High order methods for the approximation of the incompressible Navier–Stokes equations in a moving domain

In this paper we address the numerical approximation of the incompressible Navier–Stokes equations in a moving domain by the spectral element method and high order time integrators. We present the Arbitrary Lagrangian Eulerian (ALE) formulation of the incompressible Navier–Stokes equations and propose a numerical method based on the following kernels: a Lagrange basis associated with Fekete points in the spectral element method context, BDF time integrators, an ALE map of high degree, and an algebraic linear solver. In particular, the high degree ALE map is appropriate to deal with a computational domain whose boundary is described with curved elements. Finally, we apply the proposed strategy to a test case.