A locally conservative and energy-stable finite element for the Navier--Stokes problem on time-dependent domains

We present a finite element method for the incompressible Navier–Stokes problem that is locally conservative, energy-stable and pressure-robust on time-dependent domains. To achieve this, the space–time formulation of the Navier–Stokes problem is considered. The space–time domain is partitioned into space–time slabs which in turn are partitioned into space–time simplices. A combined discontinuous Galerkin method across space–time slabs, and space–time hybridized discontinuous Galerkin method within a space–time slab, results in an approximate velocity field that is $H(\mathrm{d}\mathrm{i}\mathrm{v})$-conforming and exactly divergence-free, even on time-dependent domains. Numerical examples demonstrate the convergence properties and performance of the method.