On the domain geometry dependence of the LBB condition

The LBB condition is well-known to guarantee the stability of a finite elem ent (FE) velocity - pressure pair in incompressible how calculations. To en sure the condition to be satisfied a certain constant should be positive an d mesh-independent. The paper studies the dependence of the LBB condition o n the domain geometry. For model domains such as strips and rings the subst antial dependence of this constant on geometry aspect ratios is observed. I n domains with highly anisotropic substructures this may require special ca re with numerics to a-void failures similar to those when the LBB condition is violated. In the core of the paper we prove that for any FE velocity-pr essure pair satisfying usual approximation hypotheses the mesh-independent limit in the LBB condition is not greater than its continuous counterpart, the constant from the Nei as inequality. For the latter the explicit and as ymptotically accurate estimates are proved. The analytic results are illust rated by several numerical experiments.