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

TL;DR

This paper develops and analyzes a fully discrete surface finite element method for the unsteady Navier–Stokes equations on a smooth stationary surface, enforcing the tangential velocity constraint weakly via a Lagrange multiplier. It employs a generalized Taylor–Hood $(P_{k_u},P_{k_{pr}},P_{k_})$ discretization and establishes stability and a priori error estimates that depend on the choice of $k_$ and geometry approximation order $k_g$, including a modified Ritz–Stokes projection and discrete Leray projection. The results show optimal velocity and pressure convergence when $k_=k_u$, while $k_=k_u-1$ introduces geometric-error limits that may require super-parametric geometry to regain optimal rates; numerical experiments confirm the theory and compare with penalty formulations. Overall, the work provides a rigorous, geometry-aware framework for stable, accurate simulation of surface flows with tangential constraints, offering guidance on element choices and geometry refinement for practical computations.

Abstract

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 $\mathrm{\mathbf{P}}_{k_u}$-- $\mathrm{P}_{k_{pr}}$-- $\mathrm{P}_{k_λ}$, where we enforce the tangential condition of the velocity field weakly, by introducing an extra Lagrange multiplier $λ$. 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 $k_λ=k_u$ and suboptimal with respect to the geometric approximation error when $k_λ = k_u-1$ (optimal when \emph{super-parametric finite elements} are used). For the pressure, optimal $L^2(L^2)$-norm error bounds are established when $k_λ=k_u$. Assuming further regularity assumptions for our continuous problem, we are also able to show optimal convergence (using \emph{super-parametric finite elements} again) when $k_λ=k_u-1$. Numerical simulations that confirm the established theory are provided, along with a comparative analysis against a penalty approach.

Figures (5)

• Figure 1: Varying Curvature Surface | Velocity $\mathbf{u}_h^n$ at different times for $k_g=3$, $k_u=2$, $k_{pr}=1$, $k_{\lambda}=1$, with mesh size $h=0.09$ and $\Delta t = 0.01$. The length and direction of the arrows depict the strength and orientation of the current.
• Figure 2: Varying Curvature Surface | Velocity $-$ Errors ${\mathbf e}_{\mathbf{u}}^{L^{\infty}({L^2})}$, ${\mathbf e}_{{\mathbf{P}_{\Gamma}} {\mathbf{u}}}^{L^{\infty}({L^2})}$, ${\mathbf e}_{{\mathbf{n}_{\Gamma}}}^{L^{\infty}({L^2})}$ | For different choice of $k_{\lambda}, \, k_g$| $(k_\lambda=1,\, k_g=3)$, $(k_\lambda=2,\, k_g=2)$.
• Figure 3: Varying Curvature Surface | Left: Velocity $-$ Error ${\mathbf e}_{\mathbf{u}}^{L^2(a_h)}$| Right: Pressure $-$ Error ${\mathbf e}_p^{L^2({L^2})}$ | For different choice of $k_{\lambda}, \, k_g$| $(k_\lambda=1,\, k_g=3)$, $(k_\lambda=2,\, k_g=2)$.
• Figure 4: Sphere | Velocity-pressure $L^2_{L^2}$ norm $-$ Errors | P.M.: $\{k_g=2,k_u=2,k_{pr}=1,k_p=3\}$, L.M.: $\{k_g=3,k_u=2,k_{pr}=1,k_{\lambda}=1\}$.
• Figure 5: Sphere | Velocity $\mathcal{E}_{\nabla_{\Gamma}^{cov}{\mathbf{u}}}^{L^{2}_{L^2}}-$Error (Left), $\mathcal{E}_{{\mathbf{n}_{\Gamma}}}^{L^{\infty}_{L^2}}-$Error (Right) | P.M.: $\{k_g=2,k_u=2,k_{pr}=1,k_p=3\}$, L.M.: $\{k_g=3,k_u=2,k_{pr}=1,k_{\lambda}=1\}$ .

Theorems & Definitions (93)

• Definition 2.1: Tangential derivative
• Definition 2.2: Covariant derivatives
• Definition 3.1
• Lemma 3.1: Helmholtz-Leray decomposition
• Lemma 3.2: Inf-sup Condition
• Lemma 4.1: Geometric Errors
• proof
• Lemma 4.2: Estimates for ${\mathbf B}_h$
• Lemma 4.3: Norm Equivalence
• proof