  We propose and analyze a discretization scheme that combines the
discontinuous Petrov-Galerkin and finite element methods. The underlying model
problem is of general diffusion-advection-reaction type on bounded domains,
with decomposition into two sub-domains. We propose a heterogeneous variational
formulation that is of the ultra-weak (Petrov-Galerkin) form with broken test
space in one part, and of Bubnov-Galerkin form in the other. A standard
discretization with conforming approximation spaces and appropriate test spaces
(optimal test functions for the ultra-weak part and standard test functions for
the Bubnov-Galerkin part) gives rise to a coupled DPG-FEM scheme. We prove its
well-posedness and quasi-optimal convergence. Numerical results confirm
expected convergence orders.
