The inf-sup stability of the lowest order Taylor-Hood pair on anisotropic meshes

Uniform LBB conditions are of fundamental importance for the finite element solution of problems in incompressible fluid mechanics, such as the Stokes and Navier-Stokes equations. In this work we prove a uniform inf-sup condition for the lowest order Taylor-Hood pairs Q2×Q1 and P2×P1 on a family of affine anisotropic meshes. These meshes may contain refined edge and corner patches. We identify necessary hypotheses for edge patches to allow uniform stability and sufficient conditions for corner patches. For the proof, we generalise Verf ̈urth’s trick and recent results by some of the authors. Numerical evidence confirms the theoretical results.