  We consider convection-diffusion problems in time-dependent domains and
present a space-time finite element method based on quadrature in time which is
simple to implement and avoids remeshing procedures as the domain is moving.
The evolving domain is embedded in a domain with fixed mesh and a cut finite
element method with continuous elements in space and discontinuous elements in
time is proposed. The method allows the evolving geometry to cut through the
fixed background mesh arbitrarily and thus avoids remeshing procedures.
However, the arbitrary cuts may lead to ill-conditioned algebraic systems. A
stabilization term is added to the weak form which guarantees well-conditioned
linear systems independently of the position of the geometry relative to the
fixed mesh and in addition makes it possible to use quadrature rules in time to
approximate the space-time integrals. We review here the space-time cut finite
element method presented in [13] where linear elements are used in both space
and time and extend the method to higher order elements for problems on
evolving surfaces (or interfaces). We present a new stabilization term which
also when higher order elements are used controls the condition number of the
linear systems from cut finite element methods on evolving surfaces. The new
stabilization combines the consistent ghost penalty stabilization [1] with a
term controlling normal derivatives at the interface.
