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

Barrios, Tomás P.; Behrens, Edwin M.; González, Maŕıa

doi:10.1016/j.apnum.2014.05.008

Cited by 10 publications

(12 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…0 . (a similar argument was used for elasticity problem in [4]). Using that and the fact that the H 1 (Γ i ) norm is higher than the L 2 (Γ i ) norm, and by the continuity of normal trace operator, we obtain…”

Section: An Upper Bound For the Velocity Errormentioning

confidence: 99%

A Residual a Posteriori Error Estimators for a Model for Flow in Porous Media with Fractures

et al. 2018

View full text Add to dashboard Cite

show abstract

Section: An Upper Bound For the Velocity Errormentioning

confidence: 99%

A Residual a Posteriori Error Estimators for a Model for Flow in Porous Media with Fractures

et al. 2018

View full text Add to dashboard Cite

show abstract

“…Theorem 1 If (u, p) is a solution to problem (1), then (σ, u), where σ = ν ∇u − p I, is a solution to problem (5). Conversely, if (σ, u) is a solution to problem (5), then (u, p), where p = − 1 d tr(σ), is a solution to problem (1).…”

Section: A Pseudostress-velocity Formulation Of the Model Problemmentioning

confidence: 99%

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

Barrios

Cascón

González

2017

Computer Methods in Applied Mechanics and Engineering

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…In this section, we follow the ideas given in the work of Barrios et al (see also the work of Barrios and Gatica), and develop an a posteriori error analysis for the discrete scheme using an appropriate Ritz projection of the error and a quasi Helmholtz decomposition. We first introduce some notations and results concerning the Clément and Raviart‐Thomas interpolation operators.…”

Section: An a Posteriori Error Analysismentioning

confidence: 99%

“…On the other hand, concerning linear elasticity problem, in the work of Barrios et al, the authors present an alternative a posteriori error estimator to the previous one developed in a former work . This approach is based on the Ritz projection of the error (see the work of Barrios and Gatica).…”

Section: Introductionmentioning

confidence: 99%

“…As result, in the case of homogeneous Dirichlet boundary condition, we obtain a reliable and local efficient a posteriori error estimator that only requires the computation of four residuals per element, which has a low computational cost comparing with the eleven terms included in the estimator developed in the work of Barrios et al for the same case. This kind of a posteriori error estimator, at least, has been developed satisfactorily in different directions, for example, the Poisson problem, Darcy flow, the Brinkman model, linear elasticity, and recently the Oseen equations …”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A stabilized mixed method applied to Stokes system with nonhomogeneous source terms: The stationary case Dedicated to Prof. R. Rodríguez, on the occasion of his 65th birthday

Barrios

Behrens

Bustinza

2019

Numerical Methods in Fluids

Self Cite

View full text Add to dashboard Cite

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. KEYWORDSa posteriori error estimates, augmented mixed formulation, Ritz projection of the error INTRODUCTIONIn the work of Cai et al, 1 a dual mixed finite element method for the incompressible fluid flow was introduced and analyzed. The approach there follows the ideas developed in the other work of Cai et al, 2 ie, the incompressible fluid flow is reformulated using the new variable so-called pseudostress, which is in relation with the pressure and gradient of the velocity. The main advantage of this new variable is the accurate approximation to physical quantities such as the stress and vorticity, allowing to use the pair of conforming Raviart-Thomas with discontinuous polynomial as the finite element space. Furthermore, in order to obtain more flexibility in the finite element spaces, the stabilization of this approach has been studied in the work of Figueroa et al. 3 In addition, its corresponding extension to quasi Newtonian flows and Brinkman model were developed in the works of Gatica et al 4 and Barrios et al, 5 respectively. On the other hand, concerning linear elasticity problem, in the work of Barrios et al, 6 the authors present an alternative a posteriori error estimator to the previous one developed in a former work. 7 This approach is based on the Ritz projection Int J Numer Meth Fluids. 2020;92:509-527. wileyonlinelibrary.com/journal/fld

show abstract

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

Cited by 10 publications

References 15 publications

A Residual a Posteriori Error Estimators for a Model for Flow in Porous Media with Fractures

A Residual a Posteriori Error Estimators for a Model for Flow in Porous Media with Fractures

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

A stabilized mixed method applied to Stokes system with nonhomogeneous source terms: The stationary case Dedicated to Prof. R. Rodríguez, on the occasion of his 65th birthday

Contact Info

Product

Resources

About