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

TL;DR

This paper establishes uniform inf-sup stability conditions for second order Taylor-Hood finite element pairs on anisotropic meshes, facilitating accurate solutions of fluid dynamics equations with mesh refinements.

Contribution

It proves uniform LBB conditions for $ ext{Q}_2 imes ext{Q}_1$ and $ ext{P}_2 imes ext{P}_1$ pairs on complex anisotropic meshes, extending previous results.

Findings

Proves stability for anisotropic meshes with refined patches

Generalizes Verf"urth's trick for stability analysis

Enables reliable Navier-Stokes simulations on anisotropic meshes

Abstract

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.