Higher-order surface FEM for incompressible Navier-Stokes flows on manifolds

TL;DR

This paper develops a higher-order surface finite element method for simulating incompressible Navier-Stokes flows on curved manifolds, achieving high accuracy and convergence rates for complex geometries.

Contribution

It introduces a novel higher-order surface FEM framework for Navier-Stokes equations on manifolds, including stabilization techniques and applications to benchmark problems.

Findings

Highly accurate solutions obtained

Higher-order convergence rates confirmed

Effective stabilization for high Reynolds number flows

Abstract

Stationary and instationary Stokes and Navier-Stokes flows are considered on two-dimensional manifolds, i.e., on curved surfaces in three dimensions. The higher-order surface FEM is used for the approximation of the geometry, velocities, pressure, and Lagrange multiplier to enforce tangential velocities. Individual element orders are employed for these various fields. Stream-line upwind stabilization is employed for flows at high Reynolds numbers. Applications are presented which extend classical benchmark test cases from flat domains to general manifolds. Highly accurate solutions are obtained and higher-order convergence rates are confirmed.