SFEM for the unsteady Navier-Stokes Equations on a stationary surface

In this paper we consider a fully discrete numerical method for the unsteady Navier-Stokes equations on a smooth closed stationary surface in $\mathbb{R}^3$. We use the surface finite element method (SFEM) with a generalized Taylor-Hood finite element pair $\mathrm{\mathbf{P}}_{k_u}$-- $\mathrm{P}_{k_{pr}}$-- $\mathrm{P}_{k_{\lambda}}$, where we enforce the tangential condition of the velocity field weakly, by introducing an extra Lagrange multiplier $\lambda$. Depending on the richness of the finite element space involving this extra Lagrange multiplier we present a fully discrete stability and error analysis. For the velocity, we establish optimal $L^{2}(a_h)$-norm bounds ($a_h$ - an energy norm) when $k_\lambda=k_u$ and suboptimal with respect to the geometric approximation error when $k_{\lambda} = k_u-1$ (optimal when \emph{super-parametric finite elements} are used). For the pressure, optimal $L^2(L^2)$-norm error bounds are established when $k_\lambda=k_u$. Assuming further regularity assumptions for our continuous problem, we are also able to show optimal convergence (using \emph{super-parametric finite elements} again) when $k_\lambda=k_u-1$. Numerical simulations that confirm the established theory are provided, along with a comparative analysis against a penalty approach.