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

Abstract

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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.

How to cite

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Chizhonkov, 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

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