Artículos científicos

URI permanente para esta colecciónhttp://10.0.96.45:4000/handle/11056/14723

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A mixed virtual element method for a pseudostress-based formulation of linear elasticity
(Elsevier B.V., 2019) Cáceres, Ernesto; Gatica, Gabriel N.; Sequeira, Filander
In this paper we introduce and analyze a mixed virtual element method (mixed-VEM) for a pseudostress-displacement formulation of the linear elasticity problem with non homogeneous Dirichlet boundary conditions. We follow a previous work by some of the authors, and employ a mixed formulation that does not require symmetric tensor spaces in the finite element discretization. More precisely, the main unknowns here are given by the pseudostress and the displacement, whereas other physical quantities such as the stress, the strain tensor of small deformations, and the rotation, are computed through simple postprocessing formulae in terms of the pseudostress variable. We first recall the corresponding variational formulation, and then summarize the main mixed-VEM ingredients that are required for our discrete analysis. In particular, we utilize a well known local projector onto a suitable polynomial subspace to define a calculable version of our discrete bilinear form, whose continuous version requires information of the variables on the interior of each element. Next, we show that the global discrete bilinear form satisfies the hypotheses required by the Babuška–Brezzi theory. In this way, we conclude the well-posedness of our mixed-VEM scheme and derive the associated a priori error estimates for the virtual solutions as well as for the fully computable projections of them. Furthermore, we also introduce a second element-by-element postprocessing formula for the pseudostress, which yields an optimally convergent approximation of this unknown with respect to the broken H(div)-norm. In addition, this postprocessing formula can also be applied to the postprocessed stress tensor. Finally, several numerical results illustrating the good performance of the method and confirming the theoretical rates of convergence are presented.
A mixed virtual element method for quasi-Newtonian stokes flows
(SIAM, 2018) CACERES, ERNESTO; Gatica, Gabriel; Sequeira, Filander
In this paper we introduce and analyze a virtual element method (VEM) for an augmented mixed variational formulation of a class of nonlinear Stokes models arising in quasi-Newtonian fluids. While the original unknowns are given by the pseudostress, the velocity, and the pressure, the latter is eliminated by using the incompressibility condition, and in order to handle the nonlinearity involved, the velocity gradient is set as an auxiliary one. In this way, and adding a redundant term arising from the constitutive equation relating the psdeudostress and the velocity, an augmented formulation showing a saddle point structure is obtained, whose well-posedness has been established previously by using known results from nonlinear functional analysis. Then, following the basic principles and ideas of the mixed- VEM approach, we introduce a Galerkin scheme employing generic virtual element subspaces and projectors satisfying suitable abstract conditions and derive the corresponding solvability analysis, along with the associated a priori error estimates for the virtual element solution as well as for the fully computable projection of it. Next, we provide two specific choices of subspaces and local projectors verifying the required hypotheses, one of them yielding an optimally convergent mixed- VEM for the fully nonlinear problem studied here, and the other one providing a new approach for the linear version of it, that is, for the Stokes problem. In addition, we are able to apply a second element-by-element postprocessing formula for the pseudostress, which yields an optimally convergent approximation of it with respect to the broken H(div)-norm. Finally, several numerical results illustrating the good performance of the method and confirming the theoretical rates of convergence are reported. © 2018 Society for Industrial and Applied Mathematics.
A mixed virtual element method for the boussinesq problem on polygonal meshes
(Global Science Press, 2021) Gatica, Gabriel; Munar Benitez, Edgar Mauricio; Sequeira, Filander
In this work we introduce and analyze a mixed virtual element method (mixed-VEM) for the two-dimensional stationary Boussinesq problem. The continuous formulation is based on the introduction of a pseudostress tensor depending nonlinearly on the velocity, which allows to obtain an equivalent model in which the main unknowns are given by the aforementioned pseudostress tensor, the velocity and the temperature, whereas the pressure is computed via a postprocessing formula. In addition, an augmented approach together with a fixed point strategy is used to analyze the well-posedness of the resulting continuous formulation. Regarding the discrete problem, we follow the approach employed in a previous work dealing with the Navier-Stokes equations, and couple it with a VEM for the convection-diffiusion equation modelling the temperature. More precisely, we use a mixed-VEM for the scheme associated with the uid equations in such a way that the pseudostress and the velocity are approximated on virtual element subspaces of H(div) and H1, respectively, whereas a VEM is proposed to approximate the temperature on a virtual element subspace of H1. In this way, we make use of the L2-orthogonal projectors onto suitable polynomial spaces, which allows the explicit integration of the terms that appear in the bilinear and trilinear forms involved in the scheme for the uid equations. On the other hand, in order to manipulate the bilinear form associated to the heat equations, we define a suitable projector onto a space of polynomials to deal with the fact that the diffiusion tensor, which represents the thermal conductivity, is variable. Next, the corresponding solvability analysis is performed using again appropriate fixed-point arguments. Further, Strang-type estimates are applied to derive the a priori error estimates for the components of the virtual element solution as well as for the fully computable projections of them and the postprocessed pressure. The corresponding rates of convergence are also established. Finally, several numerical examples illustrating the performance of the mixed-VEM scheme and confirming these theoretical rates are presented.
A posteriori error analysis of a mixed virtual element method for a nonlinear Brinkman model of porous media flow
(Elsevier Ltd, 2020-06-25) Munar, Mauricio; Sequeira, Filander
In this paper we present an a posteriori error analysis of a mixed-VEM discretization for a nonlinear Brinkman model of porous media flow, which has been proposed by the authors in a previous work. Therein, the system is formulated in terms of a pseudostress tensor and the velocity gradient, whereas the velocity and the pressure of the fluid are computed via postprocessing formulas. Furthermore, the well-posedness of the associated augmented formulation along with a priori error bounds for the discrete scheme also were established. We now propose reliable and efficient residualbased a posteriori error estimates for a computable approximation of the virtual solution associated to the aforementioned problem. The resulting error estimator is fully computable from the degrees of freedom of the solutions and applies on very general polygonal meshes. For the analysis we make use of a global inf–sup condition, Helmholtz decomposition, local approximation properties of interpolation operators and inverse inequalities together with localization arguments based on bubble functions. Finally, we provide some numerical results confirming the properties of our estimator and illustrating the good performance of the associated adaptive algorithm
A Priori and a Posteriori Error Analyses of an Augmented HDG Method for a Class of Quasi-Newtonian Stokes Flows
(Springer, 2016-12) Gatica, Gabriel; Sequeira, Filander
In a recent work we developed a new hybridizable discontinuous Galerkin (HDG) method for a class of nonlinear Stokes models arising in quasi-Newtonian fluids. The approach there uses the incompressibility condition to eliminate the pressure, sets the gradient of the velocity as an auxiliary unknown, and enriches the original formulation with convenient redundant equations, thus giving rise to an augmented scheme. However, the corresponding analysis, which makes use of a fixed point strategy that depends on a suitably chosen parameter, yields optimal rates of convergence for only two of the six resulting unknowns, whereas the reported numerical results, showing higher orders than predicted, support the conjecture that the a priori error estimates are not sharp. In the present paper, the main features of the aforementioned augmented formulation are maintained, but after introducing slight modifications of the finite element subspaces for the pseudostress and velocity, we are able to significantly improve our previous analyses and results. More precisely, on one hand we realize here that it suffices to choose the stabilization tensor as the identity times the meshsize, and hence neither fixed-point arguments nor related parameters are needed anymore to establish the well-posedness of the discrete scheme, and on the other hand we now prove optimally convergent approximations for all the unknowns. Furthermore, we develop a reliable and efficient residual-based a posteriori error estimator, and propose the associated adaptive algorithm for our HDG approximation of the nonlinear model problem. Finally, several numerical results illustrating the performance of the method, confirming the theoretical properties of the estimator, and showing the expected behaviour of the adaptive refinements, are presented. © 2016, Springer Science+Business Media New York
A priori and a posteriori error analyses of an HDG method for the Brinkman problem
(Elsevier, 2018-01-15) Gatica, Luis; Sequeira, Filánder
In this paper we introduce and analyze a hybridizable discontinuous Galerkin (HDG) method for the linear Brinkman model of porous media flow in two and three dimensions and with non-homogeneous Dirichlet boundary conditions. We consider a fully-mixed formulation in which the main unknowns are given by the pseudostress, the velocity and the trace of the velocity, whereas the pressure is easily recovered through a simple postprocessing. We show that the corresponding continuous and discrete schemes are well-posed. In particular, we use the projection-based error analysis in order to derive a priori error estimates. Furthermore, we develop a reliable and efficient residual-based a posteriori error estimator, and propose the associated adaptive algorithm for our HDG approximation. Finally, several numerical results illustrating the performance of the method, confirming the theoretical properties of the estimator and showing the expected behavior of the adaptive refinements are presented. © 2017 Elsevier Ltd
A RTk - P-k approximation for linear elasticity yielding a broken H(div) convergent postprocessed stress
(Universidad Nacional, Costa Rica, 2015) Gatica, Gabriel N.; Gatica, Luis F.; Sequeira, Filander
We present a non-standard mixed finite element method for the linear elasticity problem in R-n with non-homogeneous Dirichlet boundary conditions. More precisely, our approach is based on a simplified interpretation of the pseudostress displacement formulation originally proposed in Arnold and Falk (1988), which does not require symmetric tensor spaces in the finite element discretization. We apply the classical Babuska-Brezzi theory to prove that the corresponding continuous and discrete schemes are well-posed. In particular, Raviart-Thomas spaces of order k >= 0 for the pseudostress and piecewise polynomials of degree <= k for the displacement can be utilized. In addition, complementing the results in the aforementioned reference, we introduce a new postprocessing formula for the stress recovering the optimally convergent approximation of the broken H(div)-norm. Numerical results confirm our theoretical findings. (C) 2015 Elsevier Ltd. All rights reserved.
Analysis of an augmented pseudostress-based mixed formulation for a nonlinear Brinkman model of porous media flow
(Elsevier, 2015) Gatica, Gabriel; Gatica, Luis; Sequeira, Filander
In this paper we introduce and analyze an augmented mixed finite element method for the two-dimensional nonlinear Brinkman model of porous media flow with mixed boundary conditions. More precisely, we extend a previous approach for the respective linear model to the present nonlinear case, and employ a dual-mixed formulation in which the main unknowns are given by the gradient of the velocity and the pseudostress. In this way, and similarly as before, the original velocity and pressure unknowns are easily recovered through a simple postprocessing. In addition, since the Neumann boundary condition becomes essential, we impose it in a weak sense, which yields the introduction of the trace of the fluid velocity over the Neumann boundary as the associated Lagrange multiplier. We apply known results from nonlinear functional analysis to prove that the corresponding continuous and discrete schemes are well-posed. In particular, a feasible choice of finite element subspaces is given by Raviart-Thomas elements of order k >= 0 for the pseudostress, piecewise polynomials of degree <= k for the gradient of the velocity, and continuous piecewise polynomials of degree <= k + 1 for the Lagrange multiplier. We also derive a reliable and efficient residual-based a posteriori error estimator for this problem. Finally, several numerical results illustrating the performance and the robustness of the method, confirming the theoretical properties of the estimator, and showing the behavior of the associated adaptive algorithm, are provided. (C) 2015 Elsevier B.V. All rights reserved.
Introducción a la estadística descriptiva con R
(Editorial Universidad Nacional, 2020) Aguilar Fernández, Eduardo; Zamora Araya, José Andrey
El presente documento tiene por objetivo ofrecer a todas aquellas personas que dan sus primeros pasos en el lenguaje R, un primer acercamiento al entorno mediante la ejecuci on de comandos b asicos. Si bien es cierto, el p ublico meta est a representado en su mayor a por estudiantes de cursos donde se aborden temas de Estad stica Descriptiva, tambi en puede ser de utilidad para personas con mayor conocimiento como fuente de consulta, as como apoyo al personal docente que desee impartir sus clases utilizando esta herramienta. La obra est a constituida por cinco cap tulos. El primero de ellos trata sobre las principales funciones para el manejo de paquetes y datos, de tal forma que se introduce poco a poco a la persona lectora en el manejo de comandos b asicos para realizar operaciones habituales cuando se trabaja con vectores, matrices y bases de datos en general.

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