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Summary  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.

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