We discuss a family of discontinuous Petrov–Galerkin (DPG) schemes for quite generalpartial differential operators. The starting point of our analysis is the DPG methodintroduced by [Demkowicz et al., SIAM J. Numer. Anal.49 (2011) 1788–1809; Zitelli et al., J.Comput. Phys. 230 (2011) 2406–2432]. This discretization resultsin a sparse positive definite linear algebraic system which can be obtained from a saddlepoint 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 domaindecomposition techniques provide stability and a priori estimates. Toobtain efficient numerical algorithms, we use a second Schur complement reduction appliedto the trial space. This restricts the degrees of freedom to the skeleton. We construct apreconditioner for the skeleton problem, and the efficiency of the discretization and thesolution method is demonstrated by numerical examples.
