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

TL;DR

This paper develops and analyzes two fully discrete finite element schemes for solving the incompressible Navier-Stokes equations on evolving surfaces with prescribed normal velocity, providing stability and error bounds.

Contribution

It introduces two novel time-discrete schemes with stability and error analysis for Navier-Stokes on evolving surfaces, employing weak enforcement of normal velocity and new surface projection techniques.

Findings

Optimal velocity error bounds established for both schemes.

Pressure convergence proven under regularity assumptions.

Comparison with penalty approach demonstrates effectiveness.

Abstract

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.