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

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

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

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topKnobloch, 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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