Local projection FEM stabilization for the time-dependent incompressible Navier-Stokes problem

Daniel Arndt; Helene Dallmann; Gert Lube

doi:10.1002/num.21944

Local projection FEM stabilization for the time-dependent incompressible Navier-Stokes problem

Daniel Arndt, Helene Dallmann, Gert Lube

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31 Scopus citations

Abstract

We consider conforming finite element (FE) approximations of the time-dependent, incompressible Navier-Stokes problem with inf-sup stable approximation of velocity and pressure. In case of high Reynolds numbers, a local projection stabilization method is considered. In particular, the idea of streamline upwinding is combined with stabilization of the divergence-free constraint. For the arising nonlinear semidiscrete problem, a stability and convergence analysis is given. Our approach improves some results of a recent paper by Matthies and Tobiska (IMA J. Numer. Anal., to appear) for the linearized model and takes partly advantage of the analysis in Burman and Fernández, Numer. Math. 107 (2007), 39-77 for edge-stabilized FE approximation of the Navier-Stokes problem. Some numerical experiments complement the theoretical results.

Original language	English
Pages (from-to)	1224-1250
Number of pages	27
Journal	Numerical Methods for Partial Differential Equations
Volume	31
Issue number	4
DOIs	https://doi.org/10.1002/num.21944
State	Published - Jul 1 2015
Externally published	Yes

Keywords

Navier-Stokes equations
Oseen equations
Smagorinsky model
incompressible flow
local projection stabilization
stabilized finite elements

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10.1002/num.21944

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title = "Local projection FEM stabilization for the time-dependent incompressible Navier-Stokes problem",

abstract = "We consider conforming finite element (FE) approximations of the time-dependent, incompressible Navier-Stokes problem with inf-sup stable approximation of velocity and pressure. In case of high Reynolds numbers, a local projection stabilization method is considered. In particular, the idea of streamline upwinding is combined with stabilization of the divergence-free constraint. For the arising nonlinear semidiscrete problem, a stability and convergence analysis is given. Our approach improves some results of a recent paper by Matthies and Tobiska (IMA J. Numer. Anal., to appear) for the linearized model and takes partly advantage of the analysis in Burman and Fern{\'a}ndez, Numer. Math. 107 (2007), 39-77 for edge-stabilized FE approximation of the Navier-Stokes problem. Some numerical experiments complement the theoretical results.",

keywords = "Navier-Stokes equations, Oseen equations, Smagorinsky model, incompressible flow, local projection stabilization, stabilized finite elements",

author = "Daniel Arndt and Helene Dallmann and Gert Lube",

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AU - Arndt, Daniel

AU - Dallmann, Helene

AU - Lube, Gert

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N2 - We consider conforming finite element (FE) approximations of the time-dependent, incompressible Navier-Stokes problem with inf-sup stable approximation of velocity and pressure. In case of high Reynolds numbers, a local projection stabilization method is considered. In particular, the idea of streamline upwinding is combined with stabilization of the divergence-free constraint. For the arising nonlinear semidiscrete problem, a stability and convergence analysis is given. Our approach improves some results of a recent paper by Matthies and Tobiska (IMA J. Numer. Anal., to appear) for the linearized model and takes partly advantage of the analysis in Burman and Fernández, Numer. Math. 107 (2007), 39-77 for edge-stabilized FE approximation of the Navier-Stokes problem. Some numerical experiments complement the theoretical results.

AB - We consider conforming finite element (FE) approximations of the time-dependent, incompressible Navier-Stokes problem with inf-sup stable approximation of velocity and pressure. In case of high Reynolds numbers, a local projection stabilization method is considered. In particular, the idea of streamline upwinding is combined with stabilization of the divergence-free constraint. For the arising nonlinear semidiscrete problem, a stability and convergence analysis is given. Our approach improves some results of a recent paper by Matthies and Tobiska (IMA J. Numer. Anal., to appear) for the linearized model and takes partly advantage of the analysis in Burman and Fernández, Numer. Math. 107 (2007), 39-77 for edge-stabilized FE approximation of the Navier-Stokes problem. Some numerical experiments complement the theoretical results.

KW - Navier-Stokes equations

KW - Oseen equations

KW - Smagorinsky model

KW - incompressible flow

KW - local projection stabilization

KW - stabilized finite elements

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Local projection FEM stabilization for the time-dependent incompressible Navier-Stokes problem

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Keywords

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