Skip to main content

Tomás Barrios

Universidad Católica de la Santísima Concepción, Department of Applied Mathematics and Physics, Adjunct

Followers

11

Following

22

Co-authors

2

Public Views

I born and grew up in Constitución, Chile, sometime in the past century. Due to my studies I moved to Concepción in 1992, where I received my first diploma, Mathematical Engineering and Doctor in Applied Sciences degrees, all of them at University of Concepción. In 2005 I got a position at Universidad Católica de la Santísima Concepción (UCSC), Chile, where actually I'm Associate Professor in Department of Applied Mathematics and Physics, Faculty of Engineering, UCSC.

less

Université de Valenciennes et du Hainaut-Cambrésis (UVHC)

Universidad de Santiago de Chile

Omdurman Islamic University

Yonsei University

Amiya Kumar Pani

Revista Ingeniería y Ciencia

Universidad EAFIT

Laboratório Nacional de Computação Científica

InterestsView All (7)

Uploads

Papers by Tomás Barrios

An augmented discontinuous Galerkin method for elliptic problems

Comptes Rendus Mathematique, 2007

In this Note we propose an augmented discontinuous Galerkin method for elliptic linear problems i... more In this Note we propose an augmented discontinuous Galerkin method for elliptic linear problems in the plane with mixed boundary conditions. Our approach introduces Galerkin least-squares terms, arising from constitutive and equilibrium equations, which allow us to look for the flux unknown in the local Raviart-Thomas space. The unique solvability is established avoiding the introduction of lifting operators and a Céa estimate is derived, which yields the rate of convergence of error, measured in an appropriate norm, being optimal respect to the h-version. We emphasize that for practical computations, this method reduces the degrees of freedom, with respect to the classical discontinuous Galerkin method. To cite this article: T.

An Augmented DG Scheme for Porous Media Equations

Springer eBooks, Sep 19, 2008

We present an augmented local discontinuous Galerkin scheme for Darcy flow, that is obtained addi... more We present an augmented local discontinuous Galerkin scheme for Darcy flow, that is obtained adding suitable Galerkin least squares terms arising from constitutive and equilibrium equations. The well-posedness of the scheme is proved applying Lax Milgram’s theorem. Finally, we present an a posteriori error estimator, and include one numerical experiment showing that the estimator is reliable and efficient.

New a posteriori error estimator for an stabilized mixed method applied to incompressible fluid flows

Applied Mathematics and Computation, Jun 1, 2019

We consider an augmented mixed finite element method for incompressible fluid flows and develop a... more We consider an augmented mixed finite element method for incompressible fluid flows and develop a simple a posteriori error analysis. We obtain an a posteriori error estimator that is reliable and locally efficient. We provide numerical experiments that illustrate the performance of the corresponding adaptive algorithm and support its use in practice.

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, 2020

In this paper, we discuss the well-posedness of a mixed discontinuous Galerkin (DG) scheme for th... more 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.

On an adaptive stabilized mixed finite element method for the Oseen problem with mixed boundary conditions

Computer Methods in Applied Mechanics and Engineering, Jun 1, 2020

We consider the Oseen problem with nonhomogeneous Dirichlet boundary conditions on a part of the ... more We consider the Oseen problem with nonhomogeneous Dirichlet boundary conditions on a part of the boundary and a Neumann type boundary condition on the remaining part. Suitable least squares terms that arise from the constitutive law, the momentum equation and the Dirichlet boundary condition are added to a dual-mixed formulation based on the pseudostressvelocity variables. We prove that the new augmented variational formulation and the corresponding Galerkin scheme are well-posed, and a Céa estimate holds for any finite element subspaces. We also provide the rate of convergence when each row of the pseudostress is approximated by Raviart-Thomas elements and the velocity is approximated by continuous piecewise polynomials. We develop an a posteriori error analysis based on a Helmholtz-type decomposition, and derive a posteriori error indicators that consist of two residual terms per element except on those elements with a side on the Dirichlet boundary, where they both have two additional terms. We prove that these a posteriori error indicators are reliable and locally efficient. Finally, we provide several numerical experiments that support the theoretical results. c

An a posteriori error analysis of an augmented discontinuous Galerkin formulation for Darcy flow

Numerische Mathematik, Sep 28, 2011

In this paper we develop an a posteriori error analysis for an augmented discontinuous Garlerkin ... more In this paper we develop an a posteriori error analysis for an augmented discontinuous Garlerkin formulation applied to the Darcy flow. More precisely, we derive a reliable and efficient a posteriori error estimator, which consists of residual terms. Finally, we present several numerical experiments, showing the robustness of the method and the theoretical properties of the estimator, thus confirming the capability of the corresponding adaptive algorithms to localize the inner layers, the singularities and/or the large stress regions of the exact solution.

Augmented mixed finite element method for the Oseen problem: A priori and a posteriori error analyses

Computer Methods in Applied Mechanics and Engineering, 2017

We propose a new augmented dual-mixed method for the Oseen problem based on the pseudostress-velo... more We propose a new augmented dual-mixed method for the Oseen problem based on the pseudostress-velocity formulation. The stabilized formulation is obtained by adding to the dual-mixed approach suitable least squares terms that arise from the constitutive and equilibrium equations. We prove that for appropriate values of the stabilization parameters, the new variational formulation and the corresponding Galerkin scheme are well-posed, and a Céa estimate holds for any finite element subspaces. We also provide the rate of convergence when each row of the pseudostress is approximated by Raviart-Thomas or Brezzi-Douglas-Marini elements and the velocity is approximated by continuous piecewise polynomials. Moreover, we derive a simple a posteriori error estimator of residual type that consists of two residual terms and prove that it is reliable and locally efficient. Finally, we include several numerical experiments that support the theoretical results.

Numerical analysis of a stabilized scheme applied to incompressible elasticity problems with Dirichlet and with mixed boundary conditions

Advances in Computational Mathematics, Jul 1, 2022

A priori and a posteriori error analyses of an augmented discontinuous Galerkin formulation

Ima Journal of Numerical Analysis, Jun 8, 2009

We use Galerkin least squares terms to develop a more general stabilized discontinuous Galerkin m... more We use Galerkin least squares terms to develop a more general stabilized discontinuous Galerkin method for elliptic problems in the plane with mixed boundary conditions. The unique solvability and optimal rate of convergence of this scheme, with respect to the h-version, are established. Furthermore, we include the corresponding a posteriori error analysis, which results in a reliable and efficient estimator. Finally, we present several numerical examples that show the capability of the adaptive algorithm to localize the singularities, confirming the theoretical properties of the a posteriori error estimate.

A note on a priori error estimates for augmented mixed methods

Applied Mathematics Letters, Jul 1, 2016

In this note we describe a strategy that improves the a priori error bounds for augmented mixed m... more 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.

A New Stabilized Scheme Applied to Incompressible Elasticity: A-priori and A-posteriori Analysis

Mecánica Computacional, Nov 17, 2019

We develop an a-priori and an a-posteriori error analysis for a stabilised mixed finite element m... more We develop an a-priori and an a-posteriori error analysis for a stabilised mixed finite element method applied to incompressible elasticity. The stabilised formulation is obtained by adding to the standard dual-mixed approach suitable least-squares terms arising from the constitutive and equilibrium equations. It is shown that the resulting variational formulation is strongly coercive, which allows to use any pair of finite element subspaces for the corresponding discrete scheme. In particular, Raviart-Thomas spaces of lowest order for the stress tensor and piecewise linear elements for the displacement, can be used. Additionally, we derive a simple a-posteriori error estimator and prove that it is reliable and locally efficient. Finally, we include several numerical experiments that confirm the theoretical results.

New a Posteriori Error Estimator for an Augmented Mixed FEM in Linear Elasticity

Lecture notes in computational science and engineering, Oct 31, 2014

ABSTRACT We consider an augmented mixed finite element method applied to the linear elasticity pr... more ABSTRACT We consider an augmented mixed finite element method applied to the linear elasticity problem with non-homogeneous Dirichlet boundary conditions and derive an a posteriori error estimator that is simpler and easier to implement than the one available in the literature. The new a posteriori error estimator is reliable and locally efficient in interior triangles; in the remaining elements, it satisfies a quasi-efficiency bound. We provide some numerical results that illustrate the performance of the corresponding adaptive algorithm.

Low cost a posteriori error estimators for an augmented mixed FEM in linear elasticity

Applied Numerical Mathematics, Oct 1, 2014

We consider an augmented mixed finite element method applied to the linear elasticity problem and... more We consider an augmented mixed finite element method applied to the linear elasticity problem and derive a posteriori error estimators that are simpler and easier to implement than the ones available in the literature. In the case of homogeneous Dirichlet boundary conditions, the new a posteriori error estimator is reliable and locally efficient, whereas for non-homogeneous Dirichlet boundary conditions, we derive an a posteriori error estimator that is reliable and satisfies a quasi-efficiency bound. Numerical experiments illustrate the performance of the corresponding adaptive algorithms and support the theoretical results.

A posteriori error analysis of an augmented mixed finite element method for Darcy flow

Computer Methods in Applied Mechanics and Engineering, 2015

We develop an a posteriori error analysis of residual type of a stabilized mixed finite element m... more We develop an a posteriori error analysis of residual type of a stabilized mixed finite element method for Darcy flow. The stabilized formulation is obtained by adding to the standard dualmixed approach suitable residual type terms arising from Darcy's law and the mass conservation equation. We derive sufficient conditions on the stabilization parameters that guarantee that the augmented variational formulation and the corresponding Galerkin scheme are well-posed. Then, we obtain a simple a posteriori error estimator and prove that it is reliable and locally efficient. Finally, we provide several numerical experiments that illustrate the theoretical results and support the use of the corresponding adaptive algorithm in practice.

Adaptive augmented rnixed FEM for the Oseen problem with mixed boundary conditions

Adaptive Solutions of a Stabilized Mixed Finite Element Method for Porous Media Equations

Mecánica Computacional, 2014

Through numerical experiments, we explore the theoretical properties of an a-posteriori error est... more Through numerical experiments, we explore the theoretical properties of an a-posteriori error estimator of an augmented mixed method applied to Darcy law. More precisely, we show numerical evidence confirming the theoretical properties of the estimator, and illustrating the capability of the corresponding adaptive algorithm to localise the singularities and boundary layers of the solutions.

Analysis of DG approximations for Stokes problem based on velocity-pseudostress formulation

Numerical Methods for Partial Differential Equations, Mar 31, 2017

In this article, we first discuss the well posedness of a modified LDG scheme of Stokes problem, ... more 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.

A note on a posteriori error analysis for dual mixed methods with mixed boundary conditions

Numerical Methods for Partial Differential Equations

In this article, we give a description of a technique to develop an a posteriori error estimator ... more 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 theoretic...

A Posteriori Error Estimation of a Stabilized Mixed Finite Element Method for Darcy Flow

Lecture notes in computational science and engineering, 2015

We consider the augmented mixed finite element method proposed in Barrios et al. (Comput Methods ... more We consider the augmented mixed finite element method proposed in Barrios et al. (Comput Methods Appl Mech Eng 283:909–922, 2015) for Darcy flow. We develop the a priori and a posteriori error analyses taking into account the approximation of the Neumann boundary condition. We derive an a posteriori error indicator that consists of two residual terms on interior elements and an additional term that accounts for the error in the boundary condition on boundary elements. We prove that the error indicator is reliable and locally efficient on interior elements. Numerical experiments illustrate the good performance of the adaptive algorithm.

On the discontinuous Galerkin method for solving boundary value problems for the Helmholtz equation: A priori and a posteriori error analyses

We apply the local discontinuous Galerkin (LDG for short) method to solve a mixed boundary value ... more We apply the local discontinuous Galerkin (LDG for short) method to solve a mixed boundary value problems for the Helmholtz equation in bounded polygonal domain in 2D. 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) there is convergence 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 terms of the size of the wavenumber and its distance to the set of eigenvalues for the same boundary value problem for the Laplacian. We also present an a posteriori error estimator showing both the reliability and efficiency of the estimator complemented with detailed information on the dependence of the constants on the wavenumber. 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...

An augmented discontinuous Galerkin method for elliptic problems

Comptes Rendus Mathematique, 2007

In this Note we propose an augmented discontinuous Galerkin method for elliptic linear problems i... more In this Note we propose an augmented discontinuous Galerkin method for elliptic linear problems in the plane with mixed boundary conditions. Our approach introduces Galerkin least-squares terms, arising from constitutive and equilibrium equations, which allow us to look for the flux unknown in the local Raviart-Thomas space. The unique solvability is established avoiding the introduction of lifting operators and a Céa estimate is derived, which yields the rate of convergence of error, measured in an appropriate norm, being optimal respect to the h-version. We emphasize that for practical computations, this method reduces the degrees of freedom, with respect to the classical discontinuous Galerkin method. To cite this article: T.

An Augmented DG Scheme for Porous Media Equations

Springer eBooks, Sep 19, 2008

We present an augmented local discontinuous Galerkin scheme for Darcy flow, that is obtained addi... more We present an augmented local discontinuous Galerkin scheme for Darcy flow, that is obtained adding suitable Galerkin least squares terms arising from constitutive and equilibrium equations. The well-posedness of the scheme is proved applying Lax Milgram’s theorem. Finally, we present an a posteriori error estimator, and include one numerical experiment showing that the estimator is reliable and efficient.

New a posteriori error estimator for an stabilized mixed method applied to incompressible fluid flows

Applied Mathematics and Computation, Jun 1, 2019

We consider an augmented mixed finite element method for incompressible fluid flows and develop a... more We consider an augmented mixed finite element method for incompressible fluid flows and develop a simple a posteriori error analysis. We obtain an a posteriori error estimator that is reliable and locally efficient. We provide numerical experiments that illustrate the performance of the corresponding adaptive algorithm and support its use in practice.

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, 2020

In this paper, we discuss the well-posedness of a mixed discontinuous Galerkin (DG) scheme for th... more 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.

On an adaptive stabilized mixed finite element method for the Oseen problem with mixed boundary conditions

Computer Methods in Applied Mechanics and Engineering, Jun 1, 2020

We consider the Oseen problem with nonhomogeneous Dirichlet boundary conditions on a part of the ... more We consider the Oseen problem with nonhomogeneous Dirichlet boundary conditions on a part of the boundary and a Neumann type boundary condition on the remaining part. Suitable least squares terms that arise from the constitutive law, the momentum equation and the Dirichlet boundary condition are added to a dual-mixed formulation based on the pseudostressvelocity variables. We prove that the new augmented variational formulation and the corresponding Galerkin scheme are well-posed, and a Céa estimate holds for any finite element subspaces. We also provide the rate of convergence when each row of the pseudostress is approximated by Raviart-Thomas elements and the velocity is approximated by continuous piecewise polynomials. We develop an a posteriori error analysis based on a Helmholtz-type decomposition, and derive a posteriori error indicators that consist of two residual terms per element except on those elements with a side on the Dirichlet boundary, where they both have two additional terms. We prove that these a posteriori error indicators are reliable and locally efficient. Finally, we provide several numerical experiments that support the theoretical results. c

An a posteriori error analysis of an augmented discontinuous Galerkin formulation for Darcy flow

Numerische Mathematik, Sep 28, 2011

In this paper we develop an a posteriori error analysis for an augmented discontinuous Garlerkin ... more In this paper we develop an a posteriori error analysis for an augmented discontinuous Garlerkin formulation applied to the Darcy flow. More precisely, we derive a reliable and efficient a posteriori error estimator, which consists of residual terms. Finally, we present several numerical experiments, showing the robustness of the method and the theoretical properties of the estimator, thus confirming the capability of the corresponding adaptive algorithms to localize the inner layers, the singularities and/or the large stress regions of the exact solution.

Augmented mixed finite element method for the Oseen problem: A priori and a posteriori error analyses

Computer Methods in Applied Mechanics and Engineering, 2017

We propose a new augmented dual-mixed method for the Oseen problem based on the pseudostress-velo... more We propose a new augmented dual-mixed method for the Oseen problem based on the pseudostress-velocity formulation. The stabilized formulation is obtained by adding to the dual-mixed approach suitable least squares terms that arise from the constitutive and equilibrium equations. We prove that for appropriate values of the stabilization parameters, the new variational formulation and the corresponding Galerkin scheme are well-posed, and a Céa estimate holds for any finite element subspaces. We also provide the rate of convergence when each row of the pseudostress is approximated by Raviart-Thomas or Brezzi-Douglas-Marini elements and the velocity is approximated by continuous piecewise polynomials. Moreover, we derive a simple a posteriori error estimator of residual type that consists of two residual terms and prove that it is reliable and locally efficient. Finally, we include several numerical experiments that support the theoretical results.

Numerical analysis of a stabilized scheme applied to incompressible elasticity problems with Dirichlet and with mixed boundary conditions

Advances in Computational Mathematics, Jul 1, 2022

A priori and a posteriori error analyses of an augmented discontinuous Galerkin formulation

Ima Journal of Numerical Analysis, Jun 8, 2009

We use Galerkin least squares terms to develop a more general stabilized discontinuous Galerkin m... more We use Galerkin least squares terms to develop a more general stabilized discontinuous Galerkin method for elliptic problems in the plane with mixed boundary conditions. The unique solvability and optimal rate of convergence of this scheme, with respect to the h-version, are established. Furthermore, we include the corresponding a posteriori error analysis, which results in a reliable and efficient estimator. Finally, we present several numerical examples that show the capability of the adaptive algorithm to localize the singularities, confirming the theoretical properties of the a posteriori error estimate.

A note on a priori error estimates for augmented mixed methods

Applied Mathematics Letters, Jul 1, 2016

In this note we describe a strategy that improves the a priori error bounds for augmented mixed m... more 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.

A New Stabilized Scheme Applied to Incompressible Elasticity: A-priori and A-posteriori Analysis

Mecánica Computacional, Nov 17, 2019

We develop an a-priori and an a-posteriori error analysis for a stabilised mixed finite element m... more We develop an a-priori and an a-posteriori error analysis for a stabilised mixed finite element method applied to incompressible elasticity. The stabilised formulation is obtained by adding to the standard dual-mixed approach suitable least-squares terms arising from the constitutive and equilibrium equations. It is shown that the resulting variational formulation is strongly coercive, which allows to use any pair of finite element subspaces for the corresponding discrete scheme. In particular, Raviart-Thomas spaces of lowest order for the stress tensor and piecewise linear elements for the displacement, can be used. Additionally, we derive a simple a-posteriori error estimator and prove that it is reliable and locally efficient. Finally, we include several numerical experiments that confirm the theoretical results.

New a Posteriori Error Estimator for an Augmented Mixed FEM in Linear Elasticity

Lecture notes in computational science and engineering, Oct 31, 2014

ABSTRACT We consider an augmented mixed finite element method applied to the linear elasticity pr... more ABSTRACT We consider an augmented mixed finite element method applied to the linear elasticity problem with non-homogeneous Dirichlet boundary conditions and derive an a posteriori error estimator that is simpler and easier to implement than the one available in the literature. The new a posteriori error estimator is reliable and locally efficient in interior triangles; in the remaining elements, it satisfies a quasi-efficiency bound. We provide some numerical results that illustrate the performance of the corresponding adaptive algorithm.

Low cost a posteriori error estimators for an augmented mixed FEM in linear elasticity

Applied Numerical Mathematics, Oct 1, 2014

We consider an augmented mixed finite element method applied to the linear elasticity problem and... more We consider an augmented mixed finite element method applied to the linear elasticity problem and derive a posteriori error estimators that are simpler and easier to implement than the ones available in the literature. In the case of homogeneous Dirichlet boundary conditions, the new a posteriori error estimator is reliable and locally efficient, whereas for non-homogeneous Dirichlet boundary conditions, we derive an a posteriori error estimator that is reliable and satisfies a quasi-efficiency bound. Numerical experiments illustrate the performance of the corresponding adaptive algorithms and support the theoretical results.

A posteriori error analysis of an augmented mixed finite element method for Darcy flow

Computer Methods in Applied Mechanics and Engineering, 2015

We develop an a posteriori error analysis of residual type of a stabilized mixed finite element m... more We develop an a posteriori error analysis of residual type of a stabilized mixed finite element method for Darcy flow. The stabilized formulation is obtained by adding to the standard dualmixed approach suitable residual type terms arising from Darcy's law and the mass conservation equation. We derive sufficient conditions on the stabilization parameters that guarantee that the augmented variational formulation and the corresponding Galerkin scheme are well-posed. Then, we obtain a simple a posteriori error estimator and prove that it is reliable and locally efficient. Finally, we provide several numerical experiments that illustrate the theoretical results and support the use of the corresponding adaptive algorithm in practice.

Adaptive augmented rnixed FEM for the Oseen problem with mixed boundary conditions

Adaptive Solutions of a Stabilized Mixed Finite Element Method for Porous Media Equations

Mecánica Computacional, 2014

Through numerical experiments, we explore the theoretical properties of an a-posteriori error est... more Through numerical experiments, we explore the theoretical properties of an a-posteriori error estimator of an augmented mixed method applied to Darcy law. More precisely, we show numerical evidence confirming the theoretical properties of the estimator, and illustrating the capability of the corresponding adaptive algorithm to localise the singularities and boundary layers of the solutions.

Analysis of DG approximations for Stokes problem based on velocity-pseudostress formulation

Numerical Methods for Partial Differential Equations, Mar 31, 2017

In this article, we first discuss the well posedness of a modified LDG scheme of Stokes problem, ... more 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.

A note on a posteriori error analysis for dual mixed methods with mixed boundary conditions

Numerical Methods for Partial Differential Equations

In this article, we give a description of a technique to develop an a posteriori error estimator ... more 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 theoretic...

A Posteriori Error Estimation of a Stabilized Mixed Finite Element Method for Darcy Flow

Lecture notes in computational science and engineering, 2015

We consider the augmented mixed finite element method proposed in Barrios et al. (Comput Methods ... more We consider the augmented mixed finite element method proposed in Barrios et al. (Comput Methods Appl Mech Eng 283:909–922, 2015) for Darcy flow. We develop the a priori and a posteriori error analyses taking into account the approximation of the Neumann boundary condition. We derive an a posteriori error indicator that consists of two residual terms on interior elements and an additional term that accounts for the error in the boundary condition on boundary elements. We prove that the error indicator is reliable and locally efficient on interior elements. Numerical experiments illustrate the good performance of the adaptive algorithm.

On the discontinuous Galerkin method for solving boundary value problems for the Helmholtz equation: A priori and a posteriori error analyses

We apply the local discontinuous Galerkin (LDG for short) method to solve a mixed boundary value ... more We apply the local discontinuous Galerkin (LDG for short) method to solve a mixed boundary value problems for the Helmholtz equation in bounded polygonal domain in 2D. 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) there is convergence 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 terms of the size of the wavenumber and its distance to the set of eigenvalues for the same boundary value problem for the Laplacian. We also present an a posteriori error estimator showing both the reliability and efficiency of the estimator complemented with detailed information on the dependence of the constants on the wavenumber. 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...