Tomas Barrios

Universidad Católica de la Santísima Concepción | UCSC · Departamento de Matemática y Física Aplicadas (DMFA)

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 expe...

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 condi- tions. The approach considers conforming finite elements for the discrete scheme, and a quasi-Helmholtz decomposi- tion result...

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 Hdiv;Ω×L2Ω. We remark that the analysis...

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 n...

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...

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 Babu\v{s}ka-Brezzi type condition within the space H(div) x L2. It is well known that the lowest order Crouzeix-Raviart element paired with piecewise constants sa...

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-stan...

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 pseudo...

This article is concerned with the Stokes system with non homogeneous source terms and non‐homogeneous 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 analyse a stabilised formulation o...

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...

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 practi...

We consider a stabilized mixed finite element method introduced recently for the generalized Stokes problem. The method is obtained by adding suitable least squares terms to the dual-mixed variational formulation of the problem in terms of the velocity and the pseudostress. We obtain a new a posteriori error estimator of residual type and prove tha...

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 al...

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,...

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 illus...

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 ter...

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 ind...

We present in this work an a posteriori error estimator for a porous media flow problem that follows the Brinkman model. First, we introduce the pseudostress as an auxiliary unknown, which let us to eliminate the pressure and thus derive a dual-mixed formulation in velocity-pseudostress. Next, in order to circumvent an inf-sup condition for the uni...

This work presents new stabilised finite element methods for a bending moments formulation of the Reissner-Mindlin plate model. The introduction of the bending moment as an extra unknown leads to a new weak formulation, where the symmetry of this variable is imposed strongly in the space. This weak problem is proved to be well-posed, and stabilised...

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 dual-mixed approach suitable residual type terms arising from Darcy’s law and the mass conservation equation. We derive sufficient conditions on the stabilization par...

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 interi...

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, w...

This work presents new stabilised finite element methods for a bending moments formulation of the Reissner-Mindlin plate model. The introduction of the bending moment as an extra unknown leads to a new weak formulation, where the symmetry of this variable is imposed strongly in the space. This weak problem is proved to be well-posed, and stabilised...

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 wi...

In this paper we present an augmented mixed formulation applied to generalized Stokes problem and uses it as state equation in an optimal control problem. The augmented scheme is obtained adding suitable least squares terms to the corresponding velocity–pseudostress formulation of the generalized Stokes problem. To ensure the existence and uniquene...

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 the...

We develop a residual-based a posteriori error analysis for the augmented mixed methods introduced in [G. N. Gatica, ESAIM, Math. Model. Numer. Anal. 40, No. 1, 1–28 (2006; Zbl 05038390)] for the problem of linear elasticity in the plane. We prove that the proposed a posteriori error estimators are both reliable and efficient. Numerical experiments...

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...

In this talk we provide a comparison of the solution of the augmented mixed method with similar approaches. More prescisaly, we compare the solution of the augmented mixed formulation with the standar FEM, dual mixed FEM and least squares FEM. Under appropiate assumptions, we prove that they are superclose. In addition, several numerical examples c...

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 exper...

In this note we present a review of a stabilized discontinuous Galerkin method for elliptic problems in the plane with mixed boundary conditions. The stabilized scheme is obtained by adding suitable Galerkin least-squares terms. The corresponding unique solvability and optimal rates of convergence, with respect to the h –version, are established by...

We use Galerkin least-squares terms and biorthogonal wavelet bases to develop a new stabilized dual-mixed finite element method for second-order elliptic equations in divergence form with Neumann boundary conditions. The approach introduces the trace of the solution on the boundary as a new unknown that acts also as a Lagrange multiplier. We show t...

In this paper, we provide a priori and a posteriori error analyses of an augmented mixed finite element method with Lagrange multipliers applied to elliptic equations in divergence form with mixed boundary conditions. The augmented scheme is obtained by including the Galerkin least-squares terms arising from the constitutive and equilibrium equatio...

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...

In this paper, we provide a priori and a posteriori error analyses of an augmented mixed finite element method with Lagrange multipliers applied to elliptic equations in divergence form with mixed boundary conditions. The augmented scheme is obtained by including the Galerkin least-squares terms arising from the constitutive and equilibrium equatio...

In this paper we develop a residual based a posteriori error analysis for an augmented mixed finite element method applied to the problem of linear elasticity in the plane. More precisely, we derive a reliable and efficient a posteriori error estimator for the case of pure Dirichlet boundary conditions. In addition, several numerical experiments co...

We present a new stabilized mixed finite element method for second order elliptic equations in divergence form with Neumann boundary conditions. The approach introduces first the trace of the solution on the boundary as a Lagrange multiplier, which yields a corresponding residual term that is expressed in the Sobolev norm of order 1/2 by means of w...

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 corresp...

We apply a mixed finite element method to solve a nonlinear second order elliptic equation in divergence form with mixed boundary conditions. Our approach introduces the trace of the solution on the Neumann boundary as a further unknown that acts also as a Lagrange multiplier. We show that the resulting variational formulation and an associated dis...

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 yields a variational formulation with a twofold saddle point structure. We derive...