Error analysis for a finite element approximation of the steady $p(\cdot)$-Navier-Stokes equations

Abstract

In this paper, we examine a finite element approximation of the steady $p(\cdot)$-Navier-Stokes equations ($p(\cdot)$ is variable dependent) and prove orders of convergence by assuming natural fractional regularity assumptions on the velocity vector field and the kinematic pressure. Compared to previous results, we treat the convective term and employ a more practicable discretization of the power-law index $p(\cdot)$. Numerical experiments confirm the quasi-optimality of the $\textit{a priori}$ error estimates (for the velocity) with respect to fractional regularity assumptions on the velocity vector field and the kinematic pressure.

Figures (4)

• Figure 1: LEFT: intersection of $\mathcal{T}_h$ with ($\mathbb{R}\times \{\frac{1}{2}\}\times \mathbb{R}$)-plane; RIGHT: electric field $\mathbf{E}\in C^\infty(\overline{\Omega};\mathbb{R}^3)$.
• Figure 2: LEFT: mechanical body force $\widehat{\mathbf{f}}\in C^\infty(\overline{\Omega};\mathbb{R}^3)$; RIGHT: total force $\mathbf{f}\in C^\infty(\overline{\Omega};\mathbb{R}^3)$ when the electric field is applied (in a $\log$-plot).
• Figure 3: LEFT: discrete velocity vector field $\mathbf{v}_h\in (\mathbb{P}^2(\mathcal{T}_h))^3$ in (\ref{['TC1']}); RIGHT: discrete velocity vector field $\mathbf{v}_h\in (\mathbb{P}^2(\mathcal{T}_h))^3$ in (\ref{['TC2']}) (in a $\log$-plot).
• Figure 4: LEFT: discrete pressure $q_h\in \mathbb{P}^1_c(\mathcal{T}_h)$ intersected with the ($\mathbb{R}\times \{\frac{1}{2}\}\times \mathbb{R}$)-plane in (\ref{['TC1']}); RIGHT: discrete pressure $q_h\in \mathbb{P}^1_c(\mathcal{T}_h)$ intersected with the ($\mathbb{R}\times \{\frac{1}{2}\}\times \mathbb{R}$)-plane in (\ref{['TC2']}).

Theorems & Definitions (64)

• Remark 2.8
• Proposition 2.14
• proof
• Lemma 2.19
• proof
• Remark 2.22
• Lemma 2.23
• proof
• Lemma 2.24: Key estimate
• proof