Convergence Analysis of the Nonconforming Finite Element Discretization of Stokes and Navier-Stokes Equations with Nonlinear Slip Boundary Conditions

This work is concerned with the nonconforming finite approximations for the Stokes and Navier-Stokes equations driven by slip boundary condition of friction type. It is well documented that if the velocity is approximated by the Crouzeix-Raviart element of order one, whereas the discrete pressure is constant elementwise that the inequality of Korn does not hold. Hence, we propose a new formulation taking into account the curvature and the contribution of tangential velocity at the boundary. Using the maximal regularity of the weak solution, we derive a priori error estimates for the velocity and pressure by taking advantage of the enrichment mapping and the application of Babuska-Brezzi's theory for mixed problems.