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

TL;DR

This paper analyzes the finite element approximation of steady $p( abla)$-Navier-Stokes equations, establishing convergence orders under fractional regularity assumptions and validating results with numerical experiments.

Contribution

It introduces a more practical discretization of the variable exponent $p( abla)$ and handles the convective term, extending previous convergence analyses.

Findings

Numerical experiments confirm quasi-optimal error estimates.

Convergence orders are established under fractional regularity assumptions.

The discretization approach improves practicality for variable exponent models.

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.