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Conditional quasi-optimal error estimate for a finite element discretization of the $p$-Navier-Stokes equations: The case $p>2$

Alex Kaltenbach, Michael Růžička

Abstract

In this paper, we derive quasi-optimal $\textit{a priori}$ error estimates for the kinematic pressure for a Finite Element (FE) approximation of steady systems of $p$-Navier-Stokes type in the case of shear-thickening, $\textit{i.e.}$, in the case $p>2$, imposing a new mild Muckenhoupt regularity condition.

Conditional quasi-optimal error estimate for a finite element discretization of the $p$-Navier-Stokes equations: The case $p>2$

Abstract

In this paper, we derive quasi-optimal error estimates for the kinematic pressure for a Finite Element (FE) approximation of steady systems of -Navier-Stokes type in the case of shear-thickening, , in the case , imposing a new mild Muckenhoupt regularity condition.

Paper Structure

This paper contains 17 sections, 14 theorems, 68 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic Notation
  4. N-functions and Orlicz spaces
  5. Basic properties of the extra stress tensor
  6. The $p$-Navier--Stokes system
  7. Discussion of Muckenhoupt regularity condition
  8. Finite Element (FE) approximation
  9. Triangulations
  10. Finite element spaces and projectors
  11. FE formulations
  12. A priori error estimates
  13. Numerical experiments
  14. Implementation details
  15. General experimental set-up
  16. ...and 2 more sections

Key Result

Proposition 2.10

Let $\mathbf{S}$ satisfy Assumption assum:extra_stress, let $\varphi$ be defined in eq:def_phi, and let $\bfF$ be defined in eq:def_F. Then, uniformly with respect to $\bfA, \bfB \in \setR^{d \times d}$, we have that The constants depend only on the characteristics of ${\mathbf{S}}$.

Figures (3)

  • Figure 1: Plot of the strain rate $\vert\bfD \bfv\vert\colon \Omega\to \mathbb{R}_{\ge 0}$ restricted to the boundary of its support, i.e., $\partial(\textup{supp}(\vert\bfD \bfv\vert))=\bigcup_{k=1}^\infty{\partial B_{r^k}^3(\mathbf{m}^k)}$. The color map indicates that the strain rate increases when approaching the first unit vector $\mathbf{e}_1\in \mathbb{S}^2$.
  • Figure 2: TOP: tetrahedron $K_0\coloneqq \textup{conv}\{\mathbf{0},\mathbf{e}_1,\mathbf{e}_1+\mathbf{e}_2,\mathbf{e}_1+\mathbf{e}_2+\mathbf{e}_3\}$ and the transformed 24 quadrature points of the Keast rule (KEAST7) (cf. keast). The quadrature point $\mathbf{q}_0\in K_0$ closest to the first unit vector is marked in red; BOTTOM: $\overline{\bfv}^{N_0}\coloneqq \sum_{k=1}^{{N_0}}{\bfv^k}\in {(W^{1,p}_0(\Omega))^3}$, where $N_0=4$ is minimal such that $\mathbf{q}_0\in \textup{supp}(\overline{\bfv}^{N_0})$.
  • Figure 3: Plots of $E_i\coloneqq (\int_{B_{r^{N_i}}^3(\mathbf{m}^{N_i})}{\mu_{\bfD\bfv}\,\mathrm{d}\mu_{h_i}})(\int_{B_{r^{N_i}}^3(\mathbf{m}^{N_i})}{\mu_{\bfD\bfv}^{-1}\,\mathrm{d}\mu_{h_i}})$, $i=1,\ldots,6$, for $p\in \{2.25,2.5,2.75,3.0,3.25,2.5\}$, where $\mathrm{d}\mu_{h_i}$, $i=1,\ldots,6$, denote the discrete measures representing the Keast rule (KEAST7) (cf. keast), indicating that $E_i\to \infty$$(i\to \infty)$ and, thus, that the violation of the Muckenhoupt condition \ref{['violation']} is sufficiently resolved by the Keast rule (KEAST7) (cf. keast).

Theorems & Definitions (33)

  • Remark 2.7
  • Proposition 2.10
  • Remark 2.13
  • Lemma 2.14: Change of Shift
  • Lemma 2.22
  • proof
  • Theorem 2.24
  • proof
  • Remark 2.25
  • Theorem 2.26
  • ...and 23 more