A parametric finite element method for the incompressible Navier--Stokes equations on an evolving surface

TL;DR

This work develops a parametric finite element method for the incompressible Navier–Stokes equations on evolving surfaces, using $P\ell$ elements with $\ell\ge2$ to discretize the geometry and velocity, and a $P^{\ell-1}$ pressure space. A semidiscrete energy stability estimate is established, and a fully discrete scheme with a Schur-complement solver is proposed and analyzed, including existence and uniqueness under a discrete LBB condition. Numerical experiments on radially evolving spheres and complex geometries demonstrate $O(h^3)$ convergence (for appropriate parameters) and reveal stability benefits from a positive bending forcing coefficient $\alpha$, while $\alpha=0$ can induce instabilities. The approach provides a robust, rigorous framework for simulating fluid flow on moving membranes and interfaces with curvature-driven forces, supported by detailed numerical evidence and efficient solvers.

Abstract

In this paper we consider the numerical approximation of the incompressible surface Navier--Stokes equations on an evolving surface. For the discrete representation of the moving surface we use parametric finite elements of degree $\ell \geq 2$. In the semidiscrete continuous-in-time setting we are able to prove a stability estimate that mimics a corresponding result for the continuous problem. Some numerical results, including a convergence experiment, demonstrate the practicality and accuracy of the proposed method.

Figures (6)

• Figure 1: ($\rho=\mu=\alpha=1$) The surface $\Gamma^m$ at times $t=0,0.6,1,2$. Below we show plots of $E^{m}$ (left) and $E^{m}_\alpha$ (right) over time.
• Figure 2: ($\rho=10^{4}\cdot\mu=\alpha=1$) The surface $\Gamma^m$ at times $t=0,0.6,1,2$. Below we show plots of $E^{m}$ (left) and $E^{m}_\alpha$ (right) over time.
• Figure 3: ($\rho=\mu=\alpha=1$) The surface $\Gamma^m$, together with a visualization of the velocity field $\vec{U}^m$ at times $t=0,1$. Below we show plots of $E^{m}$ (left) and $E^{m}_\alpha$ (right) over time.
• Figure 4: ($\rho=\mu=\alpha=1$) The surface $\Gamma^m$ at times $t=0,0.5,1,2,10,30$. Below we show plots of $E^{m}$ (left) and $E^{m}_\alpha$ (right) over time.
• Figure 5: ($\rho=\mu=1$, $\alpha=0$) The surface $\Gamma^m$ at times $t=0,0.5,1,2$. Below we show plots of $E^{m}$ (left) and $E^{m}_\alpha$ (right) over time.

Theorems & Definitions (4)

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