Skip to main content
SpringerLink
Log in

An optimal three-field finite element approximation of the Stokes system with continuous extra stresses

  • Published:
Japan Journal of Industrial and Applied Mathematics Aims and scope Submit manuscript

Abstract

The aim of this paper is to present a mixed finite element method for solving the twodimensional Stokes system, written in terms of the variables velocity, pressure and extra stress tensor. A convenient functional framework is introduced, allowing simpler choices of the type of approximation for the third variable. This is assumed here to be continuous, in order to ensure its applicability to viscoelastic flow problems. In the particular second order Galerkin method that is studied, the dimension of the corresponding finite element subspace is reduced to a minimum, at least as far as local stability analyses are concerned.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J.H. Carneiro de Araujo, V. Ruas and M.A.M. Silva Ramos, Approximation of the three-field Stokes system via optimized quadrilateral finite elements. RAIRO Modél. Math. Anal. Numér.,27 (1993), 107–127.

    MATH  MathSciNet  Google Scholar 

  2. P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, 1976.

    Google Scholar 

  3. M. Crouzeix and P.A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations (I). RAIRO Rech. Oper.,3 (1973), 33–76.

    MathSciNet  Google Scholar 

  4. B. Dupire, Problemas Variacionais Lineares, sua Aproximação e Formulações Mistas. Doctoral dissertation, Departmento de Informática, Pontifícia Universidade Católica do Rio de Janeiro, 1985.

  5. M. Fortin and R. Pierre, On the convergence of the mixed method of Crochet and Marchal for viscoelastic flows. Comput. Methods Appl. Mech. Engrg.,73 (1989), 341–350.

    Article  MATH  MathSciNet  Google Scholar 

  6. V. Girault and P.A. Raviart, Finite Element Approximation of the Navier-Stokes Equations. Lecture Notes in Math., Springer Verlag, Berlin, 1979.

    Book  MATH  Google Scholar 

  7. P. Lesaint and P.A. Raviart, On a finite element method for solving the neutron transport equations. Mathematical Aspects of Finite Element Methods in Partial Differential Equations (ed. C. de Boor), Academic Press, N.Y., 1976.

    Google Scholar 

  8. V. Ruas Santos, Finite element solution of 3D viscous flow problems using nonstandard degrees of freedom. Japan J. Appl. Math.,2 (1985), 415–431.

    Article  MATH  MathSciNet  Google Scholar 

  9. V. Ruas, Quelques approches pour le calcul tridimensionnel en fluides visqueux incompressibles par la méthode des éléments finis. C. R. Journées NURAGIEP, Ecole Centrale de Lyon, Lyon, France, May, 1990.

  10. V. Ruas, A Convergent Three-field Quadrilateral Finite Element Method for Simulating Viscoelastic Flow on Irregular Meshes. Rev. Européenne Eléments Finis,1 (1992), 391–406.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Modélisation en Mécanique/UPMC, URA229 CNRS & Université de Saint Etienne, France

    V. Ruas

  2. Departamento de Análise, Universidade Federal Fluminense, Rio de Janeiro State, Brazil

    V. Ruas

Authors
  1. V. Ruas
    View author publications

    You can also search for this author in PubMed Google Scholar

About this article

Cite this article

Ruas, V. An optimal three-field finite element approximation of the Stokes system with continuous extra stresses. Japan J. Indust. Appl. Math. 11, 113–130 (1994). https://doi.org/10.1007/BF03167217

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03167217

Key words

Search

Navigation