  We present a cut finite element method for the heat equation on two
overlapping meshes: a stationary background mesh and an overlapping mesh that
moves around inside/"on top" of it. Here the overlapping mesh is prescribed a
simple continuous motion, meaning that its location as a function of time is
continuous and piecewise linear. For the discrete function space, we use
continuous Galerkin in space and discontinuous Galerkin in time, with the
addition of a discontinuity on the boundary between the two meshes. The finite
element formulation is based on Nitsche's method and also includes an integral
term over the space-time boundary between the two meshes that mimics the
standard discontinuous Galerkin time-jump term. The simple continuous mesh
motion results in a space-time discretization for which standard analysis
methodologies either fail or are unsuitable. We therefore employ what seems to
be a relatively new energy analysis framework that is general and robust enough
to be applicable to the current setting. The energy analysis consists of a
stability estimate that is slightly stronger than the standard basic one and an
a priori error estimate that is of optimal order with respect to both time step
and mesh size. We also present numerical results for a problem in one spatial
dimension that verify the analytic error convergence orders.
