  In this paper, we present a new numerical method for determining the
numerical solution of interface problems to optimal accuracy with respect to
the polynomial order of the Lagrangian finite element space on polytopial
meshes. We introduce the notion of a virtual interface, and on this virtual
interface we enforce that "extended" interface conditions are satisfied in the
sense of a Dirichlet--Neumann coupling. The virtual interface framework serves
to bypass geometric variational crimes incurred by the classical finite element
method. Further, this approach does not require that the geometric interfaces
are spatially matching. Our analysis indicates that this approach is
well--posed and optimally convergent in $H^1$. Numerical experiments indicate
that optimal $H^1$ and $L^2$ convergence is achieved.
