  Meshing of geometric domains having curved boundaries by affine simplices
produces a polytopial approximation of those domains. The resulting error in
the representation of the domain limits the accuracy of finite element methods
based on such meshes. On the other hand, the simplicity of affine meshes makes
them a desirable modeling tool in many applications. In this paper, we develop
and analyze higher-order accurate finite element methods that remain stable and
optimally accurate on polytopial approximations of domains with smooth
boundaries. This is achieved by constraining a judiciously chosen extension of
the finite element solution on the polytopial domain to weakly match the
prescribed boundary condition on the true geometric boundary. We provide
numerical examples that highlight key properties of the new method and that
illustrate the optimal $H^1$ and $L^2$-norm convergence rates.
