- I born and grew up in Constitución, Chile, sometime in the past century. Due to my studies I moved to Concepción in 1... moreI 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.edit
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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... 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.
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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... 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.
Research Interests: Mathematics, Applied Mathematics, Numerical Analysis, Error Analysis, Convergence Rate, and 9 morePARTIAL DIFFERENTIAL EQUATION, Numerical Analysis and Computational Mathematics, Galerkin Method, Estimator, Boundary Condition, Discontinuous Galerkin, Least squares method, Elliptic equation, and a priori and a posteriori
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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... 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.
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Research Interests: Mathematics, Applied Mathematics, Linear Elasticity, Finite element method, Elasticity, and 9 moreStabilization, Boundary Conditions, Error Analysis, Adaptive algorithms, Numerical Analysis and Computational Mathematics, Estimator, Mixed Finite Element Methods, Numerical Experiments, and a priori and a posteriori
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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... 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...
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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... 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.
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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... 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...
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In this paper, we describe an a posteriori error analysis for a conforming dual mixed scheme of the Poisson problem with non homogeneous Dirichlet boundary condition. As a result, we obtain an a posteriori error estimator, which is proven... more
In this paper, we describe an a posteriori error analysis for a conforming dual mixed scheme of the Poisson problem with non homogeneous Dirichlet boundary condition. As a result, we obtain an a posteriori error estimator, which is proven to be reliable and locally efficient with respect to the usual norm on H(div;Omega) x L^2(Omega). We remark that the analysis relies on the standard Ritz projection of the error, and take into account a kind of a quasi-Helmholtz decomposition of functions in H(div;Omega), which we have established in this work. Finally, we present one numerical example that validates the well behavior of our estimator, being able to identify the numerical singularities when they exist.
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Abstract. We consider an augmented mixed finite element method for the equations of plane linear elasticity with mixed boundary conditions. The method provides simultaneous approximations of the displacements, the stress tensor and the... more
Abstract. We consider an augmented mixed finite element method for the equations of plane linear elasticity with mixed boundary conditions. The method provides simultaneous approximations of the displacements, the stress tensor and the rotation. We develop an a posteriori error analysis based on the Ritz projection of the error and the use of an appropriate auxiliary function, and derive fully local reliable a posteriori error estimates that are locally efficient up to the elements that touch the Neumann boundary. We provide numerical experiments that illustrate the performance of the corresponding adaptive algorithm and support its use in practice.
In this talk, we discuss the well posedness of a modified LDG scheme of the Poisson problem, considering a dual mixed formulation. The difficulty here relies on the fact that the application of classical Babuška-Brezzi theory is not easy... more
In this talk, we discuss the well posedness of a modified LDG scheme of the Poisson problem, considering a dual mixed formulation. The difficulty here relies on the fact that the application of classical Babuška-Brezzi theory is not easy for low order finite elements, so we proceed in a non-standard way. We first prove uniqueness, and then we apply a discrete version of Fredholm’s alternative theorem to deduce existence, while the a priori error analysis is done by 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. As a particular case we comment our result applied to Darcy flow, where we can establish the well posedness for low order finite elements and the corresponding optimal rate of convergence is established with standard additional regularity assumption of the exact solution.
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Under some regularity assumptions, we report an a priori error analysis of a dG scheme for the Poisson and Stokes flow problem in their dual mixed formulation. Both formulations satisfy a Babuška-Brezzi type condition within the space... more
Under some regularity assumptions, we report an a priori error analysis of a dG scheme for the Poisson and Stokes flow problem in their dual mixed formulation. Both formulations satisfy a Babuška-Brezzi type condition within the space H(div) × L2 . It is well known that the lowest order Crouzeix-Raviart element paired with piecewise constants satisfies such a condition on (broken) H1 × L2 spaces. In the present article, we use this pair. The continuity of the normal component is weakly imposed by penalising jumps of the broken H(div) component. For the resulting methods, we prove well-posedness and convergence with constants independent of data and mesh size. We report error estimates in the methods natural norms and optimal local error estimates for the divergence error. In fact, our finite element solution shares for each triangle one DOF with the CR interpolant and the divergence is locally the best-approximation for any regularity. Numerical experiments support the findings and ...
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En este trabajo presentamos estimas a priori del error refinadas para formulaciones mixtas aumentadas. Espećıficamente, bajo hipótesis razonables, más la utilización de la formulación de elementos finitos tradicional y la formulación... more
En este trabajo presentamos estimas a priori del error refinadas para formulaciones mixtas aumentadas. Espećıficamente, bajo hipótesis razonables, más la utilización de la formulación de elementos finitos tradicional y la formulación mixta dual, obtenemos estimas a priori del error independientes para cada una de las incógnitas involucradas, las cuales para soluciones suficientemente suaves, mejoran el resultado conocido. Finalmente, se incluyen ejemplos numéricos que confirman el resultado teórico.
Research Interests: Applied Mathematics, Numerical Analysis, Finite element method, Applied Mathematics and Computational Science, Convergence Rate, and 12 moreNumerical Analysis and Computational Mathematics, Lagrange Multiplier, Galerkin Method, Boundary Condition, Constitutive Equation, Electrical And Electronic Engineering, Adaptive Algorithm, Variational Formulation, Least squares method, Elliptic equation, Rate of Convergence, and Mixed Finite Element
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We consider the numerical solution, via the mixed finite element method, of a non-linear elliptic partial differential equation in divergence form with Dirichlet boundary conditions. Besides the temperature u and the fluxσ, we introduce∇... more
We consider the numerical solution, via the mixed finite element method, of a non-linear elliptic partial differential equation in divergence form with Dirichlet boundary conditions. Besides the temperature u and the fluxσ, we introduce∇ u as a further unknown, which ...
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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... 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.
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Through several numerical experiments, we explore the theoretical properties of a residual based a posteriori error estimator of an augmented mixed method applied to linear elasticity problem in the plane. More precisely, we show... more
Through several numerical experiments, we explore the theoretical properties of a residual based a posteriori error estimator of an augmented mixed method applied to linear elasticity problem in the plane. More precisely, we show numerical evidence confirming the theoretical properties of the estimator, and ilustrating the capability of the corresponding adaptive algorithm to localize the singularities and the large stress regions of the solution.
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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.... more
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 ( d i v ), 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.
Research Interests:
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... 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.
Research Interests:
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... 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.
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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... 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.