  The paper introduces a new finite element numerical method for the solution
of partial differential equations on evolving domains. The approach uses a
completely Eulerian description of the domain motion. The physical domain is
embedded in a triangulated computational domain and can overlap the
time-independent background mesh in an arbitrary way. The numerical method is
based on finite difference discretizations of time derivatives and a standard
geometrically unfitted finite element method with an additional stabilization
term in the spatial domain. The performance and analysis of the method rely on
the fundamental extension result in Sobolev spaces for functions defined on
bounded domains. This paper includes a complete stability and error analysis,
which accounts for discretization errors resulting from finite difference and
finite element approximations as well as for geometric errors coming from a
possible approximate recovery of the physical domain. Several numerical
examples illustrate the theory and demonstrate the practical efficiency of the
method.
