Analysis and finite element solution of a Navier–Stokes hemivariational inequality for incompressible fluid flows with damping

This paper provides a well-posedness analysis and a mixed finite element method for a hemivariational inequality of the stationary Navier–Stokes equations with a nonlinear damping term. The Navier–Stokes hemivariational inequality describes a steady incompressible fluid flow subject to a nonsmooth slip boundary condition of friction type. The well-posedness of the Navier–Stokes hemivariational inequality is established by constructing two auxiliary problems and applying Banach fixed point arguments twice. Mixed finite element methods are introduced to solve the problem, and error estimates for the solutions are derived. The error estimates are of optimal order for low-order mixed element pairs under suitable solution regularity assumptions. An efficient iterative algorithm is presented, and numerical results are provided to verify the theoretical analysis.