We develop a framework for the design of finite element methods for
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, and is immune to
large distortions of the mesh under large deformations of the domain.
We derive error estimates for the method in the context of a model
parabolic problem, and we showcase the method on numerical examples in
one and two dimensions. The talk is based on a joint work with A. Lew.