  We present a new approach to discretizing shape optimization problems that
generalizes standard moving mesh methods to higher-order mesh deformations and
that is naturally compatible with higher-order finite element discretizations
of PDE-constraints. This shape optimization method is based on discretized
deformation diffeomorphisms and allows for arbitrarily high resolution of
shapes with arbitrary smoothness. Numerical experiments show that it allows the
solution of PDE-constrained shape optimization problems to high accuracy.
