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Volume 27, Issue 2-3
Finite Elements with Local Projection Stabilization for Incompressible Flow Problems

Malte Braack & Gert Lube

J. Comp. Math., 27 (2009), pp. 116-147.

Published online: 2009-04

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  • Abstract

In this paper we review recent developments in the analysis of finite element methods for incompressible flow problems with local projection stabilization (LPS). These methods preserve the favourable stability and approximation properties of classical residual-based stabilization (RBS) techniques but avoid the strong coupling of velocity and pressure in the stabilization terms. LPS-methods belong to the class of symmetric stabilization techniques and may be characterized as variational multiscale methods. In this work we summarize the most important a priori estimates of this class of stabilization schemes developed in the past 6 years. We consider the Stokes equations, the Oseen linearization and the Navier-Stokes equations. Furthermore, we apply it to optimal control problems with linear(ized) flow problems, since the symmetry of the stabilization leads to the nice feature that the operations "discretize" and "optimize" commute.

  • Keywords

Finite element method, Stabilization, Computational fluid dynamics, Error estimates, Navier-Stokes, Stokes.

  • AMS Subject Headings

65N06, 65B99.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-27-116, author = {}, title = {Finite Elements with Local Projection Stabilization for Incompressible Flow Problems}, journal = {Journal of Computational Mathematics}, year = {2009}, volume = {27}, number = {2-3}, pages = {116--147}, abstract = {

In this paper we review recent developments in the analysis of finite element methods for incompressible flow problems with local projection stabilization (LPS). These methods preserve the favourable stability and approximation properties of classical residual-based stabilization (RBS) techniques but avoid the strong coupling of velocity and pressure in the stabilization terms. LPS-methods belong to the class of symmetric stabilization techniques and may be characterized as variational multiscale methods. In this work we summarize the most important a priori estimates of this class of stabilization schemes developed in the past 6 years. We consider the Stokes equations, the Oseen linearization and the Navier-Stokes equations. Furthermore, we apply it to optimal control problems with linear(ized) flow problems, since the symmetry of the stabilization leads to the nice feature that the operations "discretize" and "optimize" commute.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8564.html} }
TY - JOUR T1 - Finite Elements with Local Projection Stabilization for Incompressible Flow Problems JO - Journal of Computational Mathematics VL - 2-3 SP - 116 EP - 147 PY - 2009 DA - 2009/04 SN - 27 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/8564.html KW - Finite element method, Stabilization, Computational fluid dynamics, Error estimates, Navier-Stokes, Stokes. AB -

In this paper we review recent developments in the analysis of finite element methods for incompressible flow problems with local projection stabilization (LPS). These methods preserve the favourable stability and approximation properties of classical residual-based stabilization (RBS) techniques but avoid the strong coupling of velocity and pressure in the stabilization terms. LPS-methods belong to the class of symmetric stabilization techniques and may be characterized as variational multiscale methods. In this work we summarize the most important a priori estimates of this class of stabilization schemes developed in the past 6 years. We consider the Stokes equations, the Oseen linearization and the Navier-Stokes equations. Furthermore, we apply it to optimal control problems with linear(ized) flow problems, since the symmetry of the stabilization leads to the nice feature that the operations "discretize" and "optimize" commute.

Malte Braack & Gert Lube. (2019). Finite Elements with Local Projection Stabilization for Incompressible Flow Problems. Journal of Computational Mathematics. 27 (2-3). 116-147. doi:
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