Weak imposition of boundary conditions for the Navier–Stokes equations

Abstract

We prove the convergence of a finite element method for the Navier–Stokes equations in which the no-slip condition, u· τ i =0 on Γ for i=1,2 is imposed by a penalty method and the no-penetration condition, u· n=0 on Γ, is imposed by Lagrange multipliers. This approach has been studied for the Stokes problem in (Weak imposition of boundary conditions in the Stokes problem, Ph.D. Thesis, University of Pittsburgh, PA, 1999). In most flows the Reynolds number is not negligible so the u·∇ u inertial effects are important. Thus the extension beyond the Stokes problem to the Navier–Stokes equations is critical. We show existence and uniqueness of the approximate solution and optimal order of convergence can be achieved if the computational mesh follows the real boundary. Our results for the (nonlinear) Navier–Stokes equations improve known results for this approach for the Stokes problem.