Stabilization methods of bubble type for the Q1/Q1-element applied to the incompressible Navier-Stokes equations

Petr Knobloch; Lutz Tobiska

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 34, Issue: 1, page 85-107
  • ISSN: 0764-583X

Abstract

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In this paper, a general technique is developed to enlarge the velocity space V h 1 of the unstable -element by adding spaces V h 2 such that for the extended pair the Babuska-Brezzi condition is satisfied. Examples of stable elements which can be derived in such a way imply the stability of the well-known Q2/Q1-element and the 4Q1/Q1-element. However, our new elements are much more cheaper. In particular, we shall see that more than half of the additional degrees of freedom when switching from the Q1 to the Q2 and 4Q1, respectively, element are not necessary to stabilize the Q1/Q1-element. Moreover, by using the technique of reduced discretizations and eliminating the additional degrees of freedom we show the relationship between enlarging the velocity space and stabilized methods. This relationship has been established for triangular elements but was not known for quadrilateral elements. As a result we derive new stabilized methods for the Stokes and Navier-Stokes equations. Finally, we show how the Brezzi-Pitkäranta stabilization and the SUPG method for the incompressible Navier-Stokes equations can be recovered as special cases of the general approach. In contrast to earlier papers we do not restrict ourselves to linearized versions of the Navier-Stokes equations but deal with the full nonlinear case.

How to cite

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Knobloch, Petr, and Tobiska, Lutz. "Stabilization methods of bubble type for the Q1/Q1-element applied to the incompressible Navier-Stokes equations." ESAIM: Mathematical Modelling and Numerical Analysis 34.1 (2010): 85-107. <http://eudml.org/doc/197571>.

@article{Knobloch2010,
abstract = { In this paper, a general technique is developed to enlarge the velocity space $\{\rm V\}_h^1$ of the unstable -element by adding spaces $\{\rm V\}_h^2$ such that for the extended pair the Babuska-Brezzi condition is satisfied. Examples of stable elements which can be derived in such a way imply the stability of the well-known Q2/Q1-element and the 4Q1/Q1-element. However, our new elements are much more cheaper. In particular, we shall see that more than half of the additional degrees of freedom when switching from the Q1 to the Q2 and 4Q1, respectively, element are not necessary to stabilize the Q1/Q1-element. Moreover, by using the technique of reduced discretizations and eliminating the additional degrees of freedom we show the relationship between enlarging the velocity space and stabilized methods. This relationship has been established for triangular elements but was not known for quadrilateral elements. As a result we derive new stabilized methods for the Stokes and Navier-Stokes equations. Finally, we show how the Brezzi-Pitkäranta stabilization and the SUPG method for the incompressible Navier-Stokes equations can be recovered as special cases of the general approach. In contrast to earlier papers we do not restrict ourselves to linearized versions of the Navier-Stokes equations but deal with the full nonlinear case. },
author = {Knobloch, Petr, Tobiska, Lutz},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Babuska-Brezzi condition; stabilization; Stokes equations; Navier-Stokes equations.; stabilized quadilateral element; incompressible Navier-Stokes equations},
language = {eng},
month = {3},
number = {1},
pages = {85-107},
publisher = {EDP Sciences},
title = {Stabilization methods of bubble type for the Q1/Q1-element applied to the incompressible Navier-Stokes equations},
url = {http://eudml.org/doc/197571},
volume = {34},
year = {2010},
}

TY - JOUR
AU - Knobloch, Petr
AU - Tobiska, Lutz
TI - Stabilization methods of bubble type for the Q1/Q1-element applied to the incompressible Navier-Stokes equations
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 34
IS - 1
SP - 85
EP - 107
AB - In this paper, a general technique is developed to enlarge the velocity space ${\rm V}_h^1$ of the unstable -element by adding spaces ${\rm V}_h^2$ such that for the extended pair the Babuska-Brezzi condition is satisfied. Examples of stable elements which can be derived in such a way imply the stability of the well-known Q2/Q1-element and the 4Q1/Q1-element. However, our new elements are much more cheaper. In particular, we shall see that more than half of the additional degrees of freedom when switching from the Q1 to the Q2 and 4Q1, respectively, element are not necessary to stabilize the Q1/Q1-element. Moreover, by using the technique of reduced discretizations and eliminating the additional degrees of freedom we show the relationship between enlarging the velocity space and stabilized methods. This relationship has been established for triangular elements but was not known for quadrilateral elements. As a result we derive new stabilized methods for the Stokes and Navier-Stokes equations. Finally, we show how the Brezzi-Pitkäranta stabilization and the SUPG method for the incompressible Navier-Stokes equations can be recovered as special cases of the general approach. In contrast to earlier papers we do not restrict ourselves to linearized versions of the Navier-Stokes equations but deal with the full nonlinear case.
LA - eng
KW - Babuska-Brezzi condition; stabilization; Stokes equations; Navier-Stokes equations.; stabilized quadilateral element; incompressible Navier-Stokes equations
UR - http://eudml.org/doc/197571
ER -

References

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