Weak imposition of the slip boundary condition on curved boundaries for Stokes flow

We study the finite element approximation of two methods to weakly impose a slip boundary condition for incompressible fluid flows: the Lagrange multiplier method and Nitscheʼs method. For each method, we can distinguish several formulations depending on the values of some real parameters. In the case of a spatial domain with a polygonal or polyhedral boundary, we prove convergence results of their finite element approximations, extending previous results of Verfürth [33] and we show numerical results confirming them. In the case of a spatial domain with a smooth curved boundary, numerical results show that approximations computed on polygonal domains approximating the original domain may not converge to the exact solution, depending on the values of the aforementioned parameters and on the finite element discretization. These negative results seem to highlight Babuskaʼs like paradox, due to the approximation of the boundary by polygonal ones. In particular, they seem to contradict some of Verfürthʼs theoretical convergence results.