  We present a new technique to apply finite element methods to partial
differential equations over curved domains. A change of variables along a
coordinate transformation satisfying only low regularity assumptions can
translate a Poisson problem over a curved physical domain to a Poisson problem
over a polyhedral parametric domain. This greatly simplifies both the geometric
setting and the practical implementation, at the cost of having globally rough
non-trivial coefficients and data in the parametric Poisson problem. Our main
result is that a recently developed broken Bramble-Hilbert lemma is key in
harnessing regularity in the physical problem to prove higher-order finite
element convergence rates for the parametric problem. Numerical experiments are
given which confirm the predictions of our theory.
