We discuss a family of discontinuous Petrov–Galerkin (DPG) schemes for quite general
partial differential operators. The starting point of our analysis is the DPG method
introduced by [Demkowicz et al., SIAM J. Numer. Anal.
49 (2011) 1788–1809; Zitelli et al., J.
Comput. Phys. 230 (2011) 2406–2432]. This discretization results
in a sparse positive definite linear algebraic system which can be obtained from a saddle
point problem by an element-wise Schur complement reduction applied to the test space.
Here, we show that the abstract framework of saddle point problems and domain
decomposition techniques provide stability and a priori estimates. To
obtain efficient numerical algorithms, we use a second Schur complement reduction applied
to the trial space. This restricts the degrees of freedom to the skeleton. We construct a
preconditioner for the skeleton problem, and the efficiency of the discretization and the
solution method is demonstrated by numerical examples.