# 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

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topMatthies, 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

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

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