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.
This preprint gave rise to the following definitive publication(s):
Tomás BARRIOS, Rommel BUSTINZA: An a priori error analysis for discontinuous Lagrangian finite elements applied to nonconforming dual mixed formulations: Poisson and Stokes problems. Electronic Transactions on Numerical Analysis, vol. 52, pp. 455-479, (2020).