We studied an anisotropic modified Crouzeix--Raviart finite element method for the rotational form of a stationary incompressible Navier--Stokes equation with large irrotational body forces. We present an anisotropic $H^1$ error estimate for the velocity of the modified Crouzeix--Raviart finite element method for the Navier--Stokes equation. The modified Crouzeix--Raviart finite element scheme was obtained using a lifting operator that mapped the velocity test functions to $H(\div;\Omega)$-conforming finite element spaces. Because no shape-regularity mesh conditions are imposed, anisotropic meshes can be used for the analysis. The core idea of the proof involves using the relation between the Raviart--Thomas and Crouzeix--Raviart finite element spaces. Furthermore, we present a discrete Sobolev inequality under semi-regular mesh conditions to estimate the stability of the proposed method, and confirm the results obtained through numerical experiments.

翻译：本文研究了带大无旋体积力的稳态不可压缩Navier-Stokes方程旋转形式的一种各向异性修正Crouzeix-Raviart有限元方法。我们针对Navier-Stokes方程修正Crouzeix-Raviart有限元方法的速度项给出了各向异性$H^1$误差估计。该修正Crouzeix-Raviart有限元格式通过将速度测试函数映射至$H(\div;\Omega)$协调有限元空间的提升算子构建而成。由于未施加形状正则网格条件，分析中可采用各向异性网格。证明的核心思想在于利用Raviart-Thomas与Crouzeix-Raviart有限元空间之间的关系。此外，我们在半正则网格条件下提出了离散Sobolev不等式以估计所提方法的稳定性，并通过数值实验验证了所得结果。