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

Abstract

ABSTRACTThis 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.