Numerical analysis of a characteristic stabilized finite element method for the time-dependent Navier–Stokes equations with nonlinear slip boundary conditions

Abstract

Based on a characteristic method, this work is concerned with a finite element approximation to the time-dependent Navier–Stokes equations with nonlinear slip boundary conditions. Since this slip boundary condition of friction type contains a subdifferential property, its continuous variational problem is formulated as an inequality, which can turn into an equality problem by using a powerful regularized method. Then a fully discrete characteristic scheme under the stabilized lower order finite element pairs is proposed for the equality problem. Optimal error estimates for velocity and pressure are derived under the corresponding L2,H1-norms. Finally, a smooth problem test is reported to demonstrate the theoretically predicted convergence order and the expected slip phenomena, and the simulation of a bifurcated blood flow model is displayed to illustrate the efficiency of the proposed method.