Advertisement

An a Posteriori Error Estimator for a New Stabilized Formulation of the Brinkman Problem

  • Tomás Barrios
  • Rommel BustinzaEmail author
  • Galina C. García
  • María González
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 103)

Abstract

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 unique solvability, we stabilize the scheme by adding some appropriate least squares terms. The existence and uniqueness of solution are guaranteed and we derive an a posteriori error estimator based on the Ritz projection of the error, which is reliable and efficient up to high order terms. Finally, we report one numerical example confirming the good properties of the estimator.

Keywords

Posteriori Error Unique Solvability Posteriori Error Estimator Brinkman Model Posteriori Error Analysis 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    D.N. Arnold, R.S. Falk, Well-posedness of the fundamental boundary value problems for constrained anisotropic elastic materials. Arch. Ration. Mech. Anal. 98(2), 143–165 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    T.P. Barrios, R. Bustinza, An augmented discontinuous Galerkin method for stationary Stokes problem, Preprint 2010–20, Departamento de Ingeniería Matemática, Universidad de Concepción, 2010Google Scholar
  3. 3.
    T.P. Barrios, R. Bustinza, G.C. García, E. Hernández, On stabilized mixed methods for generalized Stokes problem based on the velocity-pseudostress formulation: a priori error estimates. Comput. Methods Appl. Mech. Eng. 237–240, 78–87 (2012)CrossRefGoogle Scholar
  4. 4.
    T.P. Barrios, R. Bustinza, G.C. García, M. González, A posteriori error analyses of a velocity-pseudostress formulation of the generalized Stokes problem, Preprint 2013–04, Centro de Investigación en Ingeniería Matemática, Universidad de Concepción, 2013Google Scholar
  5. 5.
    Z. Cai, B. Lee, P. Wang, Least-squares methods for incompressible Newtonian fluid flow: linear stationary problems. SIAM J. Numer. Anal. 42, 843–859 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Z. Cai, C. Tong, P.S. Vassilevski, C. Wang, Mixed finite element methods for incompressible flow: stationary Stokes equations. Numer. Methods Partial Differ. Equ. 26, 957–978 (2010)zbMATHMathSciNetGoogle Scholar
  7. 7.
    G.N. Gatica, A. Márquez, M.A. Sánchez, Analysis of a velocity-pressure-pseudostress formulation for the stationary Stokes equations. Comput. Methods Appl. Mech. Eng. 199, 1064–1079 (2010)CrossRefzbMATHGoogle Scholar
  8. 8.
    G.N. Gatica, A. Márquez, M.A. Sánchez, A priori and a posteriori error analyses of a velocity-pseudostress formulation for a class of quasi-Newtonian Stokes flows. Comput. Methods Appl. Mech. Eng. 200, 1619–1636 (2011)CrossRefzbMATHGoogle Scholar
  9. 9.
    G.N. Gatica, A. Márquez, M.A. Sánchez, Pseudostress-based mixed finite element methods for the Stokes problem in \(\mathbb{R}^{n}\) with Dirichlet boundary conditions. I: a priori error analysis. Commun. Comput. Phys. 12, 109–134 (2012)Google Scholar
  10. 10.
    G.N. Gatica, L.F. Gatica, A. Márquez, Analysis of a pseudostress-based mixed finite element method for the Brinkman model of porous media flow. Numer. Math. 126(4), 635–677 (2014)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Tomás Barrios
    • 1
  • Rommel Bustinza
    • 2
    Email author
  • Galina C. García
    • 3
  • María González
    • 4
  1. 1.Universidad Católica de la Santísima ConcepciónConcepciónChile
  2. 2.CI2MA and Departamento de Ingeniería MatemáticaUniversidad de ConcepciónConcepciónChile
  3. 3.Universidad de Santiago de ChileSantiagoChile
  4. 4.Universidade da CoruñaA CoruñaSpain

Personalised recommendations