  With this contribution, we shed light on the relation between the discrete
adjoints of multistep backward differentiation formula (BDF) methods and the
solution of the adjoint differential equation. To this end, we develop a
functional-analytic framework based on a constrained variational problem and
introduce the notion of weak adjoint solutions. We devise a finite element
Petrov-Galerkin interpretation of the BDF method together with its discrete
adjoint scheme obtained by reverse internal numerical differentiation. We show
how the finite element approximation of the weak adjoint is computed by the
discrete adjoint scheme and prove its asymptotic convergence in the space of
normalized functions of bounded variation. We also obtain asymptotic
convergence of the discrete adjoints to the classical adjoints on the inner
time interval. Finally, we give numerical results for non-adaptive and fully
adaptive BDF schemes. The presented framework opens the way to carry over the
existing theory on global error estimation techniques from finite element
methods to BDF methods.
