Stabilized finite element schemes with LBB-stable elements for incompressible flows

Gelhard, Tobias; Lube, Gert; Olshanskii, Maxim A.; Starcke, Jan-Hendrik

doi:10.1016/j.cam.2004.09.017

Cited by 126 publications

(81 citation statements)

References 14 publications

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“…Despite the progress of the SUPG/PSPG method in theory and application, an essential drawback of this method is -in particular for higher order interpolations -that various terms need to be added to the weak formulation to guarantee the consistency of the method in a strong way (Galerkin orthogonality holds for smooth solutions). Residual-based stabilisation methods which use inf-sup stable pairs of elements reduce the number of terms which have to be added to the Galerkin formulation [17,33]. However, an additional coupling term between velocity and pressure makes their analysis difficult.…”

mentioning

confidence: 99%

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

Matthies¹,

Skrzypacz²,

Tobiska³

2007

ESAIM: M2AN

173

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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confidence: 99%

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

Matthies¹,

Skrzypacz²,

Tobiska³

2007

ESAIM: M2AN

173

183

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“…The choice of the parameters δ K = h 2 K and τ K = 1 is carried out according to [7] or [9], where h K denotes the local element length.…”

Section: Topical Problems Of Fluid Mechanics 9 ______________________mentioning

confidence: 99%

Incompressible and Compressible Viscous Flow With Low Mach Numbers

Balázsová¹,

Feistauer²,

Sváček³

et al. 2017

Topical Problems of Fluid Mechanics 2017

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In this paper we compare incompressible flow and low Mach number compressible viscous flow. Incompressible Navier-Stokes equations were solved by the classical finite element method and compressible Navier-Stokes equations were treated with the aid of discontinuous Galerkin method in space and backward difference method in time. We present numerical results for a flow in a channel which represents a simplified model of the human vocal tract. Presented numerical results give a good correspondence between the the incompressible flow and the compressible flow with low Mach numbers.

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“…The choice of the parameters δ K = h 2 K and τ K = 1 is carried out according to [5] or [13], where h K denotes the local element length. Then the stabilized discrete problem at a time instant t = t n+1 reads: Find U = (u, p) ∈ W h × Q h , p := p n+1 , u := u n+1 , such that u satisfies approximately the conditions (3 a-b) and…”

Section: The Discrete Formulation Contains the Galerkin Terms Definedmentioning

confidence: 99%

“…The aerodynamical lift force and the aerodynamical moment in equations (4) are extrapolated from the previous time levels, i.e., the approximations L n , M n , L n−1 , M n−1 are assumed to be computed using (5) and the values of u n , p n , u n−1 , p n−1 . The extrapolations of the aerodynamical lift force and the aerodynamical moment are used, i.e.…”

mentioning

confidence: 99%