The inf-sup stability of the lowest order Taylor-Hood pair on Anisotropic Meshes

Uniform LBB conditions are desirable to approximate the solution of Navier-Stokes, Oseen, and Stokes equations on anisotropic meshes and to enable anisotropic refinements. We prove such conditions for the second order Taylor-Hood pairs ${\mathbb{Q}}_2\times {\mathbb{Q}}_1$ and ${\mathbb{P}}_2\times {\mathbb{P}}_1$ on a class of anisotropic meshes. These meshes may contain refined edge and corner patches. To this end, we generalise Verf"urth’s trick and recent results by some of the authors.