Solving the incompressible surface Navier-Stokes equation by surface finite elements
This work provides a practical finite element framework for simulating fluid flows on curved surfaces, which is relevant for applications in fluid dynamics and geometry processing.
The authors present a numerical method for solving the incompressible surface Navier-Stokes equation on surfaces of arbitrary genus, using standard finite element ingredients. They validate their approach by comparing with discrete exterior calculus simulations on a torus and demonstrate topological effects via the Poincaré-Hopf theorem on n-tori.
We consider a numerical approach for the incompressible surface Navier-Stokes equation on surfaces with arbitrary genus $g(\mathcal{S})$. The approach is based on a reformulation of the equation in Cartesian coordinates of the embedding $\mathbb{R}^3$, penalization of the normal component, a Chorin projection method and discretization in space by surface finite elements for each component. The approach thus requires only standard ingredients which most finite element implementations can offer. We compare computational results with discrete exterior calculus (DEC) simulations on a torus and demonstrate the interplay of the flow field with the topology by showing realizations of the Poincaré-Hopf theorem on $n$-tori.