Courant-Dorfman algebras of differential operators and Dorfman connections of Courant algebroids

We construct an algebra and a complex of multidifferential operators on tensor products of a Courant algebroid E with values in the endomorphism bundle of a smooth vector bundle B, predual of E, extending the standard complex of the Courant-Dorfman algebra of E. Also, we study Dorfman connections of E on B, and show that the Cartan calculus, curvatures of induced connections and basic differential geometric identities of them make sense in this algebra.