A Fully Discrete Surface Finite Element Method for the Navier--Stokes equations on Evolving Surfaces with prescribed Normal Velocity

We analyze two fully time-discrete numerical schemes for the incompressible Navier-Stokes equations posed on evolving surfaces in $\mathbb{R}^3$ with prescribed normal velocity using the evolving surface finite element method (ESFEM). We employ generalized Taylor-Hood finite elements $\mathrm{\mathbf{P}}_{k_u}$-- $\mathrm{P}_{k_{pr}}$-- $\mathrm{P}_{k_\lambda}$, $k_u=k_{pr}+1 \geq 2$, $k_\lambda\geq 1$, for the spatial discretization, where the normal velocity constraint is enforced weakly via a Lagrange multiplier $\lambda$, and a backward Euler discretization for the time-stepping procedure. Depending on the approximation order of $\lambda$ and weak formulation of the Navier-Stokes equations, we present stability and error analysis for two different discrete schemes, whose difference lies in the geometric information needed. We establish optimal velocity $L^{2}_{a_h}$-norm error bounds ($a_h$ an energy norm) for both schemes when $k_\lambda=k_u$, but only for the more information intensive one when $k_\lambda=k_u-1$, using iso-parametric and super-parametric discretizations, respectively, with the help of a newly derived surface Ritz-Stokes projection. Similarly, stability and optimal convergence for the pressures is established in an $L^2_{L^2}\times L^2_{H_h^{-1}}$-norm ($H_h^{-1}$ a discrete dual space) when $k_\lambda=k_u$, using a novel Leray time-projection to ensure weakly divergence conformity for our discrete velocity solution at two different time-steps (surfaces). Assuming further regularity conditions for the more information intensive scheme, along with an almost weak divergence conformity result at two different time-steps, we establish optimal $L^2_{L^2}\times L^2_{L^2}$-norm pressure error bounds when $k_\lambda=k_u-1$, using super-parametric approximation. Simulations verifying our results are provided, along with a comparison test against a penalty approach.