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General degree divergence-free finite element methods for the Stokes problem on smooth domains

Rebecca Durst, Michael Neilan

Abstract

In this paper, we construct and analyze divergence-free finite element methods for the Stokes problem on smooth domains. The discrete spaces are based on the Scott-Vogelius finite element pair of arbitrary polynomial degree greater than two. By combining the Piola transform with the classical isoparametric framework, and with a judicious choice of degrees of freedom, we prove that the method converges with optimal order in the energy norm. We also show that the discrete velocity error converges with optimal order in the $L^2$-norm. Numerical experiments are presented, which support the theoretical results.

General degree divergence-free finite element methods for the Stokes problem on smooth domains

Abstract

In this paper, we construct and analyze divergence-free finite element methods for the Stokes problem on smooth domains. The discrete spaces are based on the Scott-Vogelius finite element pair of arbitrary polynomial degree greater than two. By combining the Piola transform with the classical isoparametric framework, and with a judicious choice of degrees of freedom, we prove that the method converges with optimal order in the energy norm. We also show that the discrete velocity error converges with optimal order in the -norm. Numerical experiments are presented, which support the theoretical results.
Paper Structure (22 sections, 16 theorems, 111 equations, 3 figures, 3 tables)

This paper contains 22 sections, 16 theorems, 111 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Isoparametric framework
  4. The mappings $F_{\tilde{T}}$ and $F_T$
  5. The boundary regions of $\Omega$ and $\Omega_h$
  6. Clough-Tocher Split
  7. The Piola transform
  8. Bounds and scaling results
  9. Local spaces
  10. Degrees of Freedom on $\boldsymbol{V}(T)$
  11. Global Spaces
  12. Weak continuity properties
  13. Inf-sup stability
  14. The Stokes System and Finite Element Method
  15. Energy estimates
  16. ...and 7 more sections

Key Result

Lemma 2.1

Let $\boldsymbol{v} \in \boldsymbol{H}^2(\Omega)$ be extended into $\mathbb{R}^2$ in a way such that $\|\boldsymbol{v}\|_{H^2(\mathbb{R}^2)}\le C \|\boldsymbol{v}\|_{H^2(\Omega)}$. Then for $h$ sufficiently small,

Figures (3)

  • Figure 1: Clough-Tocher split and degrees of freedom for the quadratic Lagrange finite element space $(k=2$). The local split is mapping to the curved element $T$ via the polynomial diffeomorphism $F_T$.
  • Figure 2: Clough-Tocher split and degrees of freedom for the quartic Lagrange finite element space ($k=4$). Edge degrees of freedom on $\hat{T}$ are placed at Gauss-Lobatto points.
  • Figure 3: Divergence of the isoparametric method with Piola transform compared to the standard isoparametric method on Scott-Vogelius $\boldsymbol{\EuScript{P}}^3 - \EuScript{P}^2$ elements.

Theorems & Definitions (29)

  • Lemma 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 3.1
  • Lemma 3.2
  • Proposition 4.1
  • Remark 4.2
  • Lemma 4.3
  • ...and 19 more