  In this paper we introduce general transfer operators between high-order and
low-order refined finite element spaces that can be used to couple high-order
and low-order simulations. Under natural restrictions on the low-order refined
space we prove that both the high-to-low-order and low-to-high-order linear
mappings are conservative, constant preserving and high-order accurate. While
the proofs apply to affine geometries, numerical experiments indicate that the
results hold for more general curved and mixed meshes. These operators also
have applications in the context of coarsening solution fields defined on
meshes with nonconforming refinement. The transfer operators for $H^1$ finite
element spaces require a globally coupled solve, for which robust and efficient
preconditioners are developed. We present several numerical results confirming
our analysis and demonstrate the utility of the new mappings in the context of
adaptive mesh refinement and conservative multi-discretization coupling.
