the different VMS methods is presented in some detail and their relation among each other and also to other discretizations is discussed. Another emphasis consists in giving an overview about known results from the numerical analysis of the VMS methods. A few results are presented in detail to highlight the used mathematical tools. Furthermore, the literature presenting numerical studies with the VMS methods is surveyed and the obtained results are summarized.

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Section: Numerical Analysismentioning

confidence: 99%

A Review of Variational Multiscale Methods for the Simulation of Turbulent Incompressible Flows

Ahmed

Rebollo

John

et al. 2015

Arch Computat Methods Eng

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“…This, as well, provides more flexibility for the choice of γ K (see [JJ+14 JJ+14] for a detailed discussion of this issue in the case of the Stokes problem).…”

Section: A-priori Estimatesmentioning

confidence: 99%

“…In [MT15 MT15] velocity stabilisation terms that satisfy (3.5 3.5) with c s ≤ Cc a are used. We have chosen to avoid that assumption, since, as it has been shown in [JJ+14 JJ+14], a large stabilisation parameter in the grad-div term may be beneficial in some cases.…”

Section: Numerical Confirmation (Part 1)mentioning

confidence: 99%

“…The introduction of this term was first proposed in [FH88 FH88] and it has been extensively used to improve the control of the discrete divergence for the Navier-Stokes equation and coupled problems (see, e.g., [LM+09 LM+09,GL+12 GL+12]). An analysis of this method can be found in [OR04 OR04], and, especially in [JJ+14 JJ+14], where a very detailed analysis and discussion on the selection of the stabilisation parameter is presented.…”

Section: Introductionmentioning

confidence: 99%

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Stabilised finite element methods for the Oseen problem on anisotropic quadrilateral meshes

Barrenechea

Wachtel

2018

ESAIM: M2AN

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This version is available at https://strathprints.strath.ac.uk/61287/ Strathprints is designed to allow users to access the research output of the University of Strathclyde. Unless otherwise explicitly stated on the manuscript, Copyright © and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. Please check the manuscript for details of any other licences that may have been applied. You may not engage in further distribution of the material for any profitmaking activities or any commercial gain. You may freely distribute both the url (https://strathprints.strath.ac.uk/) and the content of this paper for research or private study, educational, or not-for-profit purposes without prior permission or charge.Any correspondence concerning this service should be sent to the In this work we present and analyse new inf-sup stable, and stabilised, finite element methods for the Oseen equation in anisotropic quadrilateral meshes. The meshes are formed of closed parallelograms, and the analysis is restricted to two space dimensions. Starting with the lowest order Q 2 1 × P 0 pair, we first identify the pressure components that make this finite element pair to be non-inf-sup stable, especially with respect to the aspect ratio. We then propose a way to penalise them, both strongly, by directly removing them from the space, and weakly, by adding a stabilisation term based on jumps of the pressure across selected edges. Concerning the velocity stabilisation, we propose an enhanced grad-div term. Stability and optimal a priori error estimates are given, and the results are confirmed numerically.

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“…Typically, this parameter is chosen to be a global constant, and recent work in [12] gives guidance on how to choose it. However, we propose to instead choose the parameter locally and will demonstrate that doing so can produce significantly more accurate solutions.…”

Section: Introductionmentioning

confidence: 99%

Locally chosen grad-div stabilization parameters for finite element discretizations of incompressible flow problems

Heavner¹

2014

SIURO

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Grad-div stabilization has recently been found to be an important tool for finite element method simulations of incompressible flow problems, acting to improve mass conservation in solutions and reducing the effect of the pressure error on the velocity error. Typically, the associated stabilization parameter is chosen globally, but herein we consider local choices. We show that, both for an analytic test problem and for lift and drag calculations of flow around a cylinder, local choices of the grad-div stabilization parameter can provide more accurate solutions.

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On the parameter choice in grad-div stabilization for the Stokes equations

Cited by 103 publications

References 27 publications

A Review of Variational Multiscale Methods for the Simulation of Turbulent Incompressible Flows

A Review of Variational Multiscale Methods for the Simulation of Turbulent Incompressible Flows

Stabilised finite element methods for the Oseen problem on anisotropic quadrilateral meshes

Locally chosen grad-div stabilization parameters for finite element discretizations of incompressible flow problems

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