On the domain geometry dependence of the LBB condition

¹ Dept. Mechanics and Mathematics, Moscow State University, Moscow 119899, Russia. (chizhonk@mech.math.msu.su)
² Dept. Mechanics and Mathematics, Moscow State University, Moscow 119899, Russia. (ay@olshan.msk.ru)

Received: 9 September 1999
Revised: 20 March 2000

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.