Dr. Barrios-Faundez, Tomas

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 complication here lies in the fact that the classical Babuška-Brezzi theory is difficult to verify 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's 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, under standard additional regularity assumptions on the exact solution, the optimal rate of convergence of the method is guaranteed.

In this article, we first discuss the well posedness of a modified LDG scheme of Stokes problem, considering a velocity-pseudostress formulation. The difficulty here relies on the fact that the application of classical Babuška-Brezzi theory is not easy, so we proceed in a nonstandard way. For uniqueness, we apply a discrete version of Fredholm's alternative theorem, while the a priori error analysis is done introducing suitable projections of exact solution. As a result, we prove that the method is convergent, and under suitable regularity assumptions on the exact solution, the optimal rate of convergence is guaranteed. Next, we explore two stabilizations to the previous scheme, by adding least squares type terms. For these cases, well posedness and the a priori error estimates are proved by the application of standard theory. We end this work with some numerical experiments considering our third scheme, whose results are in agreement with the theoretical properties we deduce.

This paper deals with the a posteriori error analysis for an augmented mixed discontinuous formulation for the stationary Stokes problem. By considering an appropriate auxiliary problem, we derive an a posteriori error estimator. We prove that this estimator is reliable and locally efficient, and consists of just five residual terms. Numerical experiments confirm the theoretical properties of the augmented discontinuous scheme as well as of the estimator. They also show the capability of the corresponding adaptive algorithm to localize the singularities and the large stress regions of the solution.