  We present a cut finite element method for the heat equation on two
overlapping meshes: a stationary background mesh and an overlapping mesh that
evolves inside/"on top" of it. Here the overlapping mesh is prescribed a simple
discontinuous evolution, meaning that its location, size, and shape as
functions of time are discontinuous and piecewise constant. 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. The simple
discontinuous mesh evolution results in a space-time discretization with a
slabwise product structure between space and time which allows for existing
analysis methodologies to be applied with only minor modifications. We follow
the analysis methodology presented by Eriksson and Johnson in [1, 2]. The
greatest modification is the introduction of a Ritzlike "shift operator" that
is used to obtain the discrete strong stability needed for the error analysis.
The shift operator generalizes the original analysis to some methods for which
the discrete subspace at one time does not lie in the space of the stiffness
form at the subsequent time. The error analysis consists of 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.
