Stabilized low-order mixed finite element methods for a Navier-Stokes hemivariational inequality

In this paper, pressure projection stabilized low-order mixed finite element methods are studied to solve a Navier-Stokes hemivariational inequality for a boundary value problem of the Navier-Stokes equations involving a non-smooth non-monotone boundary condition. A new abstract mixed hemivariational inequality is introduced for the purpose of analyzing stabilized mixed finite element methods to solve the Navier-Stokes hemivariational inequality using velocity-pressure pairs without the discrete inf-sup condition. The well-posedness of the abstract problem is established through considerations of a related saddle-point formulation and fixed-point arguments. Then the results on the abstract problem are applied to the study of the Navier-Stokes hemivariational inequality and its stabilized mixed finite element approximations. Optimal order error estimates are derived for finite element solutions of the pressure projection stabilized lowest-order conforming pair and lowest equal order pair under appropriate solution regularity assumptions. Numerical results are reported on the performance of the pressure projection stabilized mixed finite element methods for solving the Navier-Stokes hemivariational inequality.