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.

翻译：本文研究稳态$p(\cdot)$-Navier-Stokes方程（$p(\cdot)$为变量相关）的有限元逼近方法，并在速度矢量场与运动学压力满足自然分数阶正则性假设的条件下证明了收敛阶。与已有结果相比，我们处理了对流项并采用了更实用的幂律指数$p(\cdot)$离散化方案。数值实验证实了关于速度矢量场与运动学压力的分数阶正则性假设下速度$\textit{a priori}$误差估计的拟最优性。