Note on quasi-optimal error estimates for the pressure for shear-thickening fluids

Abstract

In this paper, we derive quasi-optimal a priori error estimates for the kinematic pressure for a Local Discontinuous Galerkin (LDG) approximation of steady systems of $p$-Navier-Stokes type in the case of shear-thickening, i.e., in the case $p>2$, imposing a new mild Muckenhoupt regularity condition.

Figures (3)

• Figure 1: LEFT: 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 $\mathbf{e}_1\in \mathbb{S}^2$. RIGHT: Plot of the velocity vector field $\bfv\colon \Omega\to \mathbb{R}^3$ restricted to the boundary of its support, i.e., $\partial(\textup{supp}(\vert\bfv\vert))=\bigcup_{k=1}^\infty{\partial B_{r^k}^3(\mathbf{m}^k)}$.
• Figure 2: LEFT: Plot of the tetrahedron $K_0\coloneqq \textup{conv}\{\mathbf{0},\mathbf{e}_1,\mathbf{e}_1+\mathbf{e}_2,\mathbf{e}_1+\mathbf{e}_2+\mathbf{e}_3\}$ (in orange) and the corresponding transformed 24 quadrature points (in blue and red) 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; MIDDLE: Plot of $\overline{\bfv}^{N_0}\coloneqq \sum_{k=1}^{{N_0}}{\bfv^k}\in {(W^{1,p}_0(\Omega))^3}$, where $N_0=4$ is minimal such that $\mathbf{q}_0\in \textup{supp}(\overline{\bfv}^{N_0})$; RIGHT: Plot of $\overline{\bfv}^{N_1}\coloneqq \sum_{k=1}^{{N_1}}{\bfv^k}\in {(W^{1,p}_0(\Omega))^3}$, where $N_1=5$ is minimal such that $\mathbf{q}_1\in \textup{supp}(\overline{\bfv}^{N_1})$.
• 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 (35)

• Remark 2
• Proposition 3
• Remark 4
• Lemma 5: Change of Shift
• Lemma 6
• proof
• Theorem 7
• proof
• Remark 8
• Theorem 9