# On the domain geometry dependence of the LBB condition

Evgenii V. Chizhonkov; Maxim A. Olshanskii

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 34, Issue: 5, page 935-951
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topChizhonkov, Evgenii V., and Olshanskii, Maxim A.. " On the domain geometry dependence of the LBB condition." ESAIM: Mathematical Modelling and Numerical Analysis 34.5 (2010): 935-951. <http://eudml.org/doc/197462>.

@article{Chizhonkov2010,

abstract = {
The LBB condition is well-known to guarantee the stability of a finite
element (FE) velocity - pressure pair in incompressible flow calculations.
To ensure the condition to be satisfied a certain constant should be positive and
mesh-independent. The paper studies the dependence of the LBB condition on the
domain geometry. For model domains such as strips and rings the
substantial dependence of this constant on geometry aspect ratios is observed.
In domains with highly anisotropic substructures this may require special care
with numerics to avoid failures similar to those when
the LBB condition is violated. In the core of the paper
we prove that for any FE velocity-pressure pair satisfying usual approximation
hypotheses the mesh-independent limit in the LBB condition is not greater than
its continuous counterpart, the constant from the Nečas inequality.
For the latter the explicit and asymptotically accurate estimates are proved.
The analytic results are illustrated by several numerical experiments.
},

author = {Chizhonkov, Evgenii V., Olshanskii, Maxim A.},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {LBB; infsup condition; Stokes problem; domain geometry;
eigenvalue problem; finite elements.; domain geometry dependence; LBB condition; finite element velocity-pressure pair; mesh-independent limit; Nečas inequality},

language = {eng},

month = {3},

number = {5},

pages = {935-951},

publisher = {EDP Sciences},

title = { On the domain geometry dependence of the LBB condition},

url = {http://eudml.org/doc/197462},

volume = {34},

year = {2010},

}

TY - JOUR

AU - Chizhonkov, Evgenii V.

AU - Olshanskii, Maxim A.

TI - On the domain geometry dependence of the LBB condition

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/3//

PB - EDP Sciences

VL - 34

IS - 5

SP - 935

EP - 951

AB -
The LBB condition is well-known to guarantee the stability of a finite
element (FE) velocity - pressure pair in incompressible flow calculations.
To ensure the condition to be satisfied a certain constant should be positive and
mesh-independent. The paper studies the dependence of the LBB condition on the
domain geometry. For model domains such as strips and rings the
substantial dependence of this constant on geometry aspect ratios is observed.
In domains with highly anisotropic substructures this may require special care
with numerics to avoid failures similar to those when
the LBB condition is violated. In the core of the paper
we prove that for any FE velocity-pressure pair satisfying usual approximation
hypotheses the mesh-independent limit in the LBB condition is not greater than
its continuous counterpart, the constant from the Nečas inequality.
For the latter the explicit and asymptotically accurate estimates are proved.
The analytic results are illustrated by several numerical experiments.

LA - eng

KW - LBB; infsup condition; Stokes problem; domain geometry;
eigenvalue problem; finite elements.; domain geometry dependence; LBB condition; finite element velocity-pressure pair; mesh-independent limit; Nečas inequality

UR - http://eudml.org/doc/197462

ER -

## References

top- P.P. Aristov and E.V. Chizhonkov, On the Constant in the LBB condition for rectangular domains. Report No. 9535, Dept. of Math. Univ. of Nijmegen, The Netherlands (1995).
- I. Babuska, The finite element method with Lagrange multipliers. Numer. Math.20 (1973) 179-192.
- D. Boffi, F. Brezzi and L. Gastaldi, On the convergence of eigenvalues for mixed formulations. Ann. Sc. Norm. Sup. Pisa25 (1997) 131-154.
- D. Boffi, F. Brezzi and L. Gastaldi, On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form. Math. Comp. 69 (2000) 141-158.
- D. Braess, Finite Elemente: Theorie, schnelle Löser und Anwendungen in der Elastizitätstheorie. Springer-Verlag, Berlin, Heidelberg, New York (1997).
- J.H. Bramble and J.E. Pasciak, A preconditioning technique for indefinite systems resulting from mixed approximation of elliptic problems. Math. Comp.50 (1988) 1-17.
- F. Brezzi, (1974) On the existence, uniqueness and approximation of the saddle-point problems arising from Lagrange multipliers. Numer. Math.20 (1974) 179-192.
- F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer Series in Comp. Math.15, Springer-Verlag, New York (1991).
- E.V. Chizhonkov, Application of the Cossera spectrum to the optimization of a method for solving the Stokes Problem. Russ. J. Numer. Anal. Math. Model.9 (1994) 191-199.
- M. Crouzeix, Étude d'une méthode de linéarisation. Résolution des équations de Stokes stationaires. Application aux équations des Navier - Stokes stationaires, Cahiers de l'IRIA (1974) 139-244.
- C.M. Dafermos, Some remarks on Korn's inequality. Z. Angew. Math. Phys.19 (1968) 913-920.
- V. Girault and P.A. Raviart, Finite element methods for Navier-Stokes equations. Springer-Verlag, Berlin (1986).
- P. Grisvard, Elliptic problems in nonsmooth domains. Pitman, Boston (1985).
- M. Gunsburger, Finite element methods for viscous incompressible flows. A guide to the theory, practice and algorithms. Academic Press, London (1989).
- C.O. Horgan and L.E. Payne, On inequalities of Korn, Friedrichs and Babuska-Aziz. Arch. Ration. Mech. Anal.40 (1971) 384-402.
- G.M. Kobelkov, On equivalent norms in L2. Anal. Math. No. 3 (1977) 177-186.
- U. Langer and W. Queck, On the convergence factor of Uzawa's algorithm. J. Comp. Appl. Math.15 (1986) 191-202.
- S.G. Mikhlin, The spectrum of an operator pencil of the elasticity theory. Uspekhi Mat. Nauk28 (1973) 43-82; English translation in Russian Math. Surveys,28.
- M.A. Olshanskii, Stokes problem with model boundary conditions. Sbornik: Mathematics188 (1997) 603-620.
- M.A. Olshanskii and E.V. Chizhonkov, On the optimal constant in the inf-sup condition for rectangle. Matematicheskie Zametki67 (2000) 387-396.
- B.N. Parlett, The Symmetrical Eigenvalue Problem. Prentice-Hall, Englewood Cliffs, New Jersey (1980).
- R. Rannacher and S. Turek, A simple nonconforming quadrilateral Stokes element. Numer. Methods Partial Differential Equation8 (1992) 97-111.
- D. Silvester and A. Wathen, Fast iterative solution of stabilized Stokes systems part II: Using block preconditioners. SIAM J. Numer. Anal.31 (1994) 1352-1367.
- M. Schäfer and S. Turek, Benchmark computations of laminar flow around cylinder, in Flow Simulation with High-Performance Computers II, E.H. Hirschel Ed., Notes on Numerical Fluid Mechanics,52, Vieweg (1996) 547-566.
- G. Strang and G.I. Fix, An analysis of the finite element methods. Prentice-Hall, New-York (1973).
- S. Turek, Efficient solvers for incompressible flow problems: An algorithmic approach in view of computational aspects. LNCSE6, Springer, Heidelberg (1999).
- S. Turek and Chr. Becker, FEATFLOW: Finite element software for the incompressible Navier-Stokes equations: User Manual, Release 1.1. Univ. of Heidelberg (1998) (http://www.featflow.de).

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.