Dr. Barrios-Faundez, Tomas

2016, Dr. Barrios-Faundez, Tomas, Bustinza, Rommel, Domínguez, Víctor

We develop an a posteriori error analysis for Helmholtz problem using the local discontinuous Galerkin (LDG for short) approach. For the sake of completeness, we give a description of the main a priori results of this method. Indeed, under some assumptions on regularity of the solution of an adjoint problem, we prove that: (a) the corresponding indefinite discrete scheme is well posed; (b) the approach is convergent, with the expected convergence rates as long as the meshsize h is small enough. We give precise information on how small h has to be in term soft he size of the wave number and its distance to the set of eigenvalues for the same boundary value problem for the Laplacian. After that, we present a reliable and efficient a posteriori error estimator with detailed information on the dependence of the constants on the wave number. We finish presenting extensive numerical experiments which illustrate the theoretical results proven in this paper and suggest that stability and convergence may occur under less restrictive assumptions than those taken in the present work.

2020, Dr. Barrios-Faundez, Tomas, Bustinza, Rommel

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.

2016, Dr. Barrios-Faundez, Tomas, Dr. Behrens-Rincon, Edwin, Bustinza, Rommel

In this note we describe a strategy that improves the a priori error bounds for augmented mixed methods under appropriate hypotheses. This means that we can derive a priori error estimates for each one of the involved unknowns. Usually, the standard a priori error estimate is for the total error. Finally, a numerical example is included, that illustrates the theoretical results proven in this paper.

2017, Dr. Barrios-Faundez, Tomas, Bustinza, Rommel, Sánchez, Felipe

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.

2023, Barrios-Faundez, Tomas, Bustinza, Rommel, Campos, Camila

In this article, we give a description of a technique to develop an a posteriori error estimator for the dual mixed methods, when applied to elliptic partial differential equations with non homogeneous mixed boundary conditions. The approach considers conforming finite elements for the discrete scheme, and a quasi-Helmholtz decomposition result to obtain a residual a posteriori error estimator. After applying first a homogenization technique (for the Neumann boundary condition), we derive an a posteriori error estimator, which looks to be expensive to compute. This motivates the derivation of another a posteriori error estimator, that is fully computable. As a consequence, we establish the equivalence between the latter a posteriori error estimator and the natural norm of the error, that is, we prove the reliability and local efficiency of the aforementioned estimator. Finally, we report numerical examples showing the good properties of the estimator, in agreement with the theoretical results of this work.

2022, Dr. Barrios-Faundez, Tomas, Dr. Behrens-Rincon, Edwin, Bustinza, Rommel

We analyze a new stabilized dual-mixed method applied to incompressible linear elasticity problems, considering two kinds of data on the boundary of the domain: non homogeneous Dirichlet and mixed boundary conditions. In this approach, we circumvent the standard use of the rotation to impose weakly the symmetry of stress tensor. We prove that the new variational formulation and the corresponding Galerkin scheme are well-posed. We also provide the rate of convergence when each row of the stress is approximated by Raviart-Thomas elements and the displacement is approximated by continuous piecewise polynomials. Moreover, we derive a residual a posteriori error estimator for each situation. The corresponding analysis is quite different, depending on the type of boundary conditions. For known displacement on the whole boundary, we based our analysis on Ritz projection of the error, which requires a suitable quasi-Helmholtz decomposition of functions living in H (div; Ω). As a result, we obtain a simple a posteriori error estimator, which consists of five residual terms, and results to be reliable and locally efficient. On the other hand, when we consider mixed boundary conditions, these tools are not necessary. Then, we are able to develop an a posteriori error analysis, which provides us of an estimator consisting of three residual terms. In addition, we prove that in general this estimator is reliable, and when the traction datum is piecewise polynomial, locally efficient. In the second situation, we propose a numerical procedure to compute the numerical approximation, at a reasonable cost. Finally, we include several numerical experiments that illustrate the performance of the corresponding adaptive algorithm for each problem, and support its use in practice.

2021, Dr. Barrios-Faundez, Tomas, Dr. Behrens-Rincon, Edwin, Bustinza, Rommel

In this work, we focus our attention in the Stokes flow with nonhomogeneous source terms, formulated in dual mixed form. For the sake of completeness, we begin recalling the corresponding well-posedness at continuous and discrete levels. After that, and with the help of a kind of a quasi-Helmholtz decomposition of functions in H (div), we develop a residual type a posteriori error analysis, deducing an estimator that is reliable and locally efficient. Finally, we provide numerical experiments, which confirm our theoretical results on the a posteriori error estimator and illustrate the performance of the corresponding adaptive algorithm, supporting its use in practice.

2020, Barrios-Faundez, Tomas, Behrens-Rincon, Edwin, Bustinza, Rommel

This article is concerned with the Stokes system with nonhomogeneous source terms and nonhomogeneous Dirichlet boundary condition. First, we reformulate the problem in its dual mixed form, and then, we study its corresponding well‐posedness. Next, in order to circumvent the well‐known Babuška‐Brezzi condition, we analyze a stabilized formulation of the resulting approach. Additionally, we endow the scheme with an a posteriori error estimator that is reliable and efficient. Finally, we provide numerical experiments that illustrate the performance of the corresponding adaptive algorithm and support its use in practice.

2024, Dr. Barrios-Faundez, Tomas, Bustinza, Rommel

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.