  We propose a new discretization method for PDEs on moving domains in the
setting of unfitted finite element methods, which is provably higher-order
accurate in space and time. In the considered setting, the physical domain that
evolves essentially arbitrarily through a time-independent computational
background domain, is represented by a level set function. For the time
discretization, the application of standard time stepping schemes that are
based on finite difference approximations of the time derivative is not
directly possible, as the degrees of freedom may get active or inactive across
such a finite difference stencil in time. In [Lehrenfeld, Olshanskii. An
Eulerian finite element method for PDEs in time-dependent domains. ESAIM: M2AN,
53:585--614, 2019] this problem is overcome by extending the discrete solution
at every timestep to a sufficiently large neighborhood so that all the degrees
of freedom that are relevant at the next time step stay active. But that paper
focuses on low-order methods. We advance these results with introducing and
analyzing realizable techniques for the extension to higher order. To obtain
higher-order convergence in space and time, we combine the BDF time stepping
with the isoparametric unfitted FEM. The latter has been used and analyzed for
several stationary problems before. However, for moving domains the key
ingredient in the method, the transformation of the underlying mesh, becomes
time-dependent which gives rise to some technical issues. We treat these with
special care, carry out an a priori error analysis and two numerical
experiments.
