  This article introduces the DPG-star (from now on, denoted DPG$^*$) finite
element method. It is a method that is in some sense dual to the discontinuous
Petrov-Galerkin (DPG) method. The DPG methodology can be viewed as a means to
solve an overdetermined discretization of a boundary value problem. In the same
vein, the DPG$^*$ methodology is a means to solve an underdetermined
discretization. These two viewpoints are developed by embedding the same
operator equation into two different saddle-point problems. The analyses of the
two problems have many common elements. Comparison to other methods in the
literature round out the newly garnered perspective. Notably, DPG$^*$ and DPG
methods can be seen as generalizations of $\mathcal{L}\mathcal{L}^\ast$ and
least-squares methods, respectively. A priori error analysis and a posteriori
error control for the DPG$^*$ method are considered in detail. Reports of
several numerical experiments are provided which demonstrate the essential
features of the new method. A notable difference between the results from the
DPG$^*$ and DPG analyses is that the convergence rates of the former are
limited by the regularity of an extraneous Lagrange multiplier variable.
