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A short note on inf-sup conditions for the Taylor-Hood family $Q_k$-$Q_{k-1}$

Walter Zulehner

TL;DR

The paper analyzes stability of the generalized Taylor-Hood finite element family $Q_k$-$Q_{k-1}$ for the Stokes problem on quadrilateral/hexahedral meshes, covering 2D (general quadrilaterals) and 3D (parallelepipeds) under do-nothing boundary conditions. It introduces an element-wise $T$-coercivity framework to derive both local and global discrete inf-sup conditions, including the LBB and BP conditions, and employs Gauss-Lobatto quadrature and Verfuerth-type arguments to transfer local results to the global discrete spaces $V_h$ and $Q_h$. The work extends discrete LBB results to mixed and Neumann boundary settings, providing a comprehensive analysis under mesh restrictions and highlighting the remaining open problem of general 3D hexahedral meshes. Practical implications include improved stability guarantees for Taylor-Hood discretizations on structured quadrilateral/hexahedral grids and insights for multigrid smoothing analyses. All results are framed within precise mappings and norms, enabling direct applicability to error estimates and numerical solvers for the Stokes equations.

Abstract

We discuss two types of discrete inf-sup conditions for the Taylor-Hood family $Q_k$-$Q_{k-1}$ for all $k\in \mathbb{N}$ with $k\ge 2$ in 2D and 3D. While in 2D all results hold for a general class of hexahedral meshes, the results in 3D are restricted to meshes of parallelepipeds. The analysis is based on an element-wise technique as opposed to the widely used macroelement technique. This leads to inf-sup conditions on each element of the subdivision as well as to inf-sup conditions on the whole computational domain.

A short note on inf-sup conditions for the Taylor-Hood family $Q_k$-$Q_{k-1}$

TL;DR

The paper analyzes stability of the generalized Taylor-Hood finite element family - for the Stokes problem on quadrilateral/hexahedral meshes, covering 2D (general quadrilaterals) and 3D (parallelepipeds) under do-nothing boundary conditions. It introduces an element-wise -coercivity framework to derive both local and global discrete inf-sup conditions, including the LBB and BP conditions, and employs Gauss-Lobatto quadrature and Verfuerth-type arguments to transfer local results to the global discrete spaces and . The work extends discrete LBB results to mixed and Neumann boundary settings, providing a comprehensive analysis under mesh restrictions and highlighting the remaining open problem of general 3D hexahedral meshes. Practical implications include improved stability guarantees for Taylor-Hood discretizations on structured quadrilateral/hexahedral grids and insights for multigrid smoothing analyses. All results are framed within precise mappings and norms, enabling direct applicability to error estimates and numerical solvers for the Stokes equations.

Abstract

We discuss two types of discrete inf-sup conditions for the Taylor-Hood family - for all with in 2D and 3D. While in 2D all results hold for a general class of hexahedral meshes, the results in 3D are restricted to meshes of parallelepipeds. The analysis is based on an element-wise technique as opposed to the widely used macroelement technique. This leads to inf-sup conditions on each element of the subdivision as well as to inf-sup conditions on the whole computational domain.
Paper Structure (8 sections, 16 theorems, 109 equations)

This paper contains 8 sections, 16 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. $T$-coercivity
  3. Discrete inf-sup conditions
  4. The discrete LBB condition
  5. The discrete BP condition
  6. Conclusions and open problems
  7. The continuous LBB condition
  8. Some useful relations

Key Result

Lemma 1

Let $k \in \mathbb{N}$ with $k \ge 2$, let $K \in \mathcal{T}_h$ and assume that Condition restriction holds for $K$. Then there is a linear mapping $T_K \colon \bar{Q}_K \to \bar{V}_K$ such that and for all $q \in \bar{Q}_K$.

Theorems & Definitions (31)

  • Lemma 1
  • proof
  • Corollary 1
  • proof
  • Lemma 2
  • proof
  • Corollary 2
  • proof
  • Lemma 3
  • proof
  • ...and 21 more