Tomás Barrios, Rommel Bustinza:
An a priori error analysis for discontinuous Lagrangian finite elements applied to nonconforming dual mixed formulations: Poisson and Stokes problems
In this paper, we discuss the well posedness of a mixed discontinuous Galerkin (DG) scheme for the Poisson and Stokes problems in 2D, considering only piecewise Lagrangian finite elements. The difficulty here relies on the fact that the classical Babuska-Brezzi theory is not easy to check for low order finite elements, so we proceed in a non-standard way. First, we prove uniqueness, and then we apply a discrete version of Fredholm alternative theorem to ensure existence. The a priori error analysis is done by introducing suitable projections of the exact solution. As a result, we prove that the method is convergent and that, under standard additional regularity assumptions on the exact solution, the optimal rate of convergence of the method is guaranteed.