A Mixed-FEM for Navier–Stokes type variational inequality with nonlinear damping term

In this work we consider a mixed finite element method (FEM) for incompressible Navier–Stokes equations with nonlinear damping term and friction boundary conditions. After establishing the variational formulation, we show the well-posedness of this problem. Subsequently, we focus our attention on the mixed FEM and analyze the Galerkin approximation to this system. Then some optimal error estimates are deduced. In the end, some iterative algorithms are presented and numerical results are given to verify the theoretical analysis.