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 \(H(\mathop {\textrm{div}};\varOmega )\) -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.
