  The Galerkin difference (GD) basis is a set of continuous, piecewise
polynomials defined using a finite difference like grid of degrees of freedom.
The one dimensional GD basis functions are naturally extended to multiple
dimensions using the tensor product constructions to quadrilateral elements for
discretizing partial differential equations. Here we propose two approaches to
handling complex geometries using the GD basis within a discontinuous Galerkin
finite element setting: (1) using non-conforming, curvilinear GD elements and
(2) coupling affine GD elements with curvilinear simplicial elements. In both
cases the (semidiscrete) discontinuous Galerkin method is provably energy
stable even when variational crimes are committed and in both cases a
weight-adjusted mass matrix is used, which ensures that only the reference mass
matrix must be inverted. Additionally, we give sufficient conditions on the
treatment of metric terms for the curvilinear, nonconforming GD elements to
ensure that the scheme is both constant preserving and conservative. Numerical
experiments confirm the stability results and demonstrate the accuracy of the
coupled schemes.
