A locally conservative and energy-stable finite-element method 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(div)-conforming and exactly divergence-free, even on time-dependent domains. Numerical examples demonstrate the convergence properties and performance of the method.