Residual-Based Stabilized Higher-Order Fem for a Generalized Oseen Problem

Lube, Gert; Rapin, Gerd

doi:10.1142/s0218202506001418

Cited by 27 publications

(25 citation statements)

References 17 publications

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“…their asymptotic dependence on the local mesh size and the local Peclet number must be correct. We refer to [12] and Lube and Rapin [29] for the corresponding estimate for P r /P s elements with s < r.…”

Section: Isotropic a Priori Results Of Stabilized Schemesmentioning

confidence: 99%

A stabilized finite element scheme for the Navier-Stokes equations on quadrilateral anisotropic meshes

Braack¹

2008

ESAIM: M2AN

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Abstract.It is well known that the classical local projection method as well as residual-based stabilization techniques, as for instance streamline upwind Petrov-Galerkin (SUPG), are optimal on isotropic meshes. Here we extend the local projection stabilization for the Navier-Stokes system to anisotropic quadrilateral meshes in two spatial dimensions. We describe the new method and prove an a priori error estimate. This method leads on anisotropic meshes to qualitatively better convergence behavior than other isotropic stabilization methods. The capability of the method is illustrated by means of two numerical test problems.Mathematics Subject Classification. 35Q30, 65N30, 76D05.

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Section: Isotropic a Priori Results Of Stabilized Schemesmentioning

confidence: 99%

A stabilized finite element scheme for the Navier-Stokes equations on quadrilateral anisotropic meshes

Braack¹

2008

ESAIM: M2AN

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“…The SUPG method was introduced in [81,28] for stabilizing scalar convectiondominated convection-diffusion equations. Stabilizations of the Oseen equations and the stationary Navier-Stokes equations which contain the SUPG term were analyzed in [72,111,136,53], and extensions of the analysis can be found in [137,110,114]. Surveys of the results are provided in [22,124].…”

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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“…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

170

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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Residual-Based Stabilized Higher-Order Fem for a Generalized Oseen Problem

Cited by 27 publications

References 17 publications

A stabilized finite element scheme for the Navier-Stokes equations on quadrilateral anisotropic meshes

A stabilized finite element scheme for the Navier-Stokes equations on quadrilateral anisotropic meshes

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

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

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