  We design and analyze a coupling of a discontinuous Galerkin finite element
method with a boundary element method to solve the Helmholtz equation with
variable coefficients in three dimensions. The coupling is realized with a
mortar variable that is related to an impedance trace on a smooth interface.
The method obtained has a block structure with nonsingular subblocks. We prove
quasi-optimality of the $h$- and $p$-versions of the scheme, under a threshold
condition on the approximability properties of the discrete spaces. Amongst
others, an essential tool in the analysis is a novel
discontinuous-to-continuous reconstruction operator on tetrahedral meshes with
curved faces.
