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We consider a numerical approach for the incompressible surface Navier-Stokesequation on surfaces with arbitrary genus $g(\mathcal{S})$. The approach isbased on a reformulation of the equation in Cartesian coordinates of theembedding $\mathbb{R}^3$, penalization of the normal component, a Chorinprojection method and discretization in space by surface finite elements foreach component. The approach thus requires only standard ingredients which mostfinite element implementations can offer. We compare computational results withdiscrete exterior calculus (DEC) simulations on a torus and demonstrate theinterplay of the flow field with the topology by showing realizations of thePoincar\'e-Hopf theorem on $n$-tori.

Solving the incompressible surface Navier-Stokes equation by surface finite elements