A parametric finite element method for the incompressible Navier--Stokes equations on an evolving surface

In this paper we consider the numerical approximation of the incompressible surface Navier–Stokes equations on an evolving surface. For the discrete representation of the moving surface we use parametric finite elements of degree $\mathrm{\ell }\ge 2$. In the semidiscrete continuous-in-time setting we are able to prove a stability estimate that mimics a corresponding result for the continuous problem. Some numerical results, including a convergence experiment, demonstrate the practicality and accuracy of the proposed method.