Stokes and Navier-Stokes equations under power law slip boundary condition: Numerical analysis

Abstract

In this work, we study theoretically and numerically the equations of Stokes and Navier-Stokes under power law slip boundary condition. We establish existence of a unique solution by using the monotone operators theory for the Stokes equations whereas for the Navier-Stokes equations, we construct the solution by means of Galerkin's approximation combined with some compactness results. Next, we formulate and analyze the finite element approximations associated to these problems. We derive optimal and sub-optimal a priori error estimate for both problems depending how the monotonicity is used. Iterative schemes for solving the nonlinear problems are formulated and convergence is studied. Numerical experiments presented confirm the theoretical findings. • New power law slip boundary condition for the Stokes and Navier Stokes system. • Well posedness and optimal convergence rates for standard finite element spaces. • Formulation and analysis of the iterative scheme. • Verification of the convergence properties, simulation of the Lid driven cavity.