The Crouzeix–Raviart element for the Stokes equations with the slip boundary condition on a curved boundary

Abstract

When the curved boundary is approximated by a polygon/polyhedron’s surface, using a continuous discrete velocity to implement the slip boundary condition (SBC) of the Stokes problem may cause the variational crime. To avoid the variational crime, we apply the Crouzeix–Raviart (CR) element to discretize SBC. In error analysis, we modify the classical interpolation to satisfy the discrete SBC and obtain the interpolation error. Taking the domain perturbation into account, we derive the consistency error, and then investigate the relationship between the convergence order and the outer normal approximation for both 2D and 3D cases, where we obtain the optimal convergence for 2D cases. However, for 3D cases, our analysis elucidates that the outer normal approximation may yield a loss of the convergence rate, which is confirmed by two specific interpolation examples. The theoretical results are validated by numerical experiments.