On Finite Volume Methods for a Navier–Stokes Variational Inequality

Finite volume methods are introduced to solve a class of variational inequality of the second kind, which is governed by the incompressible Navier–Stokes equations under nonlinear slip boundary conditions of friction type. The numerical schemes are derived with the use of a proper numerical integration formula (e.g., trapezoidal rule) to approximate the subdifferential term due to the nonlinear constraints. For two particular finite volume schemes, existence and uniqueness of the numerical solutions are proved and optimal order error estimates are derived for solving the Navier–Stokes variational inequality. Numerical results are reported, focusing on the numerical convergence orders and illustrations of the tangential velocity components characterizing the potential slip phenomena.