  The paper introduces a finite element method for an Eulerian formulation of
partial differential equations governing the transport and diffusion of a
scalar quantity in a time-dependent domain. The method follows the idea from
Lehrenfeld & Olshanskii [ESAIM: M2AN, 53(2): 585-614, 2019] of a solution
extension to realise the Eulearian time-stepping scheme. However, a
reformulation of the partial differential equation is suggested to derive a
scheme which conserves the quantity under consideration exactly on the discrete
level. For the spatial discretisation, the paper considers an unfitted finite
element method. Ghost-penalty stabilisation is used to release the discrete
solution extension and gives a scheme robust against arbitrary intersections
between the mesh and geometry interface. The stability is analysed for both
first- and second-order backward differentiation formula versions of the
scheme. Several numerical examples in two and three spatial dimensions are
included to illustrate the potential of this method.
