  We propose and analyze computationally 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 potential of such cut finite element techniques for
higher order space-time finite element methods has rarely been studied in the
literature so far and deserves further elucidation. 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. Thereby, an expensive remeshing
and adaptation of the sparse matrix data structure are avoided and the
computations are accelerated. To prevent spurious oscillations caused by
irregular intersections of mesh cells, a penalization, defining also implicitly
the extension to ghost 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 parallel
implementation of the matrix assembly is described. The optimal order
convergence properties of the algorithm are illustrated in a numerical
experiment for an evolving domain. The well-known 2d benchmark of flow around a
cylinder as well as flow around moving obstacles with arising cut cells and
fictitious domains are considered further.
