On the two-step BDF finite element methods for the incompressible Navier–Stokes problem under boundary conditions of friction type

Abstract

This article concerns the numerical approximation of the two-dimensional nonstationary Navier–Stokes equations with slip boundary conditions of friction type. It studies the well-posedness and error analysis properties of the numerical scheme in L2-norm for all positive time using the two-step backward differentiation formula (BDF) in time and the finite element approximation in space. Error estimates are proved under feasible assumptions on the regularity of the solution and with the aid of different versions of discrete Grownwall lemmas. Finally, some numerical simulations are presented to illustrate our theoretical analysis.

Keywords

• Navier–Stokes equations
• Nonlinear slip boundary conditions
• Variational inequality
• Finite element method
• Two-step BDF scheme