  High-order finite element methods harbor the potential to deliver improved
accuracy per degree of freedom versus low-order methods. Their success,
however, hinges upon the use of a curvilinear mesh of not only sufficiently
high accuracy but also sufficiently high quality. In this paper, theoretical
results are presented quantifying the impact of mesh parameterization on the
accuracy of a high-order finite element approximation, and a formal definition
of shape regularity is introduced for curvilinear meshes based on these
results. This formal definition of shape regularity in turn inspires a new set
of quality metrics for curvilinear finite elements. Computable bounds are
established for these quality metrics using the Bernstein-B\'ezier form, and a
new curvilinear mesh optimization procedure is proposed based on these bounds.
Numerical results confirming the importance of shape regularity in the context
of high-order finite element methods are presented, and numerical results
demonstrating the promise of the proposed curvilinear mesh optimization
procedure are also provided. The theoretical results in this paper apply to any
piecewise-polynomial or piecewise-rational finite element method posed on a
mesh of polynomial or rational mapped simplices and hypercubes. As such, they
apply not only to classical continuous Galerkin finite element methods but also
to discontinuous Galerkin finite element methods and even isogeometric methods
based on NURBS, T-splines, or hierarchical B-splines.
