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

Gunar Matthies; Piotr Skrzypacz; Lutz Tobiska

ESAIM: Mathematical Modelling and Numerical Analysis (2007)

  • Volume: 41, Issue: 4, page 713-742
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Matthies, Gunar, Skrzypacz, Piotr, and Tobiska, Lutz. "A unified convergence analysis for local projection stabilisations applied to the Oseen problem." ESAIM: Mathematical Modelling and Numerical Analysis 41.4 (2007): 713-742. <http://eudml.org/doc/250028>.

@article{Matthies2007,
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. },
author = {Matthies, Gunar, Skrzypacz, Piotr, Tobiska, Lutz},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Stabilised finite elements; Navier-Stokes equations; equal-order interpolation.; local inf-sup condition; spurious oscillations; convection-dominated flows; optimal a priori error estimates},
language = {eng},
month = {10},
number = {4},
pages = {713-742},
publisher = {EDP Sciences},
title = {A unified convergence analysis for local projection stabilisations applied to the Oseen problem},
url = {http://eudml.org/doc/250028},
volume = {41},
year = {2007},
}

TY - JOUR
AU - Matthies, Gunar
AU - Skrzypacz, Piotr
AU - Tobiska, Lutz
TI - A unified convergence analysis for local projection stabilisations applied to the Oseen problem
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2007/10//
PB - EDP Sciences
VL - 41
IS - 4
SP - 713
EP - 742
AB - 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.
LA - eng
KW - Stabilised finite elements; Navier-Stokes equations; equal-order interpolation.; local inf-sup condition; spurious oscillations; convection-dominated flows; optimal a priori error estimates
UR - http://eudml.org/doc/250028
ER -

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Citations in EuDML Documents

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  1. Gabriel R. Barrenechea, Volker John, Petr Knobloch, A local projection stabilization finite element method with nonlinear crosswind diffusion for convection-diffusion-reaction equations
  2. Petr Knobloch, On a variant of the local projection method stable in the SUPG norm
  3. Malte Braack, Benjamin Tews, Linear-quadratic optimal control for the Oseen equations with stabilized finite elements
  4. Malte Braack, A stabilized finite element scheme for the Navier-Stokes equations on quadrilateral anisotropic meshes

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