An Eulerian finite element method for tangential Navier-Stokes equations on evolving surfaces

TL;DR

This paper presents a novel Eulerian finite element method for solving the tangential Navier-Stokes equations on evolving surfaces, combining geometric unfitted discretization with stability and convergence analysis.

Contribution

It introduces a new geometrically unfitted finite element approach for tangential Navier-Stokes equations on evolving surfaces, with comprehensive numerical analysis.

Findings

Method achieves optimal order convergence.

Stability of the numerical scheme is proven.

Numerical experiments confirm theoretical results.

Abstract

The paper introduces a geometrically unfitted finite element method for the numerical solution of the tangential Navier--Stokes equations posed on a passively evolving smooth closed surface embedded in $\mathbb{R}^3$. The discrete formulation employs finite difference and finite elements methods to handle evolution in time and variation in space, respectively. A complete numerical analysis of the method is presented, including stability, optimal order convergence, and quantification of the geometric errors. Results of numerical experiments are also provided.