Anisotropic modified Crouzeix-Raviart finite element method for the stationary Navier-Stokes equation

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.