  We introduce a framework for the design of finite element methods for
two-dimensional moving boundary problems with prescribed boundary evolution
that have arbitrarily high order of accuracy, both in space and in time. At the
core of our approach is the use of a universal mesh: a stationary background
mesh containing the domain of interest for all times that adapts to the
geometry of the immersed domain by adjusting a small number of mesh elements in
the neighborhood of the moving boundary. The resulting method maintains an
exact representation of the (prescribed) moving boundary at the discrete level,
or an approximation of the appropriate order, yet is immune to large
distortions of the mesh under large deformations of the domain. The framework
is general, making it possible to achieve any desired order of accuracy in
space and time by selecting a preferred and suitable finite-element space on
the universal mesh for the problem at hand, and a preferred and suitable time
integrator for ordinary differential equations. We illustrate our approach by
constructing a particular class of methods, and apply them to a
prescribed-boundary variant of the Stefan problem. We present numerical
evidence for the order of accuracy of our schemes in one and two dimensions.
