On a Stokes hemivariational inequality for incompressible fluid flows with damping

In this paper, a Stokes hemivariational inequality is studied for incompressible fluid flows with the damping effect. The hemivariational inequality feature is caused by the presence of a nonsmooth slip boundary condition of friction type. Well-posedness of the Stokes hemivariational inequality is established through the consideration of a minimization problem. Mixed finite element methods are introduced to solve the Stokes hemivariational inequality and error estimates are derived for the mixed finite element solutions. The error estimates are of optimal order for low-order mixed element pairs under suitable solution regularity assumptions. An efficient iterative algorithm is introduced to solve the mixed finite element system. Numerical results are reported on the performance of the proposed algorithm and the numerical convergence orders of the finite element solutions.