  This paper presents a space-time embedded-hybridized discontinuous Galerkin
(EHDG) method for the Navier--Stokes equations on moving domains. This method
uses a different hybridization compared to the space-time hybridized
discontinuous Galerkin (HDG) method we presented previously in (Int. J. Numer.
Meth. Fluids 89: 519--532, 2019). In the space-time EHDG method the velocity
trace unknown is continuous while the pressure trace unknown is discontinuous
across facets. In the space-time HDG method, all trace unknowns are
discontinuous across facets. Alternatively, we present also a space-time
embedded discontinuous Galerkin (EDG) method in which all trace unknowns are
continuous across facets. The advantage of continuous trace unknowns is that
the formulation has fewer global degrees-of-freedom for a given mesh than when
using discontinuous trace unknowns. Nevertheless, the discrete velocity field
obtained by the space-time EHDG and EDG methods, like the space-time HDG
method, is exactly divergence-free, even on moving domains. However, only the
space-time EHDG and HDG methods result in divergence-conforming velocity
fields. An immediate consequence of this is that the space-time EHDG and HDG
discretizations of the conservative form of the Navier--Stokes equations are
energy stable. The space-time EDG method, on the other hand, requires a
skew-symmetric formulation of the momentum advection term to be energy-stable.
Numerical examples will demonstrate the differences in solution obtained by the
space-time EHDG, EDG, and HDG methods.
