A two-grid stabilization method for solving the steady-state Navier-Stokes equations

Kaya, Songül; Rivière, Béatrice

doi:10.1002/num.20120

Cited by 66 publications

(46 citation statements)

References 17 publications

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“…In [10], the method is extended to compressible Navier-Stokes equations coupled with further transport equations to describe chemically reacting flows. Finally, turbulent flows including the 3D mixing layer problem are numerically computed by the local projection method in [29,31].…”

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confidence: 99%

A unified convergence analysis for local projection stabilisations applied to the Oseen problem

Matthies¹,

Skrzypacz²,

Tobiska³

2007

ESAIM: M2AN

172

183

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Abstract. The discretisation of the Oseen problem by finite element methods may suffer in general from two shortcomings. First, the discrete inf-sup (Babuška-Brezzi) condition can be violated. Second, spurious oscillations occur due to the dominating convection. One way to overcome both difficulties is the use of local projection techniques. Studying the local projection method in an abstract setting, we show that the fulfilment of a local inf-sup condition between approximation and projection spaces allows to construct an interpolation with additional orthogonality properties. Based on this special interpolation, optimal a-priori error estimates are shown with error constants independent of the Reynolds number. Applying the general theory, we extend the results of Braack and Burman for the standard two-level version of the local projection stabilisation to discretisations of arbitrary order on simplices, quadrilaterals, and hexahedra. Moreover, our general theory allows to derive a novel class of local projection stabilisation by enrichment of the approximation spaces. This class of stabilised schemes uses approximation and projection spaces defined on the same mesh and leads to much more compact stencils than in the two-level approach. Finally, on simplices, the spectral equivalence of the stabilising terms of the local projection method and the subgrid modelling introduced by Guermond is shown. This clarifies the relation of the local projection stabilisation to the variational multiscale approach.Mathematics Subject Classification. 65N12, 65N30, 76D05.

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mentioning

confidence: 99%

A unified convergence analysis for local projection stabilisations applied to the Oseen problem

Matthies¹,

Skrzypacz²,

Tobiska³

2007

ESAIM: M2AN

172

183

View full text Add to dashboard Cite

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“…Furthermore, the method helps to extend these results to Navier-Stokes equations with larger Reynolds numbers by using variation multiscale method (see [27,28]) and solve the non-stationary two-or three-dimensional Navier-Stokes equations, which will be discussed in our future articles.…”

Section: Theorem 44 Under the Assumptions Of Theorem 42 If The Somentioning

confidence: 97%

Superconvergence of discontinuous Galerkin finite element method for the stationary Navier‐Stokes equations

2006

Numerical Methods Partial

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This article focuses on discontinuous Galerkin method for the two-or three-dimensional stationary incompressible Navier-Stokes equations. The velocity field is approximated by discontinuous locally solenoidal finite element, and the pressure is approximated by the standard conforming finite element. Then, superconvergence of nonconforming finite element approximations is applied by using least-squares surface fitting for the stationary Navier-Stokes equations. The method ameliorates the two noticeable disadvantages about the given finite element pair. Finally, the superconvergence result is provided under some regular assumptions.

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“…Theoretical studies of this method began in [25] and were continued in [9,17,[20][21][22][23][24]28]. We note that it is inspired by both physical ideas in turbulence modeling and algorithmic ideas developed for simulation of non-Newtonian fluids, [4].…”

Section: Dms0207 627mentioning

confidence: 99%

A two-level variational multiscale method for convection-dominated convection–diffusion equations

John

Kaya

Layton

2006

Computer Methods in Applied Mechanics and Engineering

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100

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This paper studies the error in, the efficient implementation of and time stepping methods for a variational multiscale method (VMS) for solving convectiondominated problems. The VMS studied uses a fine mesh C 0 finite element space X h to approximate the concentration and a coarse mesh discontinuous vector finite element space L H for the large scales of the flux in the two scale discretization. Our tests show that these choices lead to an efficient VMS whose complexity is further reduced if a (locally) L 2 -orthogonal basis for L H is used. A fully implicit and a semi-implicit treatment of the terms which link effects across scales are tested and compared. The semi-implicit VMS was much more efficient. The observed global accuracy of the most straightforward VMS implementation was much better than the artificial diffusion stabilization and comparable to a streamline-diffusion finite element method in our tests.

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A two-grid stabilization method for solving the steady-state Navier-Stokes equations

Cited by 66 publications

References 17 publications

A unified convergence analysis for local projection stabilisations applied to the Oseen problem

A unified convergence analysis for local projection stabilisations applied to the Oseen problem

Superconvergence of discontinuous Galerkin finite element method for the stationary Navier‐Stokes equations

A two-level variational multiscale method for convection-dominated convection–diffusion equations

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