Intrinsic Finite Element Methods for Fluids on Riemannian Manifolds Compared with Surface FEM

2026-08-30

AI Summary

本文提出了一种在黎曼流形上解决不可压缩Navier-Stokes方程的内在有限元方法，通过与表面FEM比较验证了其在长时间流动模拟中的准确性、效率及几何透明性。

Abstract

We present an intrinsic finite element formulation for the incompressible Navier--Stokes equations on Riemannian manifolds. We derive the corresponding weak formulation and prove that the backward Euler discretisation is energy stable. The proposed framework is validated on several representative manifolds, with particular attention paid to the long-time behaviour of the flow and its convergence to steady-state solutions represented by Killing vector fields. Comprehensive comparisons are performed with the surface finite element method and a corresponding eigenvalue formulation for Killing vector fields. The numerical results demonstrate that the intrinsic formulation provides an accurate, computationally efficient, and geometrically transparent alternative to embedded surface finite element formulations, while naturally extending to higher-dimensional Riemannian manifolds.

Problem

• intrinsic finite element
• Riemannian manifolds
• Navier--Stokes equations
• Killing vector fields

Innovation

• intrinsic finite element method
• Riemannian manifolds
• energy stability
• Killing vector fields
• surface FEM