  We propose and analyze numerically a new fictitious domain method, based on
higher order space-time finite element discretizations, for the simulation of
the nonstationary, incompressible Navier--Stokes equations on evolving domains.
The physical domain is embedded into a fixed computational mesh such that
arbitrary intersections of the moving domain's boundaries with the background
mesh occur. The key ingredients of the approach are the weak formulation of
Dirichlet boundary conditions by Nitsche's method, the flexible and efficient
integration over all types of intersections of cells by moving boundaries and
the spatial extension of the discrete physical quantities to the entire
computational background mesh including fictitious (ghost) subdomains of fluid
flow. To prevent spurious oscillations caused by irregular intersections of
mesh cells, a penalization ensuring the stability of the approach and defining
implicitly the extension to host domains is added. These techniques are
embedded in an arbitrary order, discontinuous Galerkin discretization of the
time variable and an inf-sup stable discretization of the spatial variables.
The convergence and stability properties of the approach are studied, firstly,
for a benchmark problem of flow around a stationary obstacle and, secondly, for
flow around moving obstacles with arising cut cells and fictitious domains. The
parallel implementation is also addressed.
