Convergence analysis of the nonconforming finite element discretization of Stokes and Navier-Stokes equations with nonlinear slip boundary conditions

Abstract

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, while the discrete pressure is constant element wise the inequality of Korn doe 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.